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18.702 Algebra II (MIT) 18.702 Algebra II (MIT)

Description

This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory. This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

Subjects

Sylow theorems | Sylow theorems | Group Representations | Group Representations | definitions | definitions | unitary representations | unitary representations | characters | characters | Schur's Lemma | Schur's Lemma | Rings: Basic Definitions | Rings: Basic Definitions | homomorphisms | homomorphisms | fractions | fractions | Factorization | Factorization | unique factorization | unique factorization | Gauss' Lemma | Gauss' Lemma | explicit factorization | explicit factorization | maximal ideals | maximal ideals | Quadratic Imaginary Integers | Quadratic Imaginary Integers | Gauss Primes | Gauss Primes | quadratic integers | quadratic integers | ideal factorization | ideal factorization | ideal classes | ideal classes | Linear Algebra over a Ring | Linear Algebra over a Ring | free modules | free modules | integer matrices | integer matrices | generators and relations | generators and relations | structure of abelian groups | structure of abelian groups | Rings: Abstract Constructions | Rings: Abstract Constructions | relations in a ring | relations in a ring | adjoining elements | adjoining elements | Fields: Field Extensions | Fields: Field Extensions | algebraic elements | algebraic elements | degree of field extension | degree of field extension | ruler and compass | ruler and compass | symbolic adjunction | symbolic adjunction | finite fields | finite fields | Fields: Galois Theory | Fields: Galois Theory | the main theorem | the main theorem | cubic equations | cubic equations | symmetric functions | symmetric functions | primitive elements | primitive elements | quartic equations | quartic equations | quintic equations | quintic equations

License

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18.702 Algebra II (MIT) 18.702 Algebra II (MIT)

Description

The course covers group theory and its representations, and focuses on the Sylow theorem, Schur's lemma, and proof of the orthogonality relations. It also analyzes the rings, the factorization processes, and the fields. Topics such as the formal construction of integers and polynomials, homomorphisms and ideals, the Gauss' lemma, quadratic imaginary integers, Gauss primes, and finite and function fields are discussed in detail. The course covers group theory and its representations, and focuses on the Sylow theorem, Schur's lemma, and proof of the orthogonality relations. It also analyzes the rings, the factorization processes, and the fields. Topics such as the formal construction of integers and polynomials, homomorphisms and ideals, the Gauss' lemma, quadratic imaginary integers, Gauss primes, and finite and function fields are discussed in detail.

Subjects

Sylow theorems | Sylow theorems | Group Representations | Group Representations | definitions | definitions | unitary representations | unitary representations | characters | characters | Schur's Lemma | Schur's Lemma | Rings: Basic Definitions | Rings: Basic Definitions | homomorphisms | homomorphisms | fractions | fractions | Factorization | Factorization | unique factorization | unique factorization | Gauss' Lemma | Gauss' Lemma | explicit factorization | explicit factorization | maximal ideals | maximal ideals | Quadratic Imaginary Integers | Quadratic Imaginary Integers | Gauss Primes | Gauss Primes | quadratic integers | quadratic integers | ideal factorization | ideal factorization | ideal classes | ideal classes | Linear Algebra over a Ring | Linear Algebra over a Ring | free modules | free modules | integer matrices | integer matrices | generators and relations | generators and relations | structure of abelian groups | structure of abelian groups | Rings: Abstract Constructions | Rings: Abstract Constructions | relations in a ring | relations in a ring | adjoining elements | adjoining elements | Fields: Field Extensions | Fields: Field Extensions | algebraic elements | algebraic elements | degree of field extension | degree of field extension | ruler and compass | ruler and compass | symbolic adjunction | symbolic adjunction | finite fields | finite fields | Fields: Galois Theory | Fields: Galois Theory | the main theorem | the main theorem | cubic equations | cubic equations | symmetric functions | symmetric functions | primitive elements | primitive elements | quartic equations | quartic equations | quintic equations | quintic equations

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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18.702 Algebra II (MIT) 18.702 Algebra II (MIT)

Description

This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory. This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

Subjects

Sylow theorems | Sylow theorems | Group Representations | Group Representations | definitions | definitions | unitary representations | unitary representations | characters | characters | Schur's Lemma | Schur's Lemma | Rings: Basic Definitions | Rings: Basic Definitions | homomorphisms | homomorphisms | fractions | fractions | Factorization | Factorization | unique factorization | unique factorization | Gauss' Lemma | Gauss' Lemma | explicit factorization | explicit factorization | maximal ideals | maximal ideals | Quadratic Imaginary Integers | Quadratic Imaginary Integers | Gauss Primes | Gauss Primes | quadratic integers | quadratic integers | ideal factorization | ideal factorization | ideal classes | ideal classes | Linear Algebra over a Ring | Linear Algebra over a Ring | free modules | free modules | integer matrices | integer matrices | generators and relations | generators and relations | structure of abelian groups | structure of abelian groups | Rings: Abstract Constructions | Rings: Abstract Constructions | relations in a ring | relations in a ring | adjoining elements | adjoining elements | Fields: Field Extensions | Fields: Field Extensions | algebraic elements | algebraic elements | degree of field extension | degree of field extension | ruler and compass | ruler and compass | symbolic adjunction | symbolic adjunction | finite fields | finite fields | Fields: Galois Theory | Fields: Galois Theory | the main theorem | the main theorem | cubic equations | cubic equations | symmetric functions | symmetric functions | primitive elements | primitive elements | quartic equations | quartic equations | quintic equations | quintic equations

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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Readme file for Introduction to OO Programming in Java

Description

This readme file contains details of links to all the Introduction to OO Programming in Java module's material held on Jorum and information about the module as well.

Subjects

ukoer | programming task guide | programming lecture | programming reading material | software design reading material | classes guide | libraries lecture | classes reading material | classes visual aid | software objects guide | graphics reading material | attributes reading material | attributes visual guide | naming conventions reading material | code reading material | java keywords reading material | variables visual guide | arithmetic reading material | java assignment | making decisions task guide | making decisions lecture | making decisions reading material | boolean expressions visual guide | repetition reading material | while loops visual guide | methods reading material | methods practical | access modifiers reading material | objects reading material | classes assignment | classes practical | child classes task guide | inheritance task guide | extending classes lecture | inheritance reading material | inheritance visual guide | inheritance practical | graphics task guide | awt reading material | graphics visual aid | awt class library reading material | event-driven programming reading material | scrollbars reading material | reflective practice visual guide | mobile phone task guide | mobile phone lecture | fixed repitition reading material | fixed repitition visual guide | mobile phone library reading material | mobile phone reading material | arrays task guide | arrays lecture | arrays reading material | arrays visual guide | creating software objects reading material | software objects visual guide | java practical | generic array list task guide | overriding methods reading material | menu and switch task guide | multi-way decisions reading material | multi-way decisions visual guide | searching task guide | searching lecture | searching reading material | software quality task guide | software quality lecture | software quality reading material | programming assignment | applet reading material | classes visual guide | object-oriented programming | object-oriented | programming | java | problem solving | java program | software design | programming languages | computers | class task guide | class reading material | class assignment | class practical | java classes | variables | attributes | arithmetic | java class | classes and arithmetic | classes | class | decisions | boolean expression | boolean expressions | repetition | methods | aggregate classes | access modifier | access modifiers | child classes | inheritance | child class | graphics | awt class library | fixed repetition | for loop | for loops | array | arrays | iteration | software object | definite iteration | generic lists | generic array list | cast | casting | overriding method | overriding methods | generic list | menu-driven program | menu-driven programs | multi-way decisions | menu and switch | search | searching | software quality | testing | software quality and testing | assessment | computers task guide | programming languages task guide | software design task guide | java program task guide | problem-solving task guide | problem solving task guide | object-oriented programming task guide | java task guide | object-oriented task guide | object oriented task guide | computers lecture | programming languages lecture | software design lecture | java program lecture | problem solving lecture | object-oriented programming lecture | java lecture | object oriented programming lecture | object-oriented lecture | computers reading material | programming languages reading material | java program reading material | problem solving reading material | object-oriented programming reading material | java reading material | object-oriented reading material | object oriented reading material | java classes task guide | variables task guide | attributes task guide | arithmetic task guide | java class task guide | classes and arithmetic task guide | classes task guide | java classes lecture | variables lecture | attributes lecture | arithmetic lecture | java class lecture | classes and arithmetic lecture | classes lecture | class lecture | java classes reading material | variables reading material | java class reading material | classes and arithmetic reading material | java classes visual aid | variables visual aid | attributes visual aid | arithmetic visual aid | java class visual aid | classes and arithmetic visual aid | class visual aid | java visual aid | object-oriented programming visual aid | programming visual aid | object-oriented visual aid | decisions task guide | boolean expression task guide | boolean expressions task guide | repetition task guide | methods task guide | decisions lecture | boolean expression lecture | boolean expressions lecture | repetition lecture | methods lecture | decisions reading material | boolean expression reading material | boolean expressions reading material | decisions visual aid | boolean expression visual aid | boolean expressions visual aid | repetition visual aid | methods visual aid | decisions practical | boolean expression practical | boolean expressions practical | repetition practical | programming practical | object oriented programming practical | object-oriented programming practical | object-oriented practical | object oriented practical | aggregate classes task guide | access modifier task guide | access modifiers task guide | aggregate classes lecture | access modifier lecture | access modifiers lecture | aggregate classes reading material | access modifier reading material | aggregate classes assignment | java classes assignment | access modifier assignment | access modifiers assignment | object oriented programming assignment | object-oriented programming assignment | object-oriented assignment | object oriented assignment | child class task guide | child classes lecture | inheritance lecture | child class lecture | child classes reading material | child class reading material | child classes visual aid | inheritance visual aid | child class visual aid | awt class library task guide | graphics lecture | awt class library lecture | awt class library visual aid | graphics assignment | awt class library assignment | fixed repetition task guide | fixed repetition lecture | fixed repetition visual aid | fixed repetition reading material | for loop task guide | for loops task guide | array task guide | iteration task guide | software object task guide | definite iteration task guide | for loop lecture | for loops lecture | array lecture | iteration lecture | software object lecture | definite iteration lecture | for loop reading material | for loops reading material | array reading material | iteration reading material | software object reading material | definite iteration reading material | for loop visual aid | for loops visual aid | array visual aid | arrays visual aid | iteration visual aid | software object visual aid | definite iteration visual aid | generic lists task guide | cast task guide | casting task guide | overriding method task guide | overriding methods task guide | generic list task guide | generic lists lecture | generic array list lecture | cast lecture | casting lecture | overriding method lecture | overriding methods lecture | generic list lecture | generic lists reading material | generic array list reading material | cast reading material | casting reading material | overriding method reading material | generic list reading material | menu-driven program task guide | menu-driven programs task guide | multi-way decisions task guide | menu-driven program lecture | menu-driven programs lecture | multi-way decisions lecture | menu and switch lecture | menu-driven program reading material | menu-driven programs reading material | menu and switch reading material | menu-driven program visual aid | menu-driven programs visual aid | multi-way decisions visual aid | menu and switch visual aid | search task guide | search lecture | search reading material | testing task guide | software quality and testing task guide | testing lecture | software quality and testing lecture | testing reading material | software quality and testing reading material | assessment reading material | assessment assignment | fixed repetition practical | jcreator guide | g622 | oo | oop | oo programming | awt | oo programming task guide | oop task guide | oo task guide | g622 task guide | oo programming lecture | oop lecture | oo lecture | g622 lecture | oo programming reading material | oop reading material | oo reading material | g622 reading material | g622 visual aid | oop visual aid | oo visual aid | oo programming visual aid | g622 practical | oo practical | oo programming practical | oop practical | g622 assignment | oo assignment | oop assignment | oo programming assignment | awt task guide | awt lecture | awt visual aid | awt assignment | Computer science | I100

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

Description

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

Subjects

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

License

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18.156 Differential Analysis (MIT) 18.156 Differential Analysis (MIT)

Description

The main goal of this course is to give the students a solid foundation in the theory of elliptic and parabolic linear partial differential equations. It is the second semester of a two-semester, graduate-level sequence on Differential Analysis. The main goal of this course is to give the students a solid foundation in the theory of elliptic and parabolic linear partial differential equations. It is the second semester of a two-semester, graduate-level sequence on Differential Analysis.

Subjects

Sobolev spaces | Sobolev spaces | Fredholm alternative | Fredholm alternative | Variable coefficient elliptic | parabolic and hyperbolic linear partial differential equations | Variable coefficient elliptic | parabolic and hyperbolic linear partial differential equations | Variational methods | Variational methods | Viscosity solutions of fully nonlinear partial differential equations | Viscosity solutions of fully nonlinear partial differential equations | Schauder theory | Schauder theory | Holder estimates | Holder estimates | linear equations | linear equations | second derivatives | second derivatives | elliptic | elliptic | parabolic | parabolic | nonlinear partial differential equations | nonlinear partial differential equations | linear partial differential equations | linear partial differential equations | harmonic functions | harmonic functions | elliptic equations | elliptic equations | parabolic equations | parabolic equations

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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22.106 Neutron Interactions and Applications (MIT) 22.106 Neutron Interactions and Applications (MIT)

Description

This course is  a foundational study of the effects of single and multiple interactions on neutron distributions and their applications to problems across the Nuclear Engineering department - fission, fusion, and RST. Particle simulation methods are introduced to deal with complex processes that cannot be studied only experimentally or by numerical solutions of equations. Treatment will emphasize basic concepts and understanding, as well as showing the underlying scientific connections with current research areas. This course is  a foundational study of the effects of single and multiple interactions on neutron distributions and their applications to problems across the Nuclear Engineering department - fission, fusion, and RST. Particle simulation methods are introduced to deal with complex processes that cannot be studied only experimentally or by numerical solutions of equations. Treatment will emphasize basic concepts and understanding, as well as showing the underlying scientific connections with current research areas.

Subjects

neutron distributions | neutron distributions | fission | fission | fusion | fusion | RST | RST | Particle simulation methods | Particle simulation methods | complex processes | complex processes | numerical solutions of equations | numerical solutions of equations | basic concepts | basic concepts | underlying scientific connections | underlying scientific connections | current research areas | current research areas | angular distributions | angular distributions | energy distributions | energy distributions | single collision | single collision | multiple collisions | multiple collisions | neutron interactions | neutron interactions | elastic scattering | elastic scattering | inelastic scattering | inelastic scattering | MCNP | MCNP | Monte Carlo | Monte Carlo | molecular dynamics | molecular dynamics

License

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6.042J Mathematics for Computer Science (MIT) 6.042J Mathematics for Computer Science (MIT)

Description

This is an introductory course in Discrete Mathematics oriented toward Computer Science and Engineering. The course divides roughly into thirds: Fundamental Concepts of Mathematics: Definitions, Proofs, Sets, Functions, Relations Discrete Structures: Modular Arithmetic, Graphs, State Machines, Counting Discrete Probability Theory A version of this course from a previous term was also taught as part of the Singapore-MIT Alliance (SMA) programme as course number SMA 5512 (Mathematics for Computer Science). This is an introductory course in Discrete Mathematics oriented toward Computer Science and Engineering. The course divides roughly into thirds: Fundamental Concepts of Mathematics: Definitions, Proofs, Sets, Functions, Relations Discrete Structures: Modular Arithmetic, Graphs, State Machines, Counting Discrete Probability Theory A version of this course from a previous term was also taught as part of the Singapore-MIT Alliance (SMA) programme as course number SMA 5512 (Mathematics for Computer Science).

Subjects

mathematical definitions | mathematical definitions | proofs and applicable methods | proofs and applicable methods | formal logic notation | formal logic notation | proof methods | proof methods | induction | induction | well-ordering | well-ordering | sets | sets | relations | relations | elementary graph theory | elementary graph theory | integer congruences | integer congruences | asymptotic notation and growth of functions | asymptotic notation and growth of functions | permutations and combinations | counting principles | permutations and combinations | counting principles | discrete probability | discrete probability | recursive definition | recursive definition | structural induction | structural induction | state machines and invariants | state machines and invariants | recurrences | recurrences | generating functions | generating functions | permutations and combinations | permutations and combinations | counting principles | counting principles | discrete mathematics | discrete mathematics | computer science | computer science

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

Description

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

Subjects

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

License

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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18.327 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

License

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6.013 Electromagnetics and Applications (MIT) 6.013 Electromagnetics and Applications (MIT)

Description

This course explores electromagnetic phenomena in modern applications, including wireless communications, circuits, computer interconnects and peripherals, optical fiber links and components, microwave communications and radar, antennas, sensors, micro-electromechanical systems, motors, and power generation and transmission. Fundamentals covered include: quasistatic and dynamic solutions to Maxwell's equations; waves, radiation, and diffraction; coupling to media and structures; guided and unguided waves; resonance; and forces, power, and energy.The instructors of this course extend a general acknowledgment to the many students and instructors who have made major contributions to the 6.013 course materials over the years, and apologize for any residual errors that may remain in these writ This course explores electromagnetic phenomena in modern applications, including wireless communications, circuits, computer interconnects and peripherals, optical fiber links and components, microwave communications and radar, antennas, sensors, micro-electromechanical systems, motors, and power generation and transmission. Fundamentals covered include: quasistatic and dynamic solutions to Maxwell's equations; waves, radiation, and diffraction; coupling to media and structures; guided and unguided waves; resonance; and forces, power, and energy.The instructors of this course extend a general acknowledgment to the many students and instructors who have made major contributions to the 6.013 course materials over the years, and apologize for any residual errors that may remain in these writ

Subjects

electromagnetics | electromagnetics | applications | applications | wireless communications | wireless communications | circuits | circuits | computer interconnects | computer interconnects | peripherals | peripherals | optical fiber links | optical fiber links | microwave | microwave | communications | communications | radar | radar | antennas | antennas | sensors | sensors | micro-electromechanical systems | micro-electromechanical systems | power generation | power generation | power transmission | power transmission | quasistatic solutions | quasistatic solutions | dynamic solutions | dynamic solutions | Maxwell | Maxwell | Maxwell's equations | Maxwell's equations | waves | waves | radiation | radiation | diffraction | diffraction | guided waves | guided waves | unguided waves | unguided waves | resonance | resonance | forces | forces | power | power | energy | energy | microwave communications | microwave communications

License

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IV (MIT) IV (MIT)

Description

The basic objective of Unified Engineering is to give a solid understanding of the fundamental disciplines of aerospace engineering, as well as their interrelationships and applications. These disciplines are Materials and Structures (M); Computers and Programming (C); Fluid Mechanics (F); Thermodynamics (T); Propulsion (P); and Signals and Systems (S). In choosing to teach these subjects in a unified manner, the instructors seek to explain the common intellectual threads in these disciplines, as well as their combined application to solve engineering Systems Problems (SP). Throughout the year, the instructors emphasize the connections among the disciplines.Technical RequirementsMicrosoft® Excel software is recommended for viewing the .xls files The basic objective of Unified Engineering is to give a solid understanding of the fundamental disciplines of aerospace engineering, as well as their interrelationships and applications. These disciplines are Materials and Structures (M); Computers and Programming (C); Fluid Mechanics (F); Thermodynamics (T); Propulsion (P); and Signals and Systems (S). In choosing to teach these subjects in a unified manner, the instructors seek to explain the common intellectual threads in these disciplines, as well as their combined application to solve engineering Systems Problems (SP). Throughout the year, the instructors emphasize the connections among the disciplines.Technical RequirementsMicrosoft® Excel software is recommended for viewing the .xls files

Subjects

Unified | Unified | Unified Engineering | Unified Engineering | aerospace | aerospace | CDIO | CDIO | C-D-I-O | C-D-I-O | conceive | conceive | design | design | implement | implement | operate | operate | team | team | team-based | team-based | discipline | discipline | materials | materials | structures | structures | materials and structures | materials and structures | computers | computers | programming | programming | computers and programming | computers and programming | fluids | fluids | fluid mechanics | fluid mechanics | thermodynamics | thermodynamics | propulsion | propulsion | signals | signals | systems | systems | signals and systems | signals and systems | systems problems | systems problems | fundamentals | fundamentals | technical communication | technical communication | graphical communication | graphical communication | communication | communication | reading | reading | research | research | experimentation | experimentation | personal response system | personal response system | prs | prs | active learning | active learning | First law | First law | first law of thermodynamics | first law of thermodynamics | thermo-mechanical | thermo-mechanical | energy | energy | energy conversion | energy conversion | aerospace power systems | aerospace power systems | propulsion systems | propulsion systems | aerospace propulsion systems | aerospace propulsion systems | heat | heat | work | work | thermal efficiency | thermal efficiency | forms of energy | forms of energy | energy exchange | energy exchange | processes | processes | heat engines | heat engines | engines | engines | steady-flow energy equation | steady-flow energy equation | energy flow | energy flow | flows | flows | path-dependence | path-dependence | path-independence | path-independence | reversibility | reversibility | irreversibility | irreversibility | state | state | thermodynamic state | thermodynamic state | performance | performance | ideal cycle | ideal cycle | simple heat engine | simple heat engine | cycles | cycles | thermal pressures | thermal pressures | temperatures | temperatures | linear static networks | linear static networks | loop method | loop method | node method | node method | linear dynamic networks | linear dynamic networks | classical methods | classical methods | state methods | state methods | state concepts | state concepts | dynamic systems | dynamic systems | resistive circuits | resistive circuits | sources | sources | voltages | voltages | currents | currents | Thevinin | Thevinin | Norton | Norton | initial value problems | initial value problems | RLC networks | RLC networks | characteristic values | characteristic values | characteristic vectors | characteristic vectors | transfer function | transfer function | ada | ada | ada programming | ada programming | programming language | programming language | software systems | software systems | programming style | programming style | computer architecture | computer architecture | program language evolution | program language evolution | classification | classification | numerical computation | numerical computation | number representation systems | number representation systems | assembly | assembly | SimpleSIM | SimpleSIM | RISC | RISC | CISC | CISC | operating systems | operating systems | single user | single user | multitasking | multitasking | multiprocessing | multiprocessing | domain-specific classification | domain-specific classification | recursive | recursive | execution time | execution time | fluid dynamics | fluid dynamics | physical properties of a fluid | physical properties of a fluid | fluid flow | fluid flow | mach | mach | reynolds | reynolds | conservation | conservation | conservation principles | conservation principles | conservation of mass | conservation of mass | conservation of momentum | conservation of momentum | conservation of energy | conservation of energy | continuity | continuity | inviscid | inviscid | steady flow | steady flow | simple bodies | simple bodies | airfoils | airfoils | wings | wings | channels | channels | aerodynamics | aerodynamics | forces | forces | moments | moments | equilibrium | equilibrium | freebody diagram | freebody diagram | free-body | free-body | free body | free body | planar force systems | planar force systems | equipollent systems | equipollent systems | equipollence | equipollence | support reactions | support reactions | reactions | reactions | static determinance | static determinance | determinate systems | determinate systems | truss analysis | truss analysis | trusses | trusses | method of joints | method of joints | method of sections | method of sections | statically indeterminate | statically indeterminate | three great principles | three great principles | 3 great principles | 3 great principles | indicial notation | indicial notation | rotation of coordinates | rotation of coordinates | coordinate rotation | coordinate rotation | stress | stress | extensional stress | extensional stress | shear stress | shear stress | notation | notation | plane stress | plane stress | stress equilbrium | stress equilbrium | stress transformation | stress transformation | mohr | mohr | mohr's circle | mohr's circle | principal stress | principal stress | principal stresses | principal stresses | extreme shear stress | extreme shear stress | strain | strain | extensional strain | extensional strain | shear strain | shear strain | strain-displacement | strain-displacement | compatibility | compatibility | strain transformation | strain transformation | transformation of strain | transformation of strain | mohr's circle for strain | mohr's circle for strain | principal strain | principal strain | extreme shear strain | extreme shear strain | uniaxial stress-strain | uniaxial stress-strain | material properties | material properties | classes of materials | classes of materials | bulk material properties | bulk material properties | origin of elastic properties | origin of elastic properties | structures of materials | structures of materials | atomic bonding | atomic bonding | packing of atoms | packing of atoms | atomic packing | atomic packing | crystals | crystals | crystal structures | crystal structures | polymers | polymers | estimate of moduli | estimate of moduli | moduli | moduli | composites | composites | composite materials | composite materials | modulus limited design | modulus limited design | material selection | material selection | materials selection | materials selection | measurement of elastic properties | measurement of elastic properties | stress-strain | stress-strain | stress-strain relations | stress-strain relations | anisotropy | anisotropy | orthotropy | orthotropy | measurements | measurements | engineering notation | engineering notation | Hooke | Hooke | Hooke's law | Hooke's law | general hooke's law | general hooke's law | equations of elasticity | equations of elasticity | boundary conditions | boundary conditions | multi-disciplinary | multi-disciplinary | models | models | engineering systems | engineering systems | experiments | experiments | investigations | investigations | experimental error | experimental error | design evaluation | design evaluation | evaluation | evaluation | trade studies | trade studies | effects of engineering | effects of engineering | social context | social context | engineering drawings | engineering drawings | 16.01 | 16.01 | 16.02 | 16.02 | 16.03 | 16.03 | 16.04 | 16.04

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IV (MIT) IV (MIT)

Description

Includes audio/video content: AV selected lectures, AV faculty introductions, AV special element video. The basic objective of Unified Engineering is to give a solid understanding of the fundamental disciplines of aerospace engineering, as well as their interrelationships and applications. These disciplines are Materials and Structures (M); Computers and Programming (C); Fluid Mechanics (F); Thermodynamics (T); Propulsion (P); and Signals and Systems (S). In choosing to teach these subjects in a unified manner, the instructors seek to explain the common intellectual threads in these disciplines, as well as their combined application to solve engineering Systems Problems (SP). Throughout the year, the instructors emphasize the connections among the disciplines. Includes audio/video content: AV selected lectures, AV faculty introductions, AV special element video. The basic objective of Unified Engineering is to give a solid understanding of the fundamental disciplines of aerospace engineering, as well as their interrelationships and applications. These disciplines are Materials and Structures (M); Computers and Programming (C); Fluid Mechanics (F); Thermodynamics (T); Propulsion (P); and Signals and Systems (S). In choosing to teach these subjects in a unified manner, the instructors seek to explain the common intellectual threads in these disciplines, as well as their combined application to solve engineering Systems Problems (SP). Throughout the year, the instructors emphasize the connections among the disciplines.

Subjects

Unified | Unified | Unified Engineering | Unified Engineering | aerospace | aerospace | CDIO | CDIO | C-D-I-O | C-D-I-O | conceive | conceive | design | design | implement | implement | operate | operate | team | team | team-based | team-based | discipline | discipline | materials | materials | structures | structures | materials and structures | materials and structures | computers | computers | programming | programming | computers and programming | computers and programming | fluids | fluids | fluid mechanics | fluid mechanics | thermodynamics | thermodynamics | propulsion | propulsion | signals | signals | systems | systems | signals and systems | signals and systems | systems problems | systems problems | fundamentals | fundamentals | technical communication | technical communication | graphical communication | graphical communication | communication | communication | reading | reading | research | research | experimentation | experimentation | personal response system | personal response system | prs | prs | active learning | active learning | First law | First law | first law of thermodynamics | first law of thermodynamics | thermo-mechanical | thermo-mechanical | energy | energy | energy conversion | energy conversion | aerospace power systems | aerospace power systems | propulsion systems | propulsion systems | aerospace propulsion systems | aerospace propulsion systems | heat | heat | work | work | thermal efficiency | thermal efficiency | forms of energy | forms of energy | energy exchange | energy exchange | processes | processes | heat engines | heat engines | engines | engines | steady-flow energy equation | steady-flow energy equation | energy flow | energy flow | flows | flows | path-dependence | path-dependence | path-independence | path-independence | reversibility | reversibility | irreversibility | irreversibility | state | state | thermodynamic state | thermodynamic state | performance | performance | ideal cycle | ideal cycle | simple heat engine | simple heat engine | cycles | cycles | thermal pressures | thermal pressures | temperatures | temperatures | linear static networks | linear static networks | loop method | loop method | node method | node method | linear dynamic networks | linear dynamic networks | classical methods | classical methods | state methods | state methods | state concepts | state concepts | dynamic systems | dynamic systems | resistive circuits | resistive circuits | sources | sources | voltages | voltages | currents | currents | Thevinin | Thevinin | Norton | Norton | initial value problems | initial value problems | RLC networks | RLC networks | characteristic values | characteristic values | characteristic vectors | characteristic vectors | transfer function | transfer function | ada | ada | ada programming | ada programming | programming language | programming language | software systems | software systems | programming style | programming style | computer architecture | computer architecture | program language evolution | program language evolution | classification | classification | numerical computation | numerical computation | number representation systems | number representation systems | assembly | assembly | SimpleSIM | SimpleSIM | RISC | RISC | CISC | CISC | operating systems | operating systems | single user | single user | multitasking | multitasking | multiprocessing | multiprocessing | domain-specific classification | domain-specific classification | recursive | recursive | execution time | execution time | fluid dynamics | fluid dynamics | physical properties of a fluid | physical properties of a fluid | fluid flow | fluid flow | mach | mach | reynolds | reynolds | conservation | conservation | conservation principles | conservation principles | conservation of mass | conservation of mass | conservation of momentum | conservation of momentum | conservation of energy | conservation of energy | continuity | continuity | inviscid | inviscid | steady flow | steady flow | simple bodies | simple bodies | airfoils | airfoils | wings | wings | channels | channels | aerodynamics | aerodynamics | forces | forces | moments | moments | equilibrium | equilibrium | freebody diagram | freebody diagram | free-body | free-body | free body | free body | planar force systems | planar force systems | equipollent systems | equipollent systems | equipollence | equipollence | support reactions | support reactions | reactions | reactions | static determinance | static determinance | determinate systems | determinate systems | truss analysis | truss analysis | trusses | trusses | method of joints | method of joints | method of sections | method of sections | statically indeterminate | statically indeterminate | three great principles | three great principles | 3 great principles | 3 great principles | indicial notation | indicial notation | rotation of coordinates | rotation of coordinates | coordinate rotation | coordinate rotation | stress | stress | extensional stress | extensional stress | shear stress | shear stress | notation | notation | plane stress | plane stress | stress equilbrium | stress equilbrium | stress transformation | stress transformation | mohr | mohr | mohr's circle | mohr's circle | principal stress | principal stress | principal stresses | principal stresses | extreme shear stress | extreme shear stress | strain | strain | extensional strain | extensional strain | shear strain | shear strain | strain-displacement | strain-displacement | compatibility | compatibility | strain transformation | strain transformation | transformation of strain | transformation of strain | mohr's circle for strain | mohr's circle for strain | principal strain | principal strain | extreme shear strain | extreme shear strain | uniaxial stress-strain | uniaxial stress-strain | material properties | material properties | classes of materials | classes of materials | bulk material properties | bulk material properties | origin of elastic properties | origin of elastic properties | structures of materials | structures of materials | atomic bonding | atomic bonding | packing of atoms | packing of atoms | atomic packing | atomic packing | crystals | crystals | crystal structures | crystal structures | polymers | polymers | estimate of moduli | estimate of moduli | moduli | moduli | composites | composites | composite materials | composite materials | modulus limited design | modulus limited design | material selection | material selection | materials selection | materials selection | measurement of elastic properties | measurement of elastic properties | stress-strain | stress-strain | stress-strain relations | stress-strain relations | anisotropy | anisotropy | orthotropy | orthotropy | measurements | measurements | engineering notation | engineering notation | Hooke | Hooke | Hooke's law | Hooke's law | general hooke's law | general hooke's law | equations of elasticity | equations of elasticity | boundary conditions | boundary conditions | multi-disciplinary | multi-disciplinary | models | models | engineering systems | engineering systems | experiments | experiments | investigations | investigations | experimental error | experimental error | design evaluation | design evaluation | evaluation | evaluation | trade studies | trade studies | effects of engineering | effects of engineering | social context | social context | engineering drawings | engineering drawings

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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8.321 Quantum Theory I (MIT) 8.321 Quantum Theory I (MIT)

Description

8.321 is the first semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: Hilbert spaces, observables, uncertainty relations, eigenvalue problems and methods for solution thereof, time-evolution in the Schrodinger, Heisenberg, and interaction pictures, connections between classical and quantum mechanics, path integrals, quantum mechanics in EM fields, angular momentum, time-independent perturbation theory, density operators, and quantum measurement. 8.321 is the first semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: Hilbert spaces, observables, uncertainty relations, eigenvalue problems and methods for solution thereof, time-evolution in the Schrodinger, Heisenberg, and interaction pictures, connections between classical and quantum mechanics, path integrals, quantum mechanics in EM fields, angular momentum, time-independent perturbation theory, density operators, and quantum measurement.

Subjects

eigenstates | eigenstates | uncertainty relation | uncertainty relation | observables | observables | eigenvalues | eigenvalues | probabilities of the results of measurement | probabilities of the results of measurement | transformation theory | transformation theory | equations of motion | equations of motion | constants of motion | constants of motion | Symmetry in quantum mechanics | Symmetry in quantum mechanics | representations of symmetry groups | representations of symmetry groups | Variational and perturbation approximations | Variational and perturbation approximations | Systems of identical particles and applications | Systems of identical particles and applications | Time-dependent perturbation theory | Time-dependent perturbation theory | Scattering theory: phase shifts | Scattering theory: phase shifts | Born approximation | Born approximation | The quantum theory of radiation | The quantum theory of radiation | Second quantization and many-body theory | Second quantization and many-body theory | Relativistic quantum mechanics of one electron | Relativistic quantum mechanics of one electron | probability | probability | measurement | measurement | motion equations | motion equations | motion constants | motion constants | symmetry groups | symmetry groups | quantum mechanics | quantum mechanics | variational approximations | variational approximations | perturbation approximations | perturbation approximations | identical particles | identical particles | time-dependent perturbation theory | time-dependent perturbation theory | scattering theory | scattering theory | phase shifts | phase shifts | quantum theory of radiation | quantum theory of radiation | second quantization | second quantization | many-body theory | many-body theory | relativistic quantum mechanics | relativistic quantum mechanics | one electron | one electron | Hilbert spaces | Hilbert spaces | time evolution | time evolution | Schrodinger picture | Schrodinger picture | Heisenberg picture | Heisenberg picture | interaction picture | interaction picture | classical mechanics | classical mechanics | path integrals | path integrals | EM fields | EM fields | electromagnetic fields | electromagnetic fields | angular momentum | angular momentum | density operators | density operators | quantum measurement | quantum measurement | quantum statistics | quantum statistics | quantum dynamics | quantum dynamics

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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8.322 Quantum Theory II (MIT) 8.322 Quantum Theory II (MIT)

Description

8.322 is the second semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: time-dependent perturbation theory and applications to radiation, quantization of EM radiation field, adiabatic theorem and Berry's phase, symmetries in QM, many-particle systems, scattering theory, relativistic quantum mechanics, and Dirac equation. 8.322 is the second semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: time-dependent perturbation theory and applications to radiation, quantization of EM radiation field, adiabatic theorem and Berry's phase, symmetries in QM, many-particle systems, scattering theory, relativistic quantum mechanics, and Dirac equation.

Subjects

uncertainty relation | uncertainty relation | observables | observables | eigenstates | eigenstates | eigenvalues | eigenvalues | probabilities of the results of measurement | probabilities of the results of measurement | transformation theory | transformation theory | equations of motion | equations of motion | constants of motion | constants of motion | Symmetry in quantum mechanics | Symmetry in quantum mechanics | representations of symmetry groups | representations of symmetry groups | Variational and perturbation approximations | Variational and perturbation approximations | Systems of identical particles and applications | Systems of identical particles and applications | Time-dependent perturbation theory | Time-dependent perturbation theory | Scattering theory: phase shifts | Scattering theory: phase shifts | Born approximation | Born approximation | The quantum theory of radiation | The quantum theory of radiation | Second quantization and many-body theory | Second quantization and many-body theory | Relativistic quantum mechanics of one electron | Relativistic quantum mechanics of one electron | probability | probability | measurement | measurement | motion equations | motion equations | motion constants | motion constants | symmetry groups | symmetry groups | quantum mechanics | quantum mechanics | variational approximations | variational approximations | perturbation approximations | perturbation approximations | identical particles | identical particles | time-dependent perturbation theory | time-dependent perturbation theory | scattering theory | scattering theory | phase shifts | phase shifts | quantum theory of radiation | quantum theory of radiation | second quantization | second quantization | many-body theory | many-body theory | relativistic quantum mechanics | relativistic quantum mechanics | one electron | one electron | quantization | quantization | EM radiation field | EM radiation field | electromagnetic radiation field | electromagnetic radiation field | adiabatic theorem | adiabatic theorem | Berry?s phase | Berry?s phase | many-particle systems | many-particle systems | Dirac equation | Dirac equation | Hilbert spaces | Hilbert spaces | time evolution | time evolution | Schrodinger picture | Schrodinger picture | Heisenberg picture | Heisenberg picture | interaction picture | interaction picture | classical mechanics | classical mechanics | path integrals | path integrals | EM fields | EM fields | electromagnetic fields | electromagnetic fields | angular momentum | angular momentum | density operators | density operators | quantum measurement | quantum measurement | quantum statistics | quantum statistics | quantum dynamics | quantum dynamics

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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18.702 Algebra II (MIT)

Description

This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

Subjects

Sylow theorems | Group Representations | definitions | unitary representations | characters | Schur's Lemma | Rings: Basic Definitions | homomorphisms | fractions | Factorization | unique factorization | Gauss' Lemma | explicit factorization | maximal ideals | Quadratic Imaginary Integers | Gauss Primes | quadratic integers | ideal factorization | ideal classes | Linear Algebra over a Ring | free modules | integer matrices | generators and relations | structure of abelian groups | Rings: Abstract Constructions | relations in a ring | adjoining elements | Fields: Field Extensions | algebraic elements | degree of field extension | ruler and compass | symbolic adjunction | finite fields | Fields: Galois Theory | the main theorem | cubic equations | symmetric functions | primitive elements | quartic equations | quintic equations

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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18.702 Algebra II (MIT)

Description

The course covers group theory and its representations, and focuses on the Sylow theorem, Schur's lemma, and proof of the orthogonality relations. It also analyzes the rings, the factorization processes, and the fields. Topics such as the formal construction of integers and polynomials, homomorphisms and ideals, the Gauss' lemma, quadratic imaginary integers, Gauss primes, and finite and function fields are discussed in detail.

Subjects

Sylow theorems | Group Representations | definitions | unitary representations | characters | Schur's Lemma | Rings: Basic Definitions | homomorphisms | fractions | Factorization | unique factorization | Gauss' Lemma | explicit factorization | maximal ideals | Quadratic Imaginary Integers | Gauss Primes | quadratic integers | ideal factorization | ideal classes | Linear Algebra over a Ring | free modules | integer matrices | generators and relations | structure of abelian groups | Rings: Abstract Constructions | relations in a ring | adjoining elements | Fields: Field Extensions | algebraic elements | degree of field extension | ruler and compass | symbolic adjunction | finite fields | Fields: Galois Theory | the main theorem | cubic equations | symmetric functions | primitive elements | quartic equations | quintic equations

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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18.702 Algebra II (MIT)

Description

This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

Subjects

Sylow theorems | Group Representations | definitions | unitary representations | characters | Schur's Lemma | Rings: Basic Definitions | homomorphisms | fractions | Factorization | unique factorization | Gauss' Lemma | explicit factorization | maximal ideals | Quadratic Imaginary Integers | Gauss Primes | quadratic integers | ideal factorization | ideal classes | Linear Algebra over a Ring | free modules | integer matrices | generators and relations | structure of abelian groups | Rings: Abstract Constructions | relations in a ring | adjoining elements | Fields: Field Extensions | algebraic elements | degree of field extension | ruler and compass | symbolic adjunction | finite fields | Fields: Galois Theory | the main theorem | cubic equations | symmetric functions | primitive elements | quartic equations | quintic equations

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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2.29 Numerical Fluid Mechanics (MIT) 2.29 Numerical Fluid Mechanics (MIT)

Description

This course introduces students to MATLAB®. Numerical methods include number representation and errors, interpolation, differentiation, integration, systems of linear equations, and Fourier interpolation and transforms. Students will study partial and ordinary differential equations as well as elliptic and parabolic differential equations, and solutions by numerical integration, finite difference methods, finite element methods, boundary element methods, and panel methods. This course introduces students to MATLAB®. Numerical methods include number representation and errors, interpolation, differentiation, integration, systems of linear equations, and Fourier interpolation and transforms. Students will study partial and ordinary differential equations as well as elliptic and parabolic differential equations, and solutions by numerical integration, finite difference methods, finite element methods, boundary element methods, and panel methods.

Subjects

numerical methods | numerical methods | interpolation | interpolation | integration | integration | systems of linear equations | systems of linear equations | differential equations | differential equations | numerical integration | numerical integration | partial differential equations of inviscid hydrodynamics | partial differential equations of inviscid hydrodynamics | finite difference methods | finite difference methods | boundary integral equation panel methods | boundary integral equation panel methods | numerical lifting surface computations | numerical lifting surface computations | Fast Fourier Transforms | Fast Fourier Transforms | Numerical representation | Numerical representation | deterministic and random sea waves | deterministic and random sea waves | Integral boundary layer equations | Integral boundary layer equations | numerical solutions | numerical solutions

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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6.253 Convex Analysis and Optimization (MIT) 6.253 Convex Analysis and Optimization (MIT)

Description

6.253 develops the core analytical issues of continuous optimization, duality, and saddle point theory, using a handful of unifying principles that can be easily visualized and readily understood. The mathematical theory of convex sets and functions is discussed in detail, and is the basis for an intuitive, highly visual, geometrical approach to the subject. 6.253 develops the core analytical issues of continuous optimization, duality, and saddle point theory, using a handful of unifying principles that can be easily visualized and readily understood. The mathematical theory of convex sets and functions is discussed in detail, and is the basis for an intuitive, highly visual, geometrical approach to the subject.

Subjects

affine hulls | affine hulls | recession cones | recession cones | global minima | global minima | local minima | local minima | optimal solutions | optimal solutions | hyper planes | hyper planes | minimax theory | minimax theory | polyhedral convexity | polyhedral convexity | polyhedral cones | polyhedral cones | polyhedral sets | polyhedral sets | convex analysis | convex analysis | optimization | optimization | convexity | convexity | Lagrange multipliers | Lagrange multipliers | duality | duality | continuous optimization | continuous optimization | saddle point theory | saddle point theory | linear algebra | linear algebra | real analysis | real analysis | convex sets | convex sets | convex functions | convex functions | extreme points | extreme points | subgradients | subgradients | constrained optimization | constrained optimization | directional derivatives | directional derivatives | subdifferentials | subdifferentials | conical approximations | conical approximations | Fritz John optimality | Fritz John optimality | Exact penalty functions | Exact penalty functions | conjugate duality | conjugate duality | conjugate functions | conjugate functions | Fenchel duality | Fenchel duality | exact penalty functions | exact penalty functions | dual computational methods | dual computational methods

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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13.024 Numerical Marine Hydrodynamics (MIT) 13.024 Numerical Marine Hydrodynamics (MIT)

Description

This course is an introduction to numerical methods: interpolation, differentiation, integration, and systems of linear equations. It covers the solution of differential equations by numerical integration, as well as partial differential equations of inviscid hydrodynamics: finite difference methods, boundary integral equation panel methods. Also addressed are introductory numerical lifting surface computations, fast Fourier transforms, the numerical representation of deterministic and random sea waves, as well as integral boundary layer equations and numerical solutions.Technical RequirementMATLAB® software is required to run the .m files found on this course site. The .FIN and .OUT are simply data offest tables. They can be viewed with any text reader. RealOne™ This course is an introduction to numerical methods: interpolation, differentiation, integration, and systems of linear equations. It covers the solution of differential equations by numerical integration, as well as partial differential equations of inviscid hydrodynamics: finite difference methods, boundary integral equation panel methods. Also addressed are introductory numerical lifting surface computations, fast Fourier transforms, the numerical representation of deterministic and random sea waves, as well as integral boundary layer equations and numerical solutions.Technical RequirementMATLAB® software is required to run the .m files found on this course site. The .FIN and .OUT are simply data offest tables. They can be viewed with any text reader. RealOne™

Subjects

numerical methods | numerical methods | interpolation | interpolation | differentiation | differentiation | integration | integration | systems of linear equations | systems of linear equations | differential equations | differential equations | numerical integration | numerical integration | partial differential | partial differential | boundary integral equation panel methods | boundary integral equation panel methods | deterministic and random sea waves | deterministic and random sea waves | Fast Fourier Transforms | Fast Fourier Transforms | finite difference methods | finite difference methods | Integral boundary layer equations | Integral boundary layer equations | numerical lifting surface computations | numerical lifting surface computations | Numerical representation | Numerical representation | numerical solutions | numerical solutions | partial differential equations of inviscid hydrodynamics | partial differential equations of inviscid hydrodynamics | incompressible fluid mechanics | incompressible fluid mechanics | calculus | calculus | complex numbers | complex numbers | root finding | root finding | curve fitting | curve fitting | numerical differentiation | numerical differentiation | numerical errors | numerical errors | panel methods | panel methods | oscillating rigid objects | oscillating rigid objects | 2.29 | 2.29

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

License

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17.559 Comparative Security and Sustainability (MIT) 17.559 Comparative Security and Sustainability (MIT)

Description

This course focuses on the complexities associated with security and sustainability of states in international relations. Covering aspects of theory, methods and empirical analysis, the course is in three parts, and each consists of seminar sessions focusing on specific topics. This course focuses on the complexities associated with security and sustainability of states in international relations. Covering aspects of theory, methods and empirical analysis, the course is in three parts, and each consists of seminar sessions focusing on specific topics.

Subjects

security; sustainability; international relations; comparative approaches; constraints; options; strategies; policy choice; developing and industrial nations; decision; trade-offs; inter-temporal effects; technology; design systems; | security; sustainability; international relations; comparative approaches; constraints; options; strategies; policy choice; developing and industrial nations; decision; trade-offs; inter-temporal effects; technology; design systems; | security | security | sustainability | sustainability | international relations | international relations | comparative approaches | comparative approaches | constraints | constraints | options | options | strategies | strategies | policy choice | policy choice | developing and industrial nations | developing and industrial nations | decision | decision | trade-offs | trade-offs | inter-temporal effects | inter-temporal effects | technology | technology | design systems | design systems

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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14.01SC Principles of Microeconomics (MIT) 14.01SC Principles of Microeconomics (MIT)

Description

Includes audio/video content: AV lectures. 14.01 Principles of Microeconomics is an introductory undergraduate course that teaches the fundamentals of microeconomics. This course introduces microeconomic concepts and analysis, supply and demand analysis, theories of the firm and individual behavior, competition and monopoly, and welfare economics. Students will also be introduced to the use of microeconomic applications to address problems in current economic policy throughout the semester. This course is a core subject in MIT's undergraduate Energy Studies Minor. This Institute-wide program complements the deep expertise obtained in any major with a broad understanding of the interlinked realms of science, technology, and social sciences as they relate to energy and associated environmen Includes audio/video content: AV lectures. 14.01 Principles of Microeconomics is an introductory undergraduate course that teaches the fundamentals of microeconomics. This course introduces microeconomic concepts and analysis, supply and demand analysis, theories of the firm and individual behavior, competition and monopoly, and welfare economics. Students will also be introduced to the use of microeconomic applications to address problems in current economic policy throughout the semester. This course is a core subject in MIT's undergraduate Energy Studies Minor. This Institute-wide program complements the deep expertise obtained in any major with a broad understanding of the interlinked realms of science, technology, and social sciences as they relate to energy and associated environmen

Subjects

Microeconomics | Microeconomics | prices | prices | normative economics | normative economics | positive economics | positive economics | microeconomic applications | microeconomic applications | supply | supply | demand | demand | equilibrium | equilibrium | demand shift | demand shift | supply shift | supply shift | government interference | government interference | elasticity | elasticity | revenue | revenue | empirical economics | empirical economics | consumer theory | consumer theory | preference assumptions | preference assumptions | indifference curves | indifference curves | utility functions | utility functions | marginal utility | marginal utility | budget constraints | budget constraints | marginal rate of transformation | marginal rate of transformation | opportunity cost | opportunity cost | constrained utility maximization | constrained utility maximization | corner solutions | corner solutions | Engel curves | Engel curves | income effect | income effect | substitution effect | substitution effect | Giffin good | Giffin good | labor economics | labor economics | child labor | child labor | producer theory | producer theory | variable inputs | variable inputs | fixed inputs | fixed inputs | firm production functions | firm production functions | marginal rate of technical substitution | marginal rate of technical substitution | returns to scale | returns to scale | productivity | productivity | perfect competition | perfect competition | search theory | search theory | residual demand | residual demand | shutdown decisions | shutdown decisions | market equilibrium | market equilibrium | agency problem | agency problem | welfare economics | welfare economics | consumer surplus | consumer surplus | producer surplus | producer surplus | dead weight loss | dead weight loss | monopoly | monopoly | oligopoly | oligopoly | market power | market power | price discrimination | price discrimination | price regulation | price regulation | antitrust policy | antitrust policy | mergers | mergers | cartel | cartel | game theory | game theory | Nash equilibrium | Nash equilibrium | Cournot model | Cournot model | duopoly | duopoly | non-cooperative competition | non-cooperative competition | Bertrand competition | Bertrand competition | factor markets | factor markets | international trade | international trade | uncertainty | uncertainty | capital markets | capital markets | intertemporal choice | intertemporal choice | real interest rate | real interest rate | compounding | compounding | inflation | inflation | investment | investment | discount rate | discount rate | net present value | net present value | income distribution | income distribution | social welfare function | social welfare function | Utilitarianism | Utilitarianism | Raulsian criteria | Raulsian criteria | Nozickian | Nozickian | commodity egalitarianism | commodity egalitarianism | isowelfare curves | isowelfare curves | social insurance | social insurance | social security | social security | moral hazard | moral hazard | taxation | taxation | EITC | EITC | healthcare | healthcare | PPACA | PPACA

License

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2.29 Numerical Marine Hydrodynamics (13.024) (MIT) 2.29 Numerical Marine Hydrodynamics (13.024) (MIT)

Description

Includes audio/video content: AV faculty introductions. This course is an introduction to numerical methods: interpolation, differentiation, integration, and systems of linear equations. It covers the solution of differential equations by numerical integration, as well as partial differential equations of inviscid hydrodynamics: finite difference methods, boundary integral equation panel methods. Also addressed are introductory numerical lifting surface computations, fast Fourier transforms, the numerical representation of deterministic and random sea waves, as well as integral boundary layer equations and numerical solutions. This course was originally offered in Course 13 (Department of Ocean Engineering) as 13.024. In 2005, ocean engineering subjects became part of Course 2 (Department Includes audio/video content: AV faculty introductions. This course is an introduction to numerical methods: interpolation, differentiation, integration, and systems of linear equations. It covers the solution of differential equations by numerical integration, as well as partial differential equations of inviscid hydrodynamics: finite difference methods, boundary integral equation panel methods. Also addressed are introductory numerical lifting surface computations, fast Fourier transforms, the numerical representation of deterministic and random sea waves, as well as integral boundary layer equations and numerical solutions. This course was originally offered in Course 13 (Department of Ocean Engineering) as 13.024. In 2005, ocean engineering subjects became part of Course 2 (Department

Subjects

numerical methods | numerical methods | interpolation | interpolation | differentiation | differentiation | integration | integration | systems of linear equations | systems of linear equations | differential equations | differential equations | numerical integration | numerical integration | partial differential | partial differential | boundary integral equation panel methods | boundary integral equation panel methods | deterministic and random sea waves | deterministic and random sea waves | Fast Fourier Transforms | Fast Fourier Transforms | finite difference methods | finite difference methods | Integral boundary layer equations | Integral boundary layer equations | numerical lifting surface computations | numerical lifting surface computations | Numerical representation | Numerical representation | numerical solutions | numerical solutions | partial differential equations of inviscid hydrodynamics | partial differential equations of inviscid hydrodynamics | incompressible fluid mechanics | incompressible fluid mechanics | calculus | calculus | complex numbers | complex numbers | root finding | root finding | curve fitting | curve fitting | numerical differentiation | numerical differentiation | numerical errors | numerical errors | panel methods | panel methods | oscillating rigid objects | oscillating rigid objects

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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