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

Description

This course will focus on fundamental subjects in (deterministic) optimization, connected through the themes of convexity, geometric multipliers, and duality. The aim is to develop the core analytical and computational 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 will be central, and will allow an intuitive, highly visual, geometrical approach to the subject. This theory will be developed in detail and in parallel with the optimization topics. The first part of the course develops the analytical issues of convexity and duality. The second part is devoted to convex optimization algorithms, and their applications to a variety This course will focus on fundamental subjects in (deterministic) optimization, connected through the themes of convexity, geometric multipliers, and duality. The aim is to develop the core analytical and computational 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 will be central, and will allow an intuitive, highly visual, geometrical approach to the subject. This theory will be developed in detail and in parallel with the optimization topics. The first part of the course develops the analytical issues of convexity and duality. The second part is devoted to convex optimization algorithms, and their applications to a varietySubjects

convexity | convexity | optimization | optimization | geometric duality | geometric duality | Lagrangian duality | Lagrangian duality | Fenchel duality | Fenchel duality | cone programming | cone programming | semidefinite programming | semidefinite programming | subgradients | subgradients | constrained optimization | constrained optimization | gradient projection | gradient projectionLicense

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See all metadata6.253 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 methodsLicense

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See all metadata6.253 Convex Analysis and Optimization (MIT) 6.253 Convex Analysis and Optimization (MIT)

Description

This course will focus on fundamental subjects in convexity, duality, and convex optimization algorithms. The aim is to develop the core analytical and algorithmic issues of continuous optimization, duality, and saddle point theory using a handful of unifying principles that can be easily visualized and readily understood. This course will focus on fundamental subjects in convexity, duality, and convex optimization algorithms. The aim is to develop the core analytical and algorithmic issues of continuous optimization, duality, and saddle point theory using a handful of unifying principles that can be easily visualized and readily understood.Subjects

convex analysis | convex analysis | convex optimization | convex optimization | hyperplanes | hyperplanes | conjugacy | conjugacy | polyhedral convexity | polyhedral convexity | geometric duality | geometric duality | duality theory | duality theory | subgradients | subgradients | optimality conditions | optimality conditions | convex optimization algorithms | convex optimization algorithmsLicense

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See all metadata18.409 Behavior of Algorithms (MIT) 18.409 Behavior of Algorithms (MIT)

Description

This course is a study of Behavior of Algorithms and covers an area of current interest in theoretical computer science. The topics vary from term to term. During this term, we discuss rigorous approaches to explaining the typical performance of algorithms with a focus on the following approaches: smoothed analysis, condition numbers/parametric analysis, and subclassing inputs. This course is a study of Behavior of Algorithms and covers an area of current interest in theoretical computer science. The topics vary from term to term. During this term, we discuss rigorous approaches to explaining the typical performance of algorithms with a focus on the following approaches: smoothed analysis, condition numbers/parametric analysis, and subclassing inputs.Subjects

Condition number | Condition number | largest singluar value of a matrix | largest singluar value of a matrix | Smoothed analysis | Smoothed analysis | Gaussian elimination | Gaussian elimination | Growth factors of partial and complete pivoting | Growth factors of partial and complete pivoting | GE of graphs with low bandwidth or small separators | GE of graphs with low bandwidth or small separators | Spectral Partitioning of planar graphs | Spectral Partitioning of planar graphs | spectral paritioning of well-shaped meshes | spectral paritioning of well-shaped meshes | spectral paritioning of nearest neighbor graphs | spectral paritioning of nearest neighbor graphs | Turner's theorem | Turner's theorem | bandwidth of semi-random graphs. | bandwidth of semi-random graphs. | McSherry's spectral bisection algorithm | McSherry's spectral bisection algorithm | Linear Programming | Linear Programming | von Neumann's algorithm | von Neumann's algorithm | primal and dual simplex methods | and duality Strong duality theorem | primal and dual simplex methods | and duality Strong duality theorem | Renegar's condition numbers | Renegar's condition numbersLicense

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See all metadata8.821 String Theory and Holographic Duality (MIT) 8.821 String Theory and Holographic Duality (MIT)

Description

Includes audio/video content: AV lectures. This string theory course focuses on holographic duality (also known as gauge / gravity duality or AdS / CFT) as a novel method of approaching and connecting a range of diverse subjects, including quantum gravity / black holes, QCD at extreme conditions, exotic condensed matter systems, and quantum information. Includes audio/video content: AV lectures. This string theory course focuses on holographic duality (also known as gauge / gravity duality or AdS / CFT) as a novel method of approaching and connecting a range of diverse subjects, including quantum gravity / black holes, QCD at extreme conditions, exotic condensed matter systems, and quantum information.Subjects

string theory | string theory | holographic duality | holographic duality | Weinberg-Witten | Weinberg-Witten | AdS/CFT duality | AdS/CFT duality | black holes | black holes | Holographic principle | Holographic principle | Wilson loops | Wilson loops | Entanglement entropy | Entanglement entropy | Quark-gluon plasmas | Quark-gluon plasmas | quantum gravity | quantum gravity | Hamilton-Jacobi | Hamilton-Jacobi | D-branes | D-branes | Large-N Expansion | Large-N Expansion | Light-Cone Gauge | Light-Cone GaugeLicense

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See all metadata6.253 Convex Analysis and Optimization (MIT)

Description

This course will focus on fundamental subjects in (deterministic) optimization, connected through the themes of convexity, geometric multipliers, and duality. The aim is to develop the core analytical and computational 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 will be central, and will allow an intuitive, highly visual, geometrical approach to the subject. This theory will be developed in detail and in parallel with the optimization topics. The first part of the course develops the analytical issues of convexity and duality. The second part is devoted to convex optimization algorithms, and their applications to a varietySubjects

convexity | optimization | geometric duality | Lagrangian duality | Fenchel duality | cone programming | semidefinite programming | subgradients | constrained optimization | gradient projectionLicense

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See all metadata6.253 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.Subjects

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

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.htmSite sourced from

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See all metadata14.471 Public Economics I (MIT) 14.471 Public Economics I (MIT)

Description

This course is a one-semester introduction to the economic analysis of taxation. It covers both theoretical contributions, such as the theory of optimal income and commodity taxation, as well as empirical work, such as the study of how taxes affect labor supply. The course is designed to acquaint students with key questions in the economics of taxation, and to equip them to carry out their own research in this field. This course is a one-semester introduction to the economic analysis of taxation. It covers both theoretical contributions, such as the theory of optimal income and commodity taxation, as well as empirical work, such as the study of how taxes affect labor supply. The course is designed to acquaint students with key questions in the economics of taxation, and to equip them to carry out their own research in this field.Subjects

economic analysis | economic analysis | taxation | taxation | wealth | wealth | financial policy | financial policy | income | income | investment | investment | asset | asset | political economy | political economy | labor | labor | capital | capital | public policy | public policy | theory | theory | evidence | evidence | government taxation policy | government taxation policy | tax incidence | tax incidence | optimal tax theory | optimal tax theory | labor supply | labor supply | savings | savings | corrective taxes for externalities | corrective taxes for externalities | corporate behavior | corporate behavior | tax expenditure policy | tax expenditure policy | theory of optimal income | theory of optimal income | commodity taxation | commodity taxation | calculus-based microeconomic analysis | calculus-based microeconomic analysis | duality methods | duality methods | household theory | household theory | firm theory | firm theory | growth theory | growth theoryLicense

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See all metadata8.04 Quantum Physics I (MIT) 8.04 Quantum Physics I (MIT)

Description

Experimental basis of Quantum Physics: photoelectric effect, Compton scattering, photons, Franck-Hertz experiment, the Bohr atom, electron diffraction, De Broglie waves, and wave-particle duality of matter and light. Introduction to wave mechanics: Schroedinger's equation, wave functions, wave packets, probability amplitudes, stationary states, the Heisenberg uncertainty principle, and zero-point energies. Solutions to Schroedinger's equation in one dimension: transmission and reflection at a barrier, barrier penetration, potential wells, the simple harmonic oscillator. Schroedinger's equation in three dimensions: central potentials, and introduction to hydrogenic systems. Experimental basis of Quantum Physics: photoelectric effect, Compton scattering, photons, Franck-Hertz experiment, the Bohr atom, electron diffraction, De Broglie waves, and wave-particle duality of matter and light. Introduction to wave mechanics: Schroedinger's equation, wave functions, wave packets, probability amplitudes, stationary states, the Heisenberg uncertainty principle, and zero-point energies. Solutions to Schroedinger's equation in one dimension: transmission and reflection at a barrier, barrier penetration, potential wells, the simple harmonic oscillator. Schroedinger's equation in three dimensions: central potentials, and introduction to hydrogenic systems.Subjects

quantum physics: photoelectric effect | quantum physics: photoelectric effect | Compton scattering | Compton scattering | photons | photons | Franck-Hertz experiment | Franck-Hertz experiment | the Bohr atom | the Bohr atom | electron diffraction | electron diffraction | deBroglie waves | deBroglie waves | wave-particle duality of matter and light | wave-particle duality of matter and light | wave mechanics: Schroedinger's equation | wave mechanics: Schroedinger's equation | wave functions | wave functions | wave packets | wave packets | probability amplitudes | probability amplitudes | stationary states | stationary states | the Heisenberg uncertainty principle | the Heisenberg uncertainty principle | zero-point energies | zero-point energies | transmission and reflection at a barrier | transmission and reflection at a barrier | barrier penetration | barrier penetration | potential wells | potential wells | simple harmonic oscillator | simple harmonic oscillator | Schroedinger's equation in three dimensions: central potentials | Schroedinger's equation in three dimensions: central potentials | introduction to hydrogenic systems | introduction to hydrogenic systems | De Broglie waves | De Broglie wavesLicense

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See all metadataCreativity Lecture 8: Creativity as a neuroscientific mystery

Description

Prof. Margaret Boden (Philosophy, Sussex) delivers a lecture as part of the Keble College Creativity series. Creativity is likely to remain a neuroscientific mystery for many years. Of the three types of creativity (combinational, exploratory, and transformational), only the first has been significantly illuminated by neuroscience. And even that is not fully understood in neural terms. The other two are even more recalcitrant. This is due to difficulty in defining thinking styles in art or science, and in identifying the various computational processes that are involved in using them. Without doing that, helpful neuroscientific questions simply cannot arise. One key problem is that hierarchical systems -- including many creative "styles"-- cannot yet be effectively represented by (connec Wales; http://creativecommons.org/licenses/by-nc-sa/2.0/uk/Subjects

individuality | keble college | cells | neuron | creativity | schizophrenia | cell | brain | keble | neurons | human | oxford | neuroscience | individuality | keble college | cells | neuron | creativity | schizophrenia | cell | brain | keble | neurons | human | oxford | neuroscience | 2012-05-18License

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See all metadata5.111 Principles of Chemical Science (MIT) 5.111 Principles of Chemical Science (MIT)

Description

This course provides an introduction to the chemistry of biological, inorganic, and organic molecules. The emphasis is on basic principles of atomic and molecular electronic structure, thermodynamics, acid-base and redox equilibria, chemical kinetics, and catalysis. In an effort to illuminate connections between chemistry and biology, a list of the biology-, medicine-, and MIT research-related examples used in 5.111 is provided in Biology-Related Examples. Acknowledgements Development and implementation of the biology-related materials in this course were funded through an HHMI Professors grant to Prof. Catherine L. Drennan. This course provides an introduction to the chemistry of biological, inorganic, and organic molecules. The emphasis is on basic principles of atomic and molecular electronic structure, thermodynamics, acid-base and redox equilibria, chemical kinetics, and catalysis. In an effort to illuminate connections between chemistry and biology, a list of the biology-, medicine-, and MIT research-related examples used in 5.111 is provided in Biology-Related Examples. Acknowledgements Development and implementation of the biology-related materials in this course were funded through an HHMI Professors grant to Prof. Catherine L. Drennan.Subjects

introductory chemistry | introductory chemistry | atomic structure | atomic structure | molecular electronic structure | molecular electronic structure | thermodynamics | thermodynamics | acid-base equillibrium | acid-base equillibrium | titration | titration | redox | redox | chemical kinetics | chemical kinetics | catalysis | catalysis | lewis structures | lewis structures | VSEPR theory | VSEPR theory | wave-particle duality | wave-particle duality | biochemistry | biochemistry | orbitals | orbitals | periodic trends | periodic trends | general chemistry | general chemistry | valence bond theory | valence bond theory | hybridization | hybridization | free energy | free energy | reaction mechanism | reaction mechanism | Rutherford backscattering | Rutherford backscatteringLicense

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See all metadata15.084J Nonlinear Programming (MIT) 15.084J Nonlinear Programming (MIT)

Description

Includes audio/video content: AV selected lectures. This course introduces students to the fundamentals of nonlinear optimization theory and methods. Topics include unconstrained and constrained optimization, linear and quadratic programming, Lagrange and conic duality theory, interior-point algorithms and theory, Lagrangian relaxation, generalized programming, and semi-definite programming. Algorithmic methods used in the class include steepest descent, Newton's method, conditional gradient and subgradient optimization, interior-point methods and penalty and barrier methods. Includes audio/video content: AV selected lectures. This course introduces students to the fundamentals of nonlinear optimization theory and methods. Topics include unconstrained and constrained optimization, linear and quadratic programming, Lagrange and conic duality theory, interior-point algorithms and theory, Lagrangian relaxation, generalized programming, and semi-definite programming. Algorithmic methods used in the class include steepest descent, Newton's method, conditional gradient and subgradient optimization, interior-point methods and penalty and barrier methods.Subjects

unconstrained and constrained optimization | unconstrained and constrained optimization | Lagrangean relaxation | Lagrangean relaxation | generalized programming | generalized programming | Newton's method | Newton's method | conditional gradient and subgradient optimization | conditional gradient and subgradient optimization | linear and quadratic programming | linear and quadratic programming | lagrange and conic duality theory | lagrange and conic duality theory | interior-point algorithms and theory | interior-point algorithms and theory | semi-definite programming | semi-definite programming | Algorithmic methods include steepest descent | Algorithmic methods include steepest descent | interior-point methods and penalty and barrier methods | interior-point methods and penalty and barrier methods | 15.084 | 15.084 | 6.252 | 6.252License

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See all metadata6.701 Introduction to Nanoelectronics (MIT) 6.701 Introduction to Nanoelectronics (MIT)

Description

Traditionally, progress in electronics has been driven by miniaturization. But as electronic devices approach the molecular scale, classical models for device behavior must be abandoned. To prepare for the next generation of electronic devices, this class teaches the theory of current, voltage and resistance from atoms up. To describe electrons at the nanoscale, we will begin with an introduction to the principles of quantum mechanics, including quantization, the wave-particle duality, wavefunctions and Schrödinger's equation. Then we will consider the electronic properties of molecules, carbon nanotubes and crystals, including energy band formation and the origin of metals, insulators and semiconductors. Electron conduction will be taught beginning with ballistic transport and concludin Traditionally, progress in electronics has been driven by miniaturization. But as electronic devices approach the molecular scale, classical models for device behavior must be abandoned. To prepare for the next generation of electronic devices, this class teaches the theory of current, voltage and resistance from atoms up. To describe electrons at the nanoscale, we will begin with an introduction to the principles of quantum mechanics, including quantization, the wave-particle duality, wavefunctions and Schrödinger's equation. Then we will consider the electronic properties of molecules, carbon nanotubes and crystals, including energy band formation and the origin of metals, insulators and semiconductors. Electron conduction will be taught beginning with ballistic transport and concludinSubjects

nanoelectronics | nanoelectronics | quantum mechanics | quantum mechanics | wave-particle duality | wave-particle duality | Schrodinger's equation | Schrodinger's equation | electronic properties of molecules | electronic properties of molecules | energy band formation | energy band formation | electron conduction | electron conduction | ballistic transport | ballistic transport | Ohm's law | Ohm's law | fundamental limits to computation | fundamental limits to computationLicense

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See all metadata6.079 Introduction to Convex Optimization (MIT) 6.079 Introduction to Convex Optimization (MIT)

Description

This course aims to give students the tools and training to recognize convex optimization problems that arise in scientific and engineering applications, presenting the basic theory, and concentrating on modeling aspects and results that are useful in applications. Topics include convex sets, convex functions, optimization problems, least-squares, linear and quadratic programs, semidefinite programming, optimality conditions, and duality theory. Applications to signal processing, control, machine learning, finance, digital and analog circuit design, computational geometry, statistics, and mechanical engineering are presented. Students complete hands-on exercises using high-level numerical software. Acknowledgements The course materials were developed jointly by Prof. Stephen Boyd (Stanford This course aims to give students the tools and training to recognize convex optimization problems that arise in scientific and engineering applications, presenting the basic theory, and concentrating on modeling aspects and results that are useful in applications. Topics include convex sets, convex functions, optimization problems, least-squares, linear and quadratic programs, semidefinite programming, optimality conditions, and duality theory. Applications to signal processing, control, machine learning, finance, digital and analog circuit design, computational geometry, statistics, and mechanical engineering are presented. Students complete hands-on exercises using high-level numerical software. Acknowledgements The course materials were developed jointly by Prof. Stephen Boyd (StanfordSubjects

convex sets | convex sets | convex functions | convex functions | optimization problems | optimization problems | least-squares | least-squares | linear and quadratic programs | linear and quadratic programs | semidefinite programming | semidefinite programming | optimality conditions | optimality conditions | duality theory | duality theoryLicense

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See all metadata6.632 Electromagnetic Wave Theory (MIT) 6.632 Electromagnetic Wave Theory (MIT)

Description

6.632 is a graduate subject on electromagnetic wave theory, emphasizing mathematical approaches, problem solving, and physical interpretation. Topics covered include: waves in media, equivalence principle, duality and complementarity, Huygens' principle, Fresnel and Fraunhofer diffraction, dyadic Green's functions, Lorentz transformation, and Maxwell-Minkowski theory. Examples deal with limiting cases of Maxwell's theory and diffraction and scattering of electromagnetic waves. 6.632 is a graduate subject on electromagnetic wave theory, emphasizing mathematical approaches, problem solving, and physical interpretation. Topics covered include: waves in media, equivalence principle, duality and complementarity, Huygens' principle, Fresnel and Fraunhofer diffraction, dyadic Green's functions, Lorentz transformation, and Maxwell-Minkowski theory. Examples deal with limiting cases of Maxwell's theory and diffraction and scattering of electromagnetic waves.Subjects

electromagnetic wave theory | electromagnetic wave theory | waves in media | waves in media | equivalence principle | equivalence principle | duality | duality | complementarity | complementarity | Huygens' principle | Huygens' principle | Fresnel diffraction | Fresnel diffraction | Fraunhofer diffraction | Fraunhofer diffraction | dyadic Green's functions | dyadic Green's functions | Lorentz transformation | Lorentz transformation | Maxwell-Minkowski theory | Maxwell-Minkowski theory | Maxwell | Maxwell | diffraction | diffraction | scattering | scatteringLicense

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See all metadata6.252J Nonlinear Programming (MIT) 6.252J Nonlinear Programming (MIT)

Description

6.252J is a course in the department's "Communication, Control, and Signal Processing" concentration. This course provides a unified analytical and computational approach to nonlinear optimization problems. The topics covered in this course include: unconstrained optimization methods, constrained optimization methods, convex analysis, Lagrangian relaxation, nondifferentiable optimization, and applications in integer programming. There is also a comprehensive treatment of optimality conditions, Lagrange multiplier theory, and duality theory. Throughout the course, applications are drawn from control, communications, power systems, and resource allocation problems. 6.252J is a course in the department's "Communication, Control, and Signal Processing" concentration. This course provides a unified analytical and computational approach to nonlinear optimization problems. The topics covered in this course include: unconstrained optimization methods, constrained optimization methods, convex analysis, Lagrangian relaxation, nondifferentiable optimization, and applications in integer programming. There is also a comprehensive treatment of optimality conditions, Lagrange multiplier theory, and duality theory. Throughout the course, applications are drawn from control, communications, power systems, and resource allocation problems.Subjects

nonlinear programming | nonlinear programming | non-linear programming | non-linear programming | nonlinear optimization | nonlinear optimization | unconstrained optimization | unconstrained optimization | gradient | gradient | conjugate direction | conjugate direction | Newton | Newton | quasi-Newton methods | quasi-Newton methods | constrained optimization | constrained optimization | feasible directions | feasible directions | projection | projection | interior point | interior point | Lagrange multiplier | Lagrange multiplier | convex analysis | convex analysis | Lagrangian relaxation | Lagrangian relaxation | nondifferentiable optimization | nondifferentiable optimization | integer programming | integer programming | optimality conditions | optimality conditions | Lagrange multiplier theory | Lagrange multiplier theory | duality theory | duality theory | control | control | communications | communications | power systems | power systems | resource allocation | resource allocation | 6.252 | 6.252 | 15.084 | 15.084License

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See all metadata8.821 String Theory (MIT) 8.821 String Theory (MIT)

Description

This is a one-semester class about gauge/gravity duality (often called AdS/CFT) and its applications. This is a one-semester class about gauge/gravity duality (often called AdS/CFT) and its applications.Subjects

string theory | string theory | conformal field theory | conformal field theory | light-cone and covariant quantization of the relativistic bosonic string | light-cone and covariant quantization of the relativistic bosonic string | quantization and spectrum of supersymmetric 10-dimensional string theories | quantization and spectrum of supersymmetric 10-dimensional string theories | T-duality and D-branes | T-duality and D-branes | toroidal compactification and orbifolds | toroidal compactification and orbifolds | 11-dimensional supergravity and M-theory. | 11-dimensional supergravity and M-theory.License

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See all metadata18.726 Algebraic Geometry (MIT) 18.726 Algebraic Geometry (MIT)

Description

This course provides an introduction to the language of schemes, properties of morphisms, and sheaf cohomology. Together with 18.725 Algebraic Geometry, students gain an understanding of the basic notions and techniques of modern algebraic geometry. This course provides an introduction to the language of schemes, properties of morphisms, and sheaf cohomology. Together with 18.725 Algebraic Geometry, students gain an understanding of the basic notions and techniques of modern algebraic geometry.Subjects

category theory | category theory | sheaves | sheaves | abelian sheaves | abelian sheaves | shcemes | shcemes | morphisms | morphisms | projective morphisms | projective morphisms | differentials | differentials | divisors | divisors | homological algebra | homological algebra | algebraic geometry | algebraic geometry | cohomology | cohomology | quasicoherent sheaves | quasicoherent sheaves | projective spaces | projective spaces | hilbert polynomials | hilbert polynomials | gaga | gaga | serre duality | serre duality | cohen-macaulay schemes | cohen-macaulay schemes | riemann-roch | riemann-roch | etale cohomology | etale cohomologyLicense

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See all metadata8.04 Quantum Physics I (MIT) 8.04 Quantum Physics I (MIT)

Description

Includes audio/video content: AV lectures. This course covers the experimental basis of quantum physics. It introduces wave mechanics, Schrödinger's equation in a single dimension, and Schrödinger's equation in three dimensions.It is the first course in the undergraduate Quantum Physics sequence, followed by 8.05 Quantum Physics II and 8.06 Quantum Physics III. Includes audio/video content: AV lectures. This course covers the experimental basis of quantum physics. It introduces wave mechanics, Schrödinger's equation in a single dimension, and Schrödinger's equation in three dimensions.It is the first course in the undergraduate Quantum Physics sequence, followed by 8.05 Quantum Physics II and 8.06 Quantum Physics III.Subjects

quantum physics: photoelectric effect | quantum physics: photoelectric effect | Compton scattering | Compton scattering | photons | photons | Franck-Hertz experiment | Franck-Hertz experiment | the Bohr atom | the Bohr atom | electron diffraction | electron diffraction | deBroglie waves | deBroglie waves | wave-particle duality of matter and light | wave-particle duality of matter and light | wave mechanics: Schroedinger's equation | wave mechanics: Schroedinger's equation | wave functions | wave functions | wave packets | wave packets | probability amplitudes | probability amplitudes | stationary states | stationary states | the Heisenberg uncertainty principle | the Heisenberg uncertainty principle | zero-point energies | zero-point energies | transmission and reflection at a barrier | transmission and reflection at a barrier | barrier penetration | barrier penetration | potential wells | potential wells | simple harmonic oscillator | simple harmonic oscillator | Schroedinger's equation in three dimensions: central potentials | and introduction to hydrogenic systems | Schroedinger's equation in three dimensions: central potentials | and introduction to hydrogenic systemsLicense

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.htmSite sourced from

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See all metadata8.04 Quantum Physics I (MIT) 8.04 Quantum Physics I (MIT)

Description

This course covers the experimental basis of quantum physics, introduces wave mechanics, Schrödinger's equation in a single dimension, and Schrödinger's equation in three dimensions. This course covers the experimental basis of quantum physics, introduces wave mechanics, Schrödinger's equation in a single dimension, and Schrödinger's equation in three dimensions.Subjects

quantum physics: photoelectric effect | quantum physics: photoelectric effect | Compton scattering | Compton scattering | photons | photons | Franck-Hertz experiment | Franck-Hertz experiment | the Bohr atom | the Bohr atom | electron diffraction | electron diffraction | deBroglie waves | deBroglie waves | wave-particle duality of matter and light | wave-particle duality of matter and light | wave mechanics: Schroedinger's equation | wave mechanics: Schroedinger's equation | wave functions | wave functions | wave packets | wave packets | probability amplitudes | probability amplitudes | stationary states | stationary states | the Heisenberg uncertainty principle | the Heisenberg uncertainty principle | zero-point energies | zero-point energies | transmission and reflection at a barrier | transmission and reflection at a barrier | barrier penetration | barrier penetration | potential wells | potential wells | simple harmonic oscillator | simple harmonic oscillator | Schroedinger's equation in three dimensions: central potentials | Schroedinger's equation in three dimensions: central potentials | and introduction to hydrogenic systems. | and introduction to hydrogenic 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.htmSite sourced from

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See all metadata8.821 String Theory and Holographic Duality (MIT)

Description

This string theory course focuses on holographic duality (also known as gauge / gravity duality or AdS / CFT) as a novel method of approaching and connecting a range of diverse subjects, including quantum gravity / black holes, QCD at extreme conditions, exotic condensed matter systems, and quantum information.Subjects

string theory | holographic duality | Weinberg-Witten | AdS/CFT duality | black holes | Holographic principle | Wilson loops | Entanglement entropy | Quark-gluon plasmas | quantum gravity | Hamilton-Jacobi | D-branes | Large-N Expansion | Light-Cone GaugeLicense

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.htmSite sourced from

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See all metadata18.409 Behavior of Algorithms (MIT)

Description

This course is a study of Behavior of Algorithms and covers an area of current interest in theoretical computer science. The topics vary from term to term. During this term, we discuss rigorous approaches to explaining the typical performance of algorithms with a focus on the following approaches: smoothed analysis, condition numbers/parametric analysis, and subclassing inputs.Subjects

Condition number | largest singluar value of a matrix | Smoothed analysis | Gaussian elimination | Growth factors of partial and complete pivoting | GE of graphs with low bandwidth or small separators | Spectral Partitioning of planar graphs | spectral paritioning of well-shaped meshes | spectral paritioning of nearest neighbor graphs | Turner's theorem | bandwidth of semi-random graphs. | McSherry's spectral bisection algorithm | Linear Programming | von Neumann's algorithm | primal and dual simplex methods | and duality Strong duality theorem | Renegar's condition numbersLicense

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.htmSite sourced from

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See all metadata6.253 Convex Analysis and Optimization (MIT)

Description

This course will focus on fundamental subjects in convexity, duality, and convex optimization algorithms. The aim is to develop the core analytical and algorithmic issues of continuous optimization, duality, and saddle point theory using a handful of unifying principles that can be easily visualized and readily understood.Subjects

convex analysis | convex optimization | hyperplanes | conjugacy | polyhedral convexity | geometric duality | duality theory | subgradients | optimality conditions | convex optimization algorithmsLicense

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.htmSite sourced from

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See all metadataDescription

This course covers the experimental basis of quantum physics, introduces wave mechanics, Schrödinger's equation in a single dimension, and Schrödinger's equation in three dimensions.Subjects

quantum physics: photoelectric effect | Compton scattering | photons | Franck-Hertz experiment | the Bohr atom | electron diffraction | deBroglie waves | wave-particle duality of matter and light | wave mechanics: Schroedinger's equation | wave functions | wave packets | probability amplitudes | stationary states | the Heisenberg uncertainty principle | zero-point energies | transmission and reflection at a barrier | barrier penetration | potential wells | simple harmonic oscillator | Schroedinger's equation in three dimensions: central potentials | and introduction to hydrogenic 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.htmSite sourced from

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See all metadataDescription

This course covers the experimental basis of quantum physics, introduces wave mechanics, Schrödinger's equation in a single dimension, and Schrödinger's equation in three dimensions.Subjects

quantum physics: photoelectric effect | Compton scattering | photons | Franck-Hertz experiment | the Bohr atom | electron diffraction | deBroglie waves | wave-particle duality of matter and light | wave mechanics: Schroedinger's equation | wave functions | wave packets | probability amplitudes | stationary states | the Heisenberg uncertainty principle | zero-point energies | transmission and reflection at a barrier | barrier penetration | potential wells | simple harmonic oscillator | Schroedinger's equation in three dimensions: central potentials | and introduction to hydrogenic 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.htmSite sourced from

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