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18.112 Functions of a Complex Variable (MIT) 18.112 Functions of a Complex Variable (MIT)

Description

This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted.This course offers biweekly problem sets with solutions, two term tests and a final exam, all with solutions. This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted.This course offers biweekly problem sets with solutions, two term tests and a final exam, all with solutions.Subjects

functions of one complex variable | functions of one complex variable | Cauchy's theorem | Cauchy's theorem | holomorphic functions | holomorphic functions | meromorphic functions | meromorphic functions | residues | residues | contour integrals | contour integrals | conformal mapping | conformal mapping | Infinite series and products | Infinite series and products | the gamma function | the gamma function | the Mittag-Leffler theorem | the Mittag-Leffler theorem | Harmonic functions | Harmonic functions | Dirichlet's problem | Dirichlet's problem | The Riemann mapping theorem | The Riemann mapping theorem | The Riemann Zeta function | The Riemann Zeta functionLicense

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See all metadata18.112 Functions of a Complex Variable (MIT) 18.112 Functions of a Complex Variable (MIT)

Description

This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted.This course offers biweekly problem sets with solutions, two term tests and a final exam, all with solutions. This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted.This course offers biweekly problem sets with solutions, two term tests and a final exam, all with solutions.Subjects

functions of one complex variable | functions of one complex variable | Cauchy's theorem | Cauchy's theorem | holomorphic functions | holomorphic functions | meromorphic functions | meromorphic functions | residues | residues | contour integrals | contour integrals | conformal mapping | conformal mapping | Infinite series and products | Infinite series and products | the gamma function | the gamma function | the Mittag-Leffler theorem | the Mittag-Leffler theorem | Harmonic functions | Harmonic functions | Dirichlet's problem | Dirichlet's problem | The Riemann mapping theorem | The Riemann mapping theorem | The Riemann Zeta function | The Riemann Zeta functionLicense

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 metadata9.01 Neuroscience and Behavior (MIT) 9.01 Neuroscience and Behavior (MIT)

Description

Relation of structure and function at various levels of neuronal integration. Topics include: functional neuroanatomy and neurophysiology, sensory and motor systems, centrally programmed behavior, sensory systems, sleep and dreaming, motivation and reward, emotional displays of various types, "higher functions" and the neocortex, and neural processes in learning and memory. In order to improve writing skills in describing experiments and reviewing journal publications in neuroscience, students are required to complete four homework assignments and one literature review with revision. Technical RequirementsMedia player software, such as Quicktime Player, RealOne Player, or Windows Media Player, is required to run the .mp3 files found on this cou Relation of structure and function at various levels of neuronal integration. Topics include: functional neuroanatomy and neurophysiology, sensory and motor systems, centrally programmed behavior, sensory systems, sleep and dreaming, motivation and reward, emotional displays of various types, "higher functions" and the neocortex, and neural processes in learning and memory. In order to improve writing skills in describing experiments and reviewing journal publications in neuroscience, students are required to complete four homework assignments and one literature review with revision. Technical RequirementsMedia player software, such as Quicktime Player, RealOne Player, or Windows Media Player, is required to run the .mp3 files found on this couSubjects

functional neuroanatomy | functional neurophysiology | motor systems | centrally programmed behavior | sensory systems | sleep | dreaming | motivation | reward | emotional displays | higher functions" | neocortex | neural processes in learning and memory | functional neuroanatomy | functional neurophysiology | motor systems | centrally programmed behavior | sensory systems | sleep | dreaming | motivation | reward | emotional displays | higher functions" | neocortex | neural processes in learning and memory | functional neuroanatomy | functional neuroanatomy | functional neurophysiology | functional neurophysiology | motor systems | motor systems | centrally programmed behavior | centrally programmed behavior | sensory systems | sensory systems | sleep | sleep | dreaming | dreaming | motivation | motivation | reward | reward | emotional displays | emotional displays | higher functions | higher functions | neocortex | neocortex | neural processes in learning and memory | neural processes in learning and memory | Neurobehavior | NeurobehaviorLicense

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See all metadata18.112 Functions of a Complex Variable (MIT) 18.112 Functions of a Complex Variable (MIT)

Description

This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted. This course offers biweekly problem sets with solutions, two term tests and a final exam, all with solutions. This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted. This course offers biweekly problem sets with solutions, two term tests and a final exam, all with solutions.Subjects

functions of one complex variable | functions of one complex variable | Cauchy's theorem | Cauchy's theorem | holomorphic functions | holomorphic functions | meromorphic functions | meromorphic functions | residues | residues | contour integrals | contour integrals | conformal mapping | conformal mapping | Infinite series and products | Infinite series and products | the gamma function | the gamma function | the Mittag-Leffler theorem | the Mittag-Leffler theorem | Harmonic functions | Harmonic functions | Dirichlet's problem | Dirichlet's problem | The Riemann mapping theorem | The Riemann mapping theorem | The Riemann Zeta function | The Riemann Zeta functionLicense

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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 metadata8.08 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 | susceptibilitiesLicense

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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This course provides a broad theoretical basis for system identification, estimation, and learning. Students will study least squares estimation and its convergence properties, Kalman filters, noise dynamics and system representation, function approximation theory, neural nets, radial basis functions, wavelets, Volterra expansions, informative data sets, persistent excitation, asymptotic variance, central limit theorems, model structure selection, system order estimate, maximum likelihood, unbiased estimates, Cramer-Rao lower bound, Kullback-Leibler information distance, Akaike's information criterion, experiment design, and model validation. This course provides a broad theoretical basis for system identification, estimation, and learning. Students will study least squares estimation and its convergence properties, Kalman filters, noise dynamics and system representation, function approximation theory, neural nets, radial basis functions, wavelets, Volterra expansions, informative data sets, persistent excitation, asymptotic variance, central limit theorems, model structure selection, system order estimate, maximum likelihood, unbiased estimates, Cramer-Rao lower bound, Kullback-Leibler information distance, Akaike's information criterion, experiment design, and model validation.Subjects

system identification; estimation; least squares estimation; Kalman filter; noise dynamics; system representation; function approximation theory; neural nets; radial basis functions; wavelets; volterra expansions; informative data sets; persistent excitation; asymptotic variance; central limit theorem; model structure selection; system order estimate; maximum likelihood; unbiased estimates; Cramer-Rao lower bound; Kullback-Leibler information distance; Akaike?s information criterion; experiment design; model validation. | system identification; estimation; least squares estimation; Kalman filter; noise dynamics; system representation; function approximation theory; neural nets; radial basis functions; wavelets; volterra expansions; informative data sets; persistent excitation; asymptotic variance; central limit theorem; model structure selection; system order estimate; maximum likelihood; unbiased estimates; Cramer-Rao lower bound; Kullback-Leibler information distance; Akaike?s information criterion; experiment design; model validation. | system identification | system identification | estimation | estimation | least squares estimation | least squares estimation | Kalman filter | Kalman filter | noise dynamics | noise dynamics | system representation | system representation | function approximation theory | function approximation theory | neural nets | neural nets | radial basis functions | radial basis functions | wavelets | wavelets | volterra expansions | volterra expansions | informative data sets | informative data sets | persistent excitation | persistent excitation | asymptotic variance | asymptotic variance | central limit theorem | central limit theorem | model structure selection | model structure selection | system order estimate | system order estimate | maximum likelihood | maximum likelihood | unbiased estimates | unbiased estimates | Cramer-Rao lower bound | Cramer-Rao lower bound | Kullback-Leibler information distance | Kullback-Leibler information distance | Akaike?s information criterion | Akaike?s information criterion | experiment design | experiment design | model validation | model validationLicense

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 metadata18.014 Calculus with Theory I (MIT) 18.014 Calculus with Theory I (MIT)

Description

18.014, Calculus with Theory, covers the same material as 18.01 (Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus. Topics: Axioms for the real numbers; the Riemann integral; limits, theorems on continuous functions; derivatives of functions of one variable; the fundamental theorems of calculus; Taylor's theorem; infinite series, power series, rigorous treatment of the elementary functions. Dr. Lachowska wishes to acknowledge Andrew Brooke-Taylor, Natasha Bershadsky, and Alex Retakh for their help with this course web site. 18.014, Calculus with Theory, covers the same material as 18.01 (Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus. Topics: Axioms for the real numbers; the Riemann integral; limits, theorems on continuous functions; derivatives of functions of one variable; the fundamental theorems of calculus; Taylor's theorem; infinite series, power series, rigorous treatment of the elementary functions. Dr. Lachowska wishes to acknowledge Andrew Brooke-Taylor, Natasha Bershadsky, and Alex Retakh for their help with this course web site.Subjects

axioms for the real numbers | axioms for the real numbers | the Riemann integral | the Riemann integral | limits | limits | theorems on continuous functions | theorems on continuous functions | derivatives of functions of one variablethe fundamental theorems of calculus | derivatives of functions of one variablethe fundamental theorems of calculus | Taylor's theorem | Taylor's theorem | infinite series | infinite series | power series | power series | rigorous treatment of the elementary functions | rigorous treatment of the elementary functionsLicense

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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This is the second term in a two-semester course on statistical mechanics. Basic principles are examined in 8.334, such as the laws of thermodynamics and the concepts of temperature, work, heat, and entropy. Topics from modern statistical mechanics are also explored including the hydrodynamic limit and classical field theories. This is the second term in a two-semester course on statistical mechanics. Basic principles are examined in 8.334, such as the laws of thermodynamics and the concepts of temperature, work, heat, and entropy. Topics from modern statistical mechanics are also explored including the hydrodynamic limit and classical field theories.Subjects

the hydrodynamic limit and classical field theories | the hydrodynamic limit and classical field theories | Phase transitions and broken symmetries: universality | Phase transitions and broken symmetries: universality | correlation functions | and scaling theory | correlation functions | and scaling theory | The renormalization approach to collective phenomena | The renormalization approach to collective phenomena | Dynamic critical behavior | Dynamic critical behavior | Random systems | Random systems | correlation functions | correlation functions | and scaling theory | and scaling theory | Phase transitions and broken symmetries: universality | correlation functions | and scaling theory | Phase transitions and broken symmetries: universality | correlation functions | and scaling theoryLicense

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See all metadata18.100B Analysis I (MIT) 18.100B Analysis I (MIT)

Description

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

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

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See all metadata18.01 Single Variable Calculus (MIT) 18.01 Single Variable Calculus (MIT)

Description

This introductory Calculus course covers differentiation and integration of functions of one variable, with applications. Topics include:Concepts of function, limits, and continuityDifferentiation rules, application to graphing, rates, approximations, and extremum problemsDefinite and indefinite integrationFundamental theorem of calculusApplications of integration to geometry and scienceElementary functionsTechniques of integrationApproximation of definite integrals, improper integrals, and L'Hôpital's rule MATLAB® is a trademark of The MathWorks, Inc. This introductory Calculus course covers differentiation and integration of functions of one variable, with applications. Topics include:Concepts of function, limits, and continuityDifferentiation rules, application to graphing, rates, approximations, and extremum problemsDefinite and indefinite integrationFundamental theorem of calculusApplications of integration to geometry and scienceElementary functionsTechniques of integrationApproximation of definite integrals, improper integrals, and L'Hôpital's rule MATLAB® is a trademark of The MathWorks, Inc.Subjects

differentiation and integration of functions of one variable | differentiation and integration of functions of one variable | limits | limits | continuity | continuity | differentiation rules | differentiation rules | extremum problems | extremum problems | definite and indefinite integration | definite and indefinite integration | fundamental theorem of calculus | fundamental theorem of calculus | elementary | elementary | techniques of integration | techniques of integration | approximation of definite integrals | approximation of definite integrals | improper integrals | improper integrals | l'H?pital's rule | l'H?pital's rule | single variable calculus | single variable calculus | mathematical applications | mathematical applications | function | function | graphing | graphing | rates | rates | approximations | approximations | definite integration | definite integration | indefinite integration | indefinite integration | geometry | geometry | science | science | elementary functions | elementary functions | definite integrals | definite integralsLicense

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See all metadata14.01SC 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 environmenSubjects

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 | PPACALicense

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See all metadata9.01 Neuroscience and Behavior (MIT) 9.01 Neuroscience and Behavior (MIT)

Description

Includes audio/video content: AV lectures. This course covers the relation of structure and function at various levels of neuronal integration. Topics include functional neuroanatomy and neurophysiology, sensory and motor systems, centrally programmed behavior, sensory systems, sleep and dreaming, motivation and reward, emotional displays of various types, "higher functions" and the neocortex, and neural processes in learning and memory. Includes audio/video content: AV lectures. This course covers the relation of structure and function at various levels of neuronal integration. Topics include functional neuroanatomy and neurophysiology, sensory and motor systems, centrally programmed behavior, sensory systems, sleep and dreaming, motivation and reward, emotional displays of various types, "higher functions" and the neocortex, and neural processes in learning and memory.Subjects

functional neuroanatomy | functional neuroanatomy | functional neurophysiology | functional neurophysiology | motor systems | motor systems | centrally programmed behavior | centrally programmed behavior | sensory systems | sensory systems | sleep | sleep | dreaming | dreaming | motivation | motivation | reward | reward | emotional displays | emotional displays | higher functions | higher functions | neocortex | neocortex | neural processes in learning and memory | neural processes in learning and memoryLicense

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 metadata8.08 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 equationsLicense

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See all metadata9.458 Parkinson's Disease Workshop (MIT) 9.458 Parkinson's Disease Workshop (MIT)

Description

Parkinson's disease (PD) is a chronic, progressive, degenerative disease of the brain that produces movement disorders and deficits in executive functions, working memory, visuospatial functions, and internal control of attention. It is named after James Parkinson (1755-1824), the English neurologist who described the first case. This six-week summer workshop explored different aspects of PD, including clinical characteristics, structural neuroimaging, neuropathology, genetics, and cognitive function (mental status, cognitive control processes, working memory, and long-term declarative memory). The workshop did not take up the topics of motor control, nondeclarative memory, or treatment. Parkinson's disease (PD) is a chronic, progressive, degenerative disease of the brain that produces movement disorders and deficits in executive functions, working memory, visuospatial functions, and internal control of attention. It is named after James Parkinson (1755-1824), the English neurologist who described the first case. This six-week summer workshop explored different aspects of PD, including clinical characteristics, structural neuroimaging, neuropathology, genetics, and cognitive function (mental status, cognitive control processes, working memory, and long-term declarative memory). The workshop did not take up the topics of motor control, nondeclarative memory, or treatment.Subjects

Parkinson's disease | Parkinson's disease | chronic progressive degenerative disease | chronic progressive degenerative disease | central nervous system | central nervous system | movement disorders | movement disorders | executive functions | executive functions | working memory | working memory | visuospatial functions | visuospatial functions | internal control of attention | internal control of attention | James Parkinson | James Parkinson | neurologist | neurologist | pathogenic mechanisms | pathogenic mechanisms | positron emission tomography (PET) | positron emission tomography (PET) | structural and functional high-field magnetic resonance imaging (MRI) | structural and functional high-field magnetic resonance imaging (MRI)License

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See all metadata14.01 Principles of Microeconomics (MIT) 14.01 Principles of Microeconomics (MIT)

Description

This introductory course teaches the fundamentals of microeconomics. Topics include consumer theory, producer theory, the behavior of firms, market equilibrium, monopoly, and the role of the government in the economy. 14.01 is a Humanities, Arts, and Social Sciences (HASS) elective and is offered both terms. 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 environmental challenges. This introductory course teaches the fundamentals of microeconomics. Topics include consumer theory, producer theory, the behavior of firms, market equilibrium, monopoly, and the role of the government in the economy. 14.01 is a Humanities, Arts, and Social Sciences (HASS) elective and is offered both terms. 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 environmental challenges.Subjects

market | market | optimization | optimization | allocation | allocation | economic measurement | economic measurement | analysis | analysis | microeconomics | microeconomics | demand | demand | supply | supply | equilibrium | equilibrium | general equilibrium | general equilibrium | government interventions | government interventions | price elasticity of demand | price elasticity of demand | income elasticity of demand | income elasticity of demand | cross price elasticity of demand | cross price elasticity of demand | price elasticity of supply | price elasticity of supply | consumer behavior | consumer behavior | consumer preference | consumer preference | utility functions | utility functions | marginal rate of substitution | marginal rate of substitution | budget constraints | budget constraints | interior solutions | interior solutions | corner solutions | corner solutions | Engle curves | Engle curves | individual demand | individual demand | market demand | market demand | revealed preferences | revealed preferences | substitution effect | substitution effect | income effect | income effect | Giffen goods | Giffen goods | consumer surplus | consumer surplus | Irish potato famine | Irish potato famine | network externalities | network externalities | uncertainty | uncertainty | preference toward risk | preference toward risk | risk premium | risk premium | indifference curves | indifference curves | diversification | diversification | insurance | insurance | producer theory | producer theory | production functions | production functions | short run | short run | long run | long run | returns to scale | returns to scale | cost functions | cost functions | economies of scale | economies of scale | economies of scope | economies of scope | learning | learning | profit maximization | profit maximization | producer surplus | producer surplus | agricultural price support | agricultural price support | tax | tax | subsidy | subsidy | exchange economy | exchange economy | contract curves | contract curves | utility possibilities frontier | utility possibilities frontier | Edgeworth Box | Edgeworth Box | production possibilities frontier | production possibilities frontier | efficiency | efficiency | monopoly | monopoly | multiplant firm | multiplant firm | social cost | social cost | price regulation | price regulation | monopsony | monopsony | price discrimination | price discrimination | peak-load pricing | peak-load pricing | two-part tariffs | two-part tariffs | bundling | bundling | monopolistic competition | monopolistic competition | game theory | game theory | oligopoly | oligopoly | Cournot | Cournot | Stackelberg | Stackelberg | Bertrand | Bertrand | Prisoner's Dilemma | Prisoner's DilemmaLicense

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 metadata18.014 Calculus with Theory (MIT) 18.014 Calculus with Theory (MIT)

Description

18.014, Calculus with Theory, covers the same material as 18.01 (Single Variable Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus. 18.014, Calculus with Theory, covers the same material as 18.01 (Single Variable Calculus), but at a deeper and more rigorous level. It emphasizes careful reasoning and understanding of proofs. The course assumes knowledge of elementary calculus.Subjects

axioms for the real numbers | axioms for the real numbers | the Riemann integral | the Riemann integral | limits | limits | theorems on continuous functions | theorems on continuous functions | derivatives of functions of one variable | derivatives of functions of one variable | the fundamental theorems of calculus | the fundamental theorems of calculus | Taylor's theorem | Taylor's theorem | infinite series | infinite series | power series | power series | rigorous treatment of the elementary functions | rigorous treatment of the elementary functionsLicense

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 metadata18.238 Geometry and Quantum Field Theory (MIT) 18.238 Geometry and Quantum Field Theory (MIT)

Description

Geometry and Quantum Field Theory, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals. It covers the basics of classical field theory, free quantum theories and Feynman diagrams. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and String Theory. Geometry and Quantum Field Theory, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals. It covers the basics of classical field theory, free quantum theories and Feynman diagrams. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and String Theory.Subjects

perturbative quantum field theory | perturbative quantum field theory | classical field theory | classical field theory | free quantum theories | free quantum theories | Feynman diagrams | Feynman diagrams | Renormalization theory | Renormalization theory | Local operators | Local operators | Operator product expansion | Operator product expansion | Renormalization group equation | Renormalization group equation | classical | classical | field | field | theory | theory | Feynman | Feynman | diagrams | diagrams | free | free | quantum | quantum | theories | theories | local | local | operators | operators | product | product | expansion | expansion | perturbative | perturbative | renormalization | renormalization | group | group | equations | equations | functional | functional | function | function | intergrals | intergrals | operator | operator | QFT | QFT | string | string | physics | physics | mathematics | mathematics | geometry | geometry | geometric | geometric | algebraic | algebraic | topology | topology | number | number | 0-dimensional | 0-dimensional | 1-dimensional | 1-dimensional | d-dimensional | d-dimensional | supergeometry | supergeometry | supersymmetry | supersymmetry | conformal | conformal | stationary | stationary | phase | phase | formula | formula | calculus | calculus | combinatorics | combinatorics | matrix | matrix | mechanics | mechanics | lagrangians | lagrangians | hamiltons | hamiltons | least | least | action | action | principle | principle | limits | limits | formalism | formalism | Feynman-Kac | Feynman-Kac | current | current | charges | charges | Noether?s | Noether?s | theorem | theorem | path | path | integral | integral | approach | approach | divergences | divergences | functional integrals | functional integrals | fee quantum theories | fee quantum theories | renormalization theory | renormalization theory | local operators | local operators | operator product expansion | operator product expansion | renormalization group equation | renormalization group equation | mathematical language | mathematical language | string theory | string theory | 0-dimensional QFT | 0-dimensional QFT | Stationary Phase Formula | Stationary Phase Formula | Matrix Models | Matrix Models | Large N Limits | Large N Limits | 1-dimensional QFT | 1-dimensional QFT | Classical Mechanics | Classical Mechanics | Least Action Principle | Least Action Principle | Path Integral Approach | Path Integral Approach | Quantum Mechanics | Quantum Mechanics | Perturbative Expansion using Feynman Diagrams | Perturbative Expansion using Feynman Diagrams | Operator Formalism | Operator Formalism | Feynman-Kac Formula | Feynman-Kac Formula | d-dimensional QFT | d-dimensional QFT | Formalism of Classical Field Theory | Formalism of Classical Field Theory | Currents | Currents | Noether?s Theorem | Noether?s Theorem | Path Integral Approach to QFT | Path Integral Approach to QFT | Perturbative Expansion | Perturbative Expansion | Renormalization Theory | Renormalization Theory | Conformal Field Theory | Conformal Field Theory | algebraic topology | algebraic topology | algebraic geometry | algebraic geometry | number theory | number theoryLicense

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 metadataHeuristic local search tutorial Heuristic local search tutorial

Description

The Problem: Real-world problems are usually (if not always) considered hard to be solved because: * Problems cannot always be represented and solved with a straightforward mathematical approach. * A lot of parameters and constraints are involved. * The number of possible solutions to a problem can be huge. * Good solutions need to be found fast * Checking every possible solution, for finding the best one, is time consuming and sometimes not even feasible. * The quality of a solution may vary according to time, thereby; more than one different solution might be required. Heuristic Search: Heuristic search refers to techniques with the aim of finding ‘good’ solutions for a very hard optimization and decision within a reasonable amount of computation time. Heuristic Local Search: The Problem: Real-world problems are usually (if not always) considered hard to be solved because: * Problems cannot always be represented and solved with a straightforward mathematical approach. * A lot of parameters and constraints are involved. * The number of possible solutions to a problem can be huge. * Good solutions need to be found fast * Checking every possible solution, for finding the best one, is time consuming and sometimes not even feasible. * The quality of a solution may vary according to time, thereby; more than one different solution might be required. Heuristic Search: Heuristic search refers to techniques with the aim of finding ‘good’ solutions for a very hard optimization and decision within a reasonable amount of computation time. Heuristic Local Search:Subjects

UNow | UNow | ukoer | ukoer | combinatorial | combinatorial | objective functions | objective functions | evalutaion | evalutaion | functions | functions | evaluation functions | evaluation functions | search space size | search space size | the knapsack problem | the knapsack problemLicense

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

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See all metadata18.112 Functions of a Complex Variable (MIT)

Description

This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted.This course offers biweekly problem sets with solutions, two term tests and a final exam, all with solutions.Subjects

functions of one complex variable | Cauchy's theorem | holomorphic functions | meromorphic functions | residues | contour integrals | conformal mapping | Infinite series and products | the gamma function | the Mittag-Leffler theorem | Harmonic functions | Dirichlet's problem | The Riemann mapping theorem | The Riemann Zeta functionLicense

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 metadata18.112 Functions of a Complex Variable (MIT)

Description

This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted.This course offers biweekly problem sets with solutions, two term tests and a final exam, all with solutions.Subjects

functions of one complex variable | Cauchy's theorem | holomorphic functions | meromorphic functions | residues | contour integrals | conformal mapping | Infinite series and products | the gamma function | the Mittag-Leffler theorem | Harmonic functions | Dirichlet's problem | The Riemann mapping theorem | The Riemann Zeta functionLicense

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 metadata9.01 Neuroscience and Behavior (MIT)

Description

Relation of structure and function at various levels of neuronal integration. Topics include: functional neuroanatomy and neurophysiology, sensory and motor systems, centrally programmed behavior, sensory systems, sleep and dreaming, motivation and reward, emotional displays of various types, "higher functions" and the neocortex, and neural processes in learning and memory. In order to improve writing skills in describing experiments and reviewing journal publications in neuroscience, students are required to complete four homework assignments and one literature review with revision. Technical RequirementsMedia player software, such as Quicktime Player, RealOne Player, or Windows Media Player, is required to run the .mp3 files found on this couSubjects

functional neuroanatomy | functional neurophysiology | motor systems | centrally programmed behavior | sensory systems | sleep | dreaming | motivation | reward | emotional displays | higher functions" | neocortex | neural processes in learning and memory | functional neuroanatomy | functional neurophysiology | motor systems | centrally programmed behavior | sensory systems | sleep | dreaming | motivation | reward | emotional displays | higher functions | neocortex | neural processes in learning and memory | NeurobehaviorLicense

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 metadata18.112 Functions of a Complex Variable (MIT)

Description

This is an advanced undergraduate course dealing with calculus in one complex variable with geometric emphasis. Since the course Analysis I (18.100B) is a prerequisite, topological notions like compactness, connectedness, and related properties of continuous functions are taken for granted. This course offers biweekly problem sets with solutions, two term tests and a final exam, all with solutions.Subjects

functions of one complex variable | Cauchy's theorem | holomorphic functions | meromorphic functions | residues | contour integrals | conformal mapping | Infinite series and products | the gamma function | the Mittag-Leffler theorem | Harmonic functions | Dirichlet's problem | The Riemann mapping theorem | The Riemann Zeta functionLicense

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

6.241 examines linear, discrete- and continuous-time, and multi-input-output systems in control and related areas. Least squares and matrix perturbation problems are considered. Topics covered include: state-space models, modes, stability, controllability, observability, transfer function matrices, poles and zeros, minimality, internal stability of interconnected systems, feedback compensators, state feedback, optimal regulation, observers, observer-based compensators, measures of control performance, and robustness issues using singular values of transfer functions. Nonlinear systems are also introduced. 6.241 examines linear, discrete- and continuous-time, and multi-input-output systems in control and related areas. Least squares and matrix perturbation problems are considered. Topics covered include: state-space models, modes, stability, controllability, observability, transfer function matrices, poles and zeros, minimality, internal stability of interconnected systems, feedback compensators, state feedback, optimal regulation, observers, observer-based compensators, measures of control performance, and robustness issues using singular values of transfer functions. Nonlinear systems are also introduced.Subjects

control | control | linear | linear | discrete | discrete | continuous-time | continuous-time | multi-input-output | multi-input-output | least squares | least squares | matrix perturbation | matrix perturbation | state-space models | stability | controllability | observability | transfer function matrices | poles | state-space models | stability | controllability | observability | transfer function matrices | poles | zeros | zeros | minimality | minimality | feedback | feedback | compensators | compensators | state feedback | state feedback | optimal regulation | optimal regulation | observers | transfer functions | observers | transfer functions | nonlinear systems | nonlinear systems | linear systems | linear 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 metadata18.022 Calculus (MIT) 18.022 Calculus (MIT)

Description

This is an undergraduate course on calculus of several variables. It covers all of the topics covered in Calculus II (18.02), but presents them in greater depth. These topics are vector algebra in 3-space, determinants, matrices, vector-valued functions of one variable, space motion, scalar functions of several variables, partial differentiation, gradient, optimization techniques, double integrals, line integrals in the plane, exact differentials, conservative fields, Green's theorem, triple integrals, line and surface integrals in space, the divergence theorem, and Stokes' theorem. Additional topics covered in 18.022 are geometry, vector fields, and linear algebra. This is an undergraduate course on calculus of several variables. It covers all of the topics covered in Calculus II (18.02), but presents them in greater depth. These topics are vector algebra in 3-space, determinants, matrices, vector-valued functions of one variable, space motion, scalar functions of several variables, partial differentiation, gradient, optimization techniques, double integrals, line integrals in the plane, exact differentials, conservative fields, Green's theorem, triple integrals, line and surface integrals in space, the divergence theorem, and Stokes' theorem. Additional topics covered in 18.022 are geometry, vector fields, and linear algebra.Subjects

vector algebra | vector algebra | determinant | determinant | matrix | matrix | matrices | matrices | vector-valued | vector-valued | functions | functions | space motion | space motion | scalar functions | scalar functions | partial differentiation | partial differentiation | gradient | gradient | optimization techniques | optimization techniques | double integrals | double integrals | line integrals | line integrals | exact differentials | exact differentials | conservative fields | conservative fields | Green's theorem | Green's theorem | triple integrals | triple integrals | surface integrals | surface integrals | divergence theorem | divergence theorem | Stokes' theorem | Stokes' theorem | geometry | geometry | vector fields | vector fields | linear algebra | linear algebraLicense

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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