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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 function

License

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

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9.01 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 cou

Subjects

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 | Neurobehavior

License

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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 function

License

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

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18.112 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 function

License

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

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18.014 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 functions

License

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

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11.437 Financing Economic Development (MIT) 11.437 Financing Economic Development (MIT)

Description

This course focuses on financing tools and program models to support local economic development. It includes an overview of private capital markets and financing sources to understand capital market imperfections that constrain economic development; business accounting; financial statement analysis; federal economic development programs; and public finance tools. Program models covered include revolving loan funds, guarantee programs, venture capital funds, bank holding companies, community development loan funds and credit unions, micro enterprise funds, and the use of the Community Reinvestment Act to leverage bank financing.Technical RequirementsMicrosoft® Excel software is recommended for viewing the .xls files found on this course site. Free Microsoft® Excel This course focuses on financing tools and program models to support local economic development. It includes an overview of private capital markets and financing sources to understand capital market imperfections that constrain economic development; business accounting; financial statement analysis; federal economic development programs; and public finance tools. Program models covered include revolving loan funds, guarantee programs, venture capital funds, bank holding companies, community development loan funds and credit unions, micro enterprise funds, and the use of the Community Reinvestment Act to leverage bank financing.Technical RequirementsMicrosoft® Excel software is recommended for viewing the .xls files found on this course site. Free Microsoft® Excel

Subjects

financing tools | financing tools | program models to support local economic development | program models to support local economic development | private capital markets | private capital markets | financing sources | financing sources | capital market imperfections | capital market imperfections | economic development | economic development | business accounting | business accounting | financial statement analysis | financial statement analysis | federal economic development programs | federal economic development programs | public finance tools | public finance tools | funds | funds | guarantee programs | guarantee programs | venture capital funds | venture capital funds | bank holding companies | bank holding companies | community development loan funds | community development loan funds | credit unions | credit unions | micro enterprise funds | micro enterprise funds | use of the Community Reinvestment Act to leverage bank financing | use of the Community Reinvestment Act to leverage bank financing | Community Reinvestment Act | Community Reinvestment Act | bank financing | bank financing | program management | program management

License

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

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11.437 Financing Economic Development (MIT) 11.437 Financing Economic Development (MIT)

Description

This course focuses on financing tools and program models to support local economic development. It includes an overview of private capital markets and financing sources to understand capital market imperfections that constrain economic development; business accounting; financial statement analysis; federal economic development programs; and public finance tools. Program models covered include revolving loan funds, guarantee programs, venture capital funds, bank holding companies, community development loan funds and credit unions, micro enterprise funds, and the use of the Community Reinvestment Act to leverage bank financing. This course focuses on financing tools and program models to support local economic development. It includes an overview of private capital markets and financing sources to understand capital market imperfections that constrain economic development; business accounting; financial statement analysis; federal economic development programs; and public finance tools. Program models covered include revolving loan funds, guarantee programs, venture capital funds, bank holding companies, community development loan funds and credit unions, micro enterprise funds, and the use of the Community Reinvestment Act to leverage bank financing.

Subjects

financing tools | financing tools | program models to support local economic development | program models to support local economic development | private capital markets | private capital markets | financing sources | financing sources | capital market imperfections | capital market imperfections | economic development | economic development | business accounting | business accounting | financial statement analysis | financial statement analysis | federal economic development programs | federal economic development programs | public finance tools | public finance tools | funds | funds | guarantee programs | guarantee programs | venture capital funds | venture capital funds | bank holding companies | bank holding companies | community development loan funds | community development loan funds | credit unions | credit unions | micro enterprise funds | micro enterprise funds | use of the Community Reinvestment Act to leverage bank financing | use of the Community Reinvestment Act to leverage bank financing

License

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

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Exposição do Mundo Português (1940), Lisboa, Portugal Exposição do Mundo Português (1940), Lisboa, Portugal

Description

Subjects

fundaçãocaloustegulbenkian | fundaçãocaloustegulbenkian | gulbenkian | gulbenkian | bibliotecadearte | bibliotecadearte | biblioteca | biblioteca | arte | arte | márionovais | márionovais | novais | novais | exposiçãodomundoportuguês | exposiçãodomundoportuguês | exposição | exposição | mundo | mundo | português | português | 1940 | 1940 | passagem | passagem | portadafundação | portadafundação | porta | porta | fundação | fundação | ligação | ligação | pavilhãodafundação | pavilhãodafundação | pavilhão | pavilhão | formaçãoeconquista | formaçãoeconquista | formação | formação | conquista | conquista | arquitectura | arquitectura | cottinellitelmo | cottinellitelmo | cottinelli | cottinelli | telmo | telmo | raulrodrigueslima | raulrodrigueslima | raul | raul | rodrigues | rodrigues | lima | lima | lisboa | lisboa | belém | belém

License

No known copyright restrictions

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18.01 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 integrals

License

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18.014 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 functions

License

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

Description

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

Subjects

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

License

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2.160 Identification, Estimation, and Learning (MIT) 2.160 Identification, Estimation, and Learning (MIT)

Description

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 validation

License

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

Description

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

Subjects

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

License

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

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18.S996 Category Theory for Scientists (MIT) 18.S996 Category Theory for Scientists (MIT)

Description

The goal of this class is to prove that category theory is a powerful language for understanding and formalizing common scientific models. The power of the language will be tested by its ability to penetrate into taken-for-granted ideas, either by exposing existing weaknesses or flaws in our understanding, or by highlighting hidden commonalities across scientific fields. The goal of this class is to prove that category theory is a powerful language for understanding and formalizing common scientific models. The power of the language will be tested by its ability to penetrate into taken-for-granted ideas, either by exposing existing weaknesses or flaws in our understanding, or by highlighting hidden commonalities across scientific fields.

Subjects

Sets | Sets | functions | functions | commutative diagrams | commutative diagrams | products | products | coproducts | coproducts | finite limits | finite limits | monoids | monoids | groups | groups | graphs | graphs | orders | orders | schemas | schemas | instances | instances | databases | databases | categories | categories | functors | functors | mathematics | mathematics | natural transformations | natural transformations | limits | limits | colimits | colimits | adjoint functors | adjoint functors | monads | monads | operads | operads | isomorphism | isomorphism | molecular dynamics | molecular dynamics | olog | olog

License

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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.

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 function

License

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

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14.01 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 Dilemma

License

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

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Exposição do Mundo Português (1940), Lisboa, Portugal Exposição do Mundo Português (1940), Lisboa, Portugal

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Subjects

fundaçãocaloustegulbenkian | fundaçãocaloustegulbenkian | gulbenkian | gulbenkian | bibliotecadearte | bibliotecadearte | biblioteca | biblioteca | arte | arte | márionovais | márionovais | novais | novais | exposiçãodomundoportuguês | exposiçãodomundoportuguês | exposição | exposição | mundoportuguês | mundoportuguês | 1940 | 1940 | pavilhãodafundação | pavilhãodafundação | pavilhão | pavilhão | fundação | fundação | saladosdocumentos | saladosdocumentos | documentos | documentos | carta | carta | doação | doação | infantedafonsohenriques | infantedafonsohenriques | infante | infante | dafonsohenriques | dafonsohenriques | henriques | henriques | egasramires | egasramires | egas | egas | ramires | ramires | igreja | igreja | sãobartolomeudecampelo | sãobartolomeudecampelo | sãobartolomeu | sãobartolomeu | campelo | campelo | arquitectura | arquitectura | raulrodrigueslima | raulrodrigueslima | lisboa | lisboa | belém | belém

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Exposição do Mundo Português (1940), Lisboa, Portugal Exposição do Mundo Português (1940), Lisboa, Portugal

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Subjects

fundaçãocaloustegulbenkian | fundaçãocaloustegulbenkian | gulbenkian | gulbenkian | bibliotecadearte | bibliotecadearte | biblioteca | biblioteca | arte | arte | márionovais | márionovais | novais | novais | exposiçãodomundoportuguês | exposiçãodomundoportuguês | exposição | exposição | mundoportuguês | mundoportuguês | 1940 | 1940 | pavilhãodafundação | pavilhãodafundação | pavilhão | pavilhão | fundação | fundação | saladosdocumentos | saladosdocumentos | documentos | documentos | carta | carta | doação | doação | infantedafonsohenriques | infantedafonsohenriques | infante | infante | dafonsohenriques | dafonsohenriques | henriques | henriques | egasramires | egasramires | egas | egas | ramires | ramires | igreja | igreja | sãobartolomeudecampelo | sãobartolomeudecampelo | sãobartolomeu | sãobartolomeu | campelo | campelo | arquitectura | arquitectura | raulrodrigueslima | raulrodrigueslima | lisboa | lisboa | belém | belém

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Exposição do Mundo Português (1940), Lisboa, Portugal Exposição do Mundo Português (1940), Lisboa, Portugal

Description

Subjects

fundaçãocaloustegulbenkian | fundaçãocaloustegulbenkian | gulbenkian | gulbenkian | bibliotecadearte | bibliotecadearte | biblioteca | biblioteca | arte | arte | márionovais | márionovais | novais | novais | exposiçãodomundoportuguês | exposiçãodomundoportuguês | exposição | exposição | mundoportuguês | mundoportuguês | 1940 | 1940 | pavilhãodafundação | pavilhãodafundação | pavilhão | pavilhão | fundação | fundação | saladdinis | saladdinis | ddinis | ddinis | trovador | trovador | lavrador | lavrador | arquitectura | arquitectura | raulrodrigueslima | raulrodrigueslima | lisboa | lisboa | belém | belém

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Parnell's Grave, Glasnevin, Co. Dublin Parnell's Grave, Glasnevin, Co. Dublin

Description

Subjects

ireland | ireland | grave | grave | 19thcentury | 19thcentury | funeral | funeral | parnell | parnell | nationalist | nationalist | glassnegative | glassnegative | glasnevin | glasnevin | leinster | leinster | codublin | codublin | politicalleader | politicalleader | glasnevincemetery | glasnevincemetery | charlesstewartparnell | charlesstewartparnell | robertfrench | robertfrench | williamlawrence | williamlawrence | nationallibraryofireland | nationallibraryofireland | tolkariver | tolkariver | lawrencecollection | lawrencecollection | katherineoshea | katherineoshea | cholerapit | cholerapit | irishparliamentaryparty | irishparliamentaryparty | landreformagitator | landreformagitator | largefuneral | largefuneral | dublinleinster | dublinleinster | lawrencephotographicstudio | lawrencephotographicstudio | thelawrencephotographcollection | thelawrencephotographcollection | cathalstiúbhardpharnell | cathalstiúbhardpharnell | 27june1846 | 27june1846 | 6october1891 | 6october1891 | irishlandlord | irishlandlord | anniversaryprocessions | anniversaryprocessions | handelsfuneralmarch | handelsfuneralmarch | divorcecrisis | divorcecrisis

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Exposição do Mundo Português (1940), Lisboa, Portugal Exposição do Mundo Português (1940), Lisboa, Portugal

Description

Subjects

fundaçãocaloustegulbenkian | fundaçãocaloustegulbenkian | gulbenkian | gulbenkian | bibliotecadearte | bibliotecadearte | biblioteca | biblioteca | arte | arte | márionovais | márionovais | novais | novais | exposiçãodomundoportuguês | exposiçãodomundoportuguês | exposição | exposição | mundoportuguês | mundoportuguês | 1940 | 1940 | pavilhãodafundação | pavilhãodafundação | pavilhão | pavilhão | fundação | fundação | saladdinis | saladdinis | ddinis | ddinis | trovador | trovador | lavrador | lavrador | arquitectura | arquitectura | raulrodrigueslima | raulrodrigueslima | lisboa | lisboa | belém | belém

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9.01 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 memory

License

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

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

Description

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

Subjects

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

License

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

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18.112 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 function

License

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

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Exposição do Mundo Português (1940), Lisboa, Portugal Exposição do Mundo Português (1940), Lisboa, Portugal

Description

Subjects

fundaçãocaloustegulbenkian | fundaçãocaloustegulbenkian | gulbenkian | gulbenkian | bibliotecadearte | bibliotecadearte | biblioteca | biblioteca | arte | arte | márionovais | márionovais | novais | novais | exposiçãodomundoportuguês | exposiçãodomundoportuguês | exposição | exposição | mundoportuguês | mundoportuguês | 1940 | 1940 | pavilhãodafundação | pavilhãodafundação | pavilhão | pavilhão | fundação | fundação | saladafonsohenriques | saladafonsohenriques | dafonsohenriques | dafonsohenriques | djoãopeculiar | djoãopeculiar | gualdimpais | gualdimpais | arquitectura | arquitectura | raulrodrigueslima | raulrodrigueslima | lisboa | lisboa | belém | belém

License

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