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18.969 Topics in Geometry: Dirac Geometry (MIT) 18.969 Topics in Geometry: Dirac Geometry (MIT)

Description

This is an introductory (i.e. first year graduate students are welcome and expected) course in generalized geometry, with a special emphasis on Dirac geometry, as developed by Courant, Weinstein, and Severa, as well as generalized complex geometry, as introduced by Hitchin. Dirac geometry is based on the idea of unifying the geometry of a Poisson structure with that of a closed 2-form, whereas generalized complex geometry unifies complex and symplectic geometry. For this reason, the latter is intimately related to the ideas of mirror symmetry. This is an introductory (i.e. first year graduate students are welcome and expected) course in generalized geometry, with a special emphasis on Dirac geometry, as developed by Courant, Weinstein, and Severa, as well as generalized complex geometry, as introduced by Hitchin. Dirac geometry is based on the idea of unifying the geometry of a Poisson structure with that of a closed 2-form, whereas generalized complex geometry unifies complex and symplectic geometry. For this reason, the latter is intimately related to the ideas of mirror symmetry.

Subjects

generalized geometry | generalized geometry | Dirac geometry | Dirac geometry | Gerbes | Gerbes | B-fields | B-fields | Courant algebroids | Courant algebroids | sigma models | sigma models | baby String theory | baby String theory | linear algebra | linear algebra | pure spinors | pure spinors | Riemannian structures | Riemannian structures | Hodge star | Hodge star | integrability | integrability | Dirac structures | Dirac structures | Lie algebroids and bialgebroids | Lie algebroids and bialgebroids | holomorphic bundles | holomorphic bundles | Picard group | Picard group | Kodaira-Spencer-Kuranishi deformation theory | Kodaira-Spencer-Kuranishi deformation theory | Kahler geometry | Kahler geometry | Hermitian geometry | Hermitian geometry | Calabi-Yau structures | Calabi-Yau structures | D-branes | D-branes

License

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18.950 Differential Geometry (MIT) 18.950 Differential Geometry (MIT)

Description

This course is an introduction to differential geometry. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature. This course is an introduction to differential geometry. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature.

Subjects

differential geometry | differential geometry | geometry of plane curves | geometry of plane curves | hypersurfaces | hypersurfaces | geometry of lengths and distances | geometry of lengths and distances

License

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18.966 Geometry of Manifolds (MIT) 18.966 Geometry of Manifolds (MIT)

Description

This is a second-semester graduate course on the geometry of manifolds. The main emphasis is on the geometry of symplectic manifolds, but the material also includes long digressions into complex geometry and the geometry of 4-manifolds, with special emphasis on topological considerations. This is a second-semester graduate course on the geometry of manifolds. The main emphasis is on the geometry of symplectic manifolds, but the material also includes long digressions into complex geometry and the geometry of 4-manifolds, with special emphasis on topological considerations.

Subjects

Differential forms | Differential forms | Lie groups | Lie groups | DeRham | DeRham | Riemannian manifolds | Riemannian manifolds | curvature | curvature | Hodge | Hodge | Hodge theory | Hodge theory | manifolds | manifolds | Riemannian geometry | Riemannian geometry | holonomy | holonomy | symplectic geometry | symplectic geometry | complex geometry | complex geometry | Hodge-Kahler theory | Hodge-Kahler theory | smooth manifold topology | smooth manifold topology

License

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18.238 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 theory

License

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3.012 Fundamentals of Materials Science (MIT) 3.012 Fundamentals of Materials Science (MIT)

Description

This course focuses on the fundamentals of structure, energetics, and bonding that underpin materials science. It is the introductory lecture class for sophomore students in Materials Science and Engineering, taken with 3.014 and 3.016 to create a unified introduction to the subject. Topics include: an introduction to thermodynamic functions and laws governing equilibrium properties, relating macroscopic behavior to atomistic and molecular models of materials; the role of electronic bonding in determining the energy, structure, and stability of materials; quantum mechanical descriptions of interacting electrons and atoms; materials phenomena, such as heat capacities, phase transformations, and multiphase equilibria to chemical reactions and magnetism; symmetry properties of molecules and s This course focuses on the fundamentals of structure, energetics, and bonding that underpin materials science. It is the introductory lecture class for sophomore students in Materials Science and Engineering, taken with 3.014 and 3.016 to create a unified introduction to the subject. Topics include: an introduction to thermodynamic functions and laws governing equilibrium properties, relating macroscopic behavior to atomistic and molecular models of materials; the role of electronic bonding in determining the energy, structure, and stability of materials; quantum mechanical descriptions of interacting electrons and atoms; materials phenomena, such as heat capacities, phase transformations, and multiphase equilibria to chemical reactions and magnetism; symmetry properties of molecules and s

Subjects

bonding | bonding | energetics | energetics | structure | structure | antibonding | antibonding | hydrogen | hydrogen | Quantum mechanics | Quantum mechanics | electron | electron | atom | atom | molecule | molecule | molecular dynamics | molecular dynamics | MD | MD | Symmetry properties | Symmetry properties | solid | solid | gas | gas | liquid | liquid | phase | phase | matter; molecular geometry | matter; molecular geometry | complex and disordered materials | complex and disordered materials | thermodynamics | thermodynamics | equilibrium property | equilibrium property | macroscopic behavior | macroscopic behavior | molecular model | molecular model | heat capacity | heat capacity | phase transformation | phase transformation | multiphase equilibria | multiphase equilibria | chemical reaction | chemical reaction | magnetism | magnetism | engineered alloy | engineered alloy | electronic and magnetic material | electronic and magnetic material | ionic solid | ionic solid | network solid | network solid | polymer | polymer | biomaterial | biomaterial | glass | glass | liquid crystal | liquid crystal | LCD | LCD | matter | matter | molecular geometry | molecular geometry

License

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18.319 Geometric Combinatorics (MIT) 18.319 Geometric Combinatorics (MIT)

Description

This course offers an introduction to discrete and computational geometry. Emphasis is placed on teaching methods in combinatorial geometry. Many results presented are recent, and include open (as yet unsolved) problems. This course offers an introduction to discrete and computational geometry. Emphasis is placed on teaching methods in combinatorial geometry. Many results presented are recent, and include open (as yet unsolved) problems.

Subjects

discrete geometry | discrete geometry | computational geometry | computational geometry | convex partitions | convex partitions | binary space partitions | binary space partitions | art gallery problems | art gallery problems | Planar graphs | Planar graphs | pseudo-triangulations | pseudo-triangulations | encompassing graphs | encompassing graphs | geometric graphs | geometric graphs | crossing numbers | crossing numbers | extremal graph theory | extremal graph theory | Gallai-Sylvester problems | Gallai-Sylvester problems

License

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2.700 Principles of Naval Architecture (MIT) 2.700 Principles of Naval Architecture (MIT)

Description

This course presents principles of naval architecture, ship geometry, hydrostatics, calculation and drawing of curves of form, intact and damage stability, hull structure strength calculations and ship resistance. It introduces computer-aided naval ship design and analysis tools. Projects include analysis of ship lines drawings, calculation of ship hydrostatic characteristics, analysis of intact and damaged stability, ship model testing, and hull structure strength calculations. This course presents principles of naval architecture, ship geometry, hydrostatics, calculation and drawing of curves of form, intact and damage stability, hull structure strength calculations and ship resistance. It introduces computer-aided naval ship design and analysis tools. Projects include analysis of ship lines drawings, calculation of ship hydrostatic characteristics, analysis of intact and damaged stability, ship model testing, and hull structure strength calculations.

Subjects

naval architecture | naval architecture | ship geometry | ship geometry | geometry of ships | geometry of ships | ship resistance | ship resistance | flow | flow | hydrostatics | hydrostatics | intact stability | intact stability | damage stability | damage stability | general stability | general stability | hull | hull | hydrostatic | hydrostatic | ship model testing | ship model testing | hull structure | hull structure | Resistance | Resistance | Propulsion | Propulsion | Vibration | Vibration | submarine | submarine | hull subdivision | hull subdivision | midsection | midsection

License

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18.969 Topics in Geometry: Dirac Geometry (MIT)

Description

This is an introductory (i.e. first year graduate students are welcome and expected) course in generalized geometry, with a special emphasis on Dirac geometry, as developed by Courant, Weinstein, and Severa, as well as generalized complex geometry, as introduced by Hitchin. Dirac geometry is based on the idea of unifying the geometry of a Poisson structure with that of a closed 2-form, whereas generalized complex geometry unifies complex and symplectic geometry. For this reason, the latter is intimately related to the ideas of mirror symmetry.

Subjects

generalized geometry | Dirac geometry | Gerbes | B-fields | Courant algebroids | sigma models | baby String theory | linear algebra | pure spinors | Riemannian structures | Hodge star | integrability | Dirac structures | Lie algebroids and bialgebroids | holomorphic bundles | Picard group | Kodaira-Spencer-Kuranishi deformation theory | Kahler geometry | Hermitian geometry | Calabi-Yau structures | D-branes

License

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

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18.950 Differential Geometry (MIT)

Description

This course is an introduction to differential geometry. The course itself is mathematically rigorous, but still emphasizes concrete aspects of geometry, centered on the notion of curvature.

Subjects

differential geometry | geometry of plane curves | hypersurfaces | geometry of lengths and distances

License

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

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18.966 Geometry of Manifolds (MIT)

Description

This is a second-semester graduate course on the geometry of manifolds. The main emphasis is on the geometry of symplectic manifolds, but the material also includes long digressions into complex geometry and the geometry of 4-manifolds, with special emphasis on topological considerations.

Subjects

Differential forms | Lie groups | DeRham | Riemannian manifolds | curvature | Hodge | Hodge theory | manifolds | Riemannian geometry | holonomy | symplectic geometry | complex geometry | Hodge-Kahler theory | smooth manifold topology

License

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

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18.238 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.

Subjects

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

License

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

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18.022 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 algebra

License

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15.083J Integer Programming and Combinatorial Optimization (MIT) 15.083J Integer Programming and Combinatorial Optimization (MIT)

Description

The course is a comprehensive introduction to the theory, algorithms and applications of integer optimization and is organized in four parts: formulations and relaxations, algebra and geometry of integer optimization, algorithms for integer optimization, and extensions of integer optimization. The course is a comprehensive introduction to the theory, algorithms and applications of integer optimization and is organized in four parts: formulations and relaxations, algebra and geometry of integer optimization, algorithms for integer optimization, and extensions of integer optimization.

Subjects

theory | theory | algorithms | algorithms | integer optimization | integer optimization | formulations and relaxations | formulations and relaxations | algebra and geometry of integer optimization | algebra and geometry of integer optimization | algorithms for integer optimization | algorithms for integer optimization | extensions of integer optimization | extensions of integer optimization | 15.083 | 15.083

License

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18.06 Linear Algebra (MIT) 18.06 Linear Algebra (MIT)

Description

This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices. This is a basic subject on matrix theory and linear algebra. Emphasis is given to topics that will be useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices.

Subjects

Generalized spaces | Generalized spaces | Linear algebra | Linear algebra | Algebra | Universal | Algebra | Universal | Mathematical analysis | Mathematical analysis | Calculus of operations | Calculus of operations | Line geometry | Line geometry | Topology | Topology | matrix theory | matrix theory | systems of equations | systems of equations | vector spaces | vector spaces | systems determinants | systems determinants | eigen values | eigen values | positive definite matrices | positive definite matrices | Markov processes | Markov processes | Fourier transforms | Fourier transforms | differential equations | differential equations | linear algebra | linear algebra | determinants | determinants | eigenvalues | eigenvalues | similarity | similarity | least-squares approximations | least-squares approximations | stability of differential equations | stability of differential equations | networks | networks

License

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18.409 Topics in Theoretical Computer Science: An Algorithmist's Toolkit (MIT) 18.409 Topics in Theoretical Computer Science: An Algorithmist's Toolkit (MIT)

Description

This course covers a collection of geometric techniques that apply broadly in modern algorithm design. This course covers a collection of geometric techniques that apply broadly in modern algorithm design.

Subjects

Spectral graph theory | Spectral graph theory | Iterative methods for linear algebra | Iterative methods for linear algebra | Convex geometry | Convex geometry | Lattices and basis reduction | Lattices and basis reduction | LPs and SDPs for approximating NP-hard problems | LPs and SDPs for approximating NP-hard problems | Graph Laplacians | Graph Laplacians | Cheeger inequalities | Cheeger inequalities | Fritz John?s theorem | Fritz John?s theorem

License

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SP.2H3 Ancient Philosophy and Mathematics (MIT) SP.2H3 Ancient Philosophy and Mathematics (MIT)

Description

Western philosophy and theoretical mathematics were born together, and the cross-fertilization of ideas in the two disciplines was continuously acknowledged throughout antiquity. In this course, we read works of ancient Greek philosophy and mathematics, and investigate the way in which ideas of definition, reason, argument and proof, rationality and irrationality, number, quality and quantity, truth, and even the idea of an idea were shaped by the interplay of philosophic and mathematical inquiry. Western philosophy and theoretical mathematics were born together, and the cross-fertilization of ideas in the two disciplines was continuously acknowledged throughout antiquity. In this course, we read works of ancient Greek philosophy and mathematics, and investigate the way in which ideas of definition, reason, argument and proof, rationality and irrationality, number, quality and quantity, truth, and even the idea of an idea were shaped by the interplay of philosophic and mathematical inquiry.

Subjects

mathematics | mathematics | geometry | geometry | history | history | philosophy | philosophy | Greek philosophy | Greek philosophy | Plato | Plato | Euclid | Euclid | Aristotle | Aristotle | Rene Descartes | Rene Descartes | Nicomachus | Nicomachus | Francis Bacon | Francis Bacon

License

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6.046J Introduction to Algorithms (SMA 5503) (MIT) 6.046J Introduction to Algorithms (SMA 5503) (MIT)

Description

This course teaches techniques for the design and analysis of efficient algorithms, emphasizing methods useful in practice. Topics covered include: sorting; search trees, heaps, and hashing; divide-and-conquer; dynamic programming; amortized analysis; graph algorithms; shortest paths; network flow; computational geometry; number-theoretic algorithms; polynomial and matrix calculations; caching; and parallel computing.This course was also taught as part of the Singapore-MIT Alliance (SMA) programme as course number SMA 5503 (Analysis and Design of Algorithms). This course teaches techniques for the design and analysis of efficient algorithms, emphasizing methods useful in practice. Topics covered include: sorting; search trees, heaps, and hashing; divide-and-conquer; dynamic programming; amortized analysis; graph algorithms; shortest paths; network flow; computational geometry; number-theoretic algorithms; polynomial and matrix calculations; caching; and parallel computing.This course was also taught as part of the Singapore-MIT Alliance (SMA) programme as course number SMA 5503 (Analysis and Design of Algorithms).

Subjects

algorithms | algorithms | efficient algorithms | efficient algorithms | sorting | sorting | search trees | search trees | heaps | heaps | hashing | hashing | divide-and-conquer | divide-and-conquer | dynamic programming | dynamic programming | amortized analysis | amortized analysis | graph algorithms | graph algorithms | shortest paths | shortest paths | network flow | network flow | computational geometry | computational geometry | number-theoretic algorithms | number-theoretic algorithms | polynomial and matrix calculations | polynomial and matrix calculations | caching | caching | parallel computing | parallel computing | SMA 5503 | SMA 5503 | 6.046 | 6.046

License

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13.400 Introduction to Naval Architecture (MIT) 13.400 Introduction to Naval Architecture (MIT)

Description

This course is an introduction to principles of naval architecture, ship geometry, hydrostatics, calculation and drawing of curves of form. It also explores concepts of  intact and damaged stability, hull structure strength calculations and ship resistance. Projects include analysis of ship lines drawings and ship model testing. This course is an introduction to principles of naval architecture, ship geometry, hydrostatics, calculation and drawing of curves of form. It also explores concepts of  intact and damaged stability, hull structure strength calculations and ship resistance. Projects include analysis of ship lines drawings and ship model testing.

Subjects

elementary principles of Naval Architecture | elementary principles of Naval Architecture | naval architecture tools | naval architecture tools | ship geometry | ship geometry | hydrostatics | hydrostatics | calculation | calculation | drawing | drawing | curves of form | curves of form | intact and damaged stability | intact and damaged stability | hull structure strength calculations | hull structure strength calculations | ship resistance | ship resistance | ship model testing | ship model testing | 2.701 | 2.701

License

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6.854J Advanced Algorithms (MIT) 6.854J Advanced Algorithms (MIT)

Description

6.854J is a first-year graduate course in algorithms, continuing where 6.046J left off. The course emphasizes fundamental algorithms and advanced methods of algorithmic design, analysis, and implementation. Topics include: data structures, network flows, linear programming, computational geometry, approximation algorithms. 6.854J is a first-year graduate course in algorithms, continuing where 6.046J left off. The course emphasizes fundamental algorithms and advanced methods of algorithmic design, analysis, and implementation. Topics include: data structures, network flows, linear programming, computational geometry, approximation algorithms.

Subjects

algorithm design and analysis | algorithm design and analysis | algorithms | algorithms | fundamental algorithms | fundamental algorithms | advanced methods of algorithmic design | advanced methods of algorithmic design | analysis | analysis | implementation | implementation | data structures | data structures | network flows | network flows | linear programming | linear programming | computational geometry | computational geometry | approximation algorithms | approximation algorithms | algorithmic design | algorithmic design | algorithmic analysis | algorithmic analysis | string algorithms | string algorithms | maximum flows | maximum flows | online algorithms | online algorithms | scheduling | scheduling | external memory algorithms | external memory algorithms | 6.854 | 6.854 | 18.415 | 18.415

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8.962 General Relativity (MIT) 8.962 General Relativity (MIT)

Description

This course covers the basic principles of Einstein's general theory of relativity. Also discussed are differential geometry, gravitomagnetism, gravitational radiation, experimental tests of general relativity, black holes, and cosmology. This course covers the basic principles of Einstein's general theory of relativity. Also discussed are differential geometry, gravitomagnetism, gravitational radiation, experimental tests of general relativity, black holes, and cosmology.

Subjects

Einstein's general theory of relativity | Einstein's general theory of relativity | Einstein | Einstein | relativity | relativity | differential geometry | differential geometry | general relativity | general relativity | black holes | black holes | cosmology | cosmology | Hamiltonian Dynamics | Hamiltonian Dynamics | Curvature | Curvature | Acceleration | Acceleration | Hilbert action | Hilbert action | Orthonormal bases | Orthonormal bases | White dwarfs | White dwarfs | neutron stars | neutron stars | Kruskal coordinates | Kruskal coordinates | Wormholes | Wormholes | Hawking radiation | Hawking radiation | Kerr solution | Kerr solution

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6.046J Introduction to Algorithms (MIT) 6.046J Introduction to Algorithms (MIT)

Description

This course teaches techniques for the design and analysis of efficient algorithms, emphasizing methods useful in practice. Topics covered include: sorting; search trees, heaps, and hashing; divide-and-conquer; dynamic programming; amortized analysis; graph algorithms; shortest paths; network flow; computational geometry; number-theoretic algorithms; polynomial and matrix calculations; caching; and parallel computing. This course teaches techniques for the design and analysis of efficient algorithms, emphasizing methods useful in practice. Topics covered include: sorting; search trees, heaps, and hashing; divide-and-conquer; dynamic programming; amortized analysis; graph algorithms; shortest paths; network flow; computational geometry; number-theoretic algorithms; polynomial and matrix calculations; caching; and parallel computing.

Subjects

algorithms | algorithms | efficient algorithms | efficient algorithms | sorting | sorting | search trees | search trees | heaps | heaps | hashing | hashing | divide-and-conquer | divide-and-conquer | dynamic programming | dynamic programming | amortized analysis | amortized analysis | graph algorithms | graph algorithms | shortest paths | shortest paths | network flow | network flow | computational geometry | computational geometry | number-theoretic algorithms | number-theoretic algorithms | polynomial and matrix calculations | polynomial and matrix calculations | caching | caching | parallel computing | parallel computing | 6.046 | 6.046 | 18.410 | 18.410

License

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13.472J Computational Geometry (MIT) 13.472J Computational Geometry (MIT)

Description

Topics in surface modeling: b-splines, non-uniform rational b-splines, physically based deformable surfaces, sweeps and generalized cylinders, offsets, blending and filleting surfaces. Non-linear solvers and intersection problems. Solid modeling: constructive solid geometry, boundary representation, non-manifold and mixed-dimension boundary representation models, octrees. Robustness of geometric computations. Interval methods. Finite and boundary element discretization methods for continuum mechanics problems. Scientific visualization. Variational geometry. Tolerances. Inspection methods. Feature representation and recognition. Shape interrogation for design, analysis, and manufacturing. Involves analytical and programming assignments. Topics in surface modeling: b-splines, non-uniform rational b-splines, physically based deformable surfaces, sweeps and generalized cylinders, offsets, blending and filleting surfaces. Non-linear solvers and intersection problems. Solid modeling: constructive solid geometry, boundary representation, non-manifold and mixed-dimension boundary representation models, octrees. Robustness of geometric computations. Interval methods. Finite and boundary element discretization methods for continuum mechanics problems. Scientific visualization. Variational geometry. Tolerances. Inspection methods. Feature representation and recognition. Shape interrogation for design, analysis, and manufacturing. Involves analytical and programming assignments.

Subjects

surface modeling | surface modeling | b-splines | b-splines | deformable surfaces | deformable surfaces | generalized cylinders | generalized cylinders | offsets | offsets | filleting surfaces | filleting surfaces | Non-linear solvers and intersection problems | Non-linear solvers and intersection problems | Solid modeling | Solid modeling | boundary representation | boundary representation | non-manifold and mixed-dimension boundary representation models | non-manifold and mixed-dimension boundary representation models | octrees | octrees | Interval methods | Interval methods | discretization methods | discretization methods | Scientific visualization | Scientific visualization | Variational geometry | Variational geometry | Tolerances | Tolerances | Inspection methods | Inspection methods | Shape interrogation | Shape interrogation | 2.158J | 2.158J | 1.128J | 1.128J | 16.940J | 16.940J | 13.472 | 13.472 | 2.158 | 2.158 | 1.128 | 1.128 | 16.940 | 16.940

License

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18.06 Linear Algebra (MIT) 18.06 Linear Algebra (MIT)

Description

Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices. Applications to least-squares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses MATLAB®. Compared with 18.700 [also Linear Algebra], more emphasis on matrix algorithms and many applications. MATLAB® is a trademark of The MathWorks, Inc. Basic subject on matrix theory and linear algebra, emphasizing topics useful in other disciplines, including systems of equations, vector spaces, determinants, eigenvalues, similarity, and positive definite matrices. Applications to least-squares approximations, stability of differential equations, networks, Fourier transforms, and Markov processes. Uses MATLAB®. Compared with 18.700 [also Linear Algebra], more emphasis on matrix algorithms and many applications. MATLAB® is a trademark of The MathWorks, Inc.

Subjects

Generalized spaces | Generalized spaces | Linear algebra | Linear algebra | Algebra | Universal | Algebra | Universal | Mathematical analysis | Mathematical analysis | Calculus of operations | Calculus of operations | Line geometry | Line geometry | Topology | Topology | matrix theory | matrix theory | systems of equations | systems of equations | vector spaces | vector spaces | systems determinants | systems determinants | eigen values | eigen values | positive definite matrices | positive definite matrices | Markov processes | Markov processes | Fourier transforms | Fourier transforms | differential equations | differential equations

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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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4.105 Geometric Disciplines and Architecture Skills: Reciprocal Methodologies (MIT) 4.105 Geometric Disciplines and Architecture Skills: Reciprocal Methodologies (MIT)

Description

This course is an intensive introduction to architectural design tools and process, and is taught through a series of short exercises. The conceptual basis of each exercise is in the interrogation of the geometric principles that lie at the core of each skill. Skills covered in this course range from techniques of hand drafting, to generation of 3D computer models, physical model-building, sketching, and diagramming. Weekly lectures and pin-ups address the conventions associated with modes of architectural representation and their capacity to convey ideas. This course is tailored and offered only to first-year M.Arch students. This course is an intensive introduction to architectural design tools and process, and is taught through a series of short exercises. The conceptual basis of each exercise is in the interrogation of the geometric principles that lie at the core of each skill. Skills covered in this course range from techniques of hand drafting, to generation of 3D computer models, physical model-building, sketching, and diagramming. Weekly lectures and pin-ups address the conventions associated with modes of architectural representation and their capacity to convey ideas. This course is tailored and offered only to first-year M.Arch students.

Subjects

geometry | geometry | representation | representation | architecture | architecture | drawing | drawing | projection | projection | perspective | perspective | planes | planes | axonometric | axonometric | stereotomy | stereotomy | volume | volume | surface | surface | curvature | curvature | curves | curves | discretization | discretization | generation | generation | construction | construction | publication | publication | presentation | presentation

License

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