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

Description

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

Subjects

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

License

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

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Geometría Gráfica Informática en Arquitectura I Geometría Gráfica Informática en Arquitectura I

Description

Teoría geométrica del objeto arquitectónico con herramientas informáticas. Esta asignatura se ocupa del estudio de las formas espaciales relacionadas con la arquitectura y de su representación, mediante el uso de los medios informáticos. Puede considerarse, en parte, como una profundización y ampliación de los conocimientos adquiridos por el alumno en Geometría Descriptiva; por otro lado, supone la aplicación, según los medios informáticos, de conceptos referentes a la expresión gráfica aprendidos en otras asignaturas de este mismo Área. Teoría geométrica del objeto arquitectónico con herramientas informáticas. Esta asignatura se ocupa del estudio de las formas espaciales relacionadas con la arquitectura y de su representación, mediante el uso de los medios informáticos. Puede considerarse, en parte, como una profundización y ampliación de los conocimientos adquiridos por el alumno en Geometría Descriptiva; por otro lado, supone la aplicación, según los medios informáticos, de conceptos referentes a la expresión gráfica aprendidos en otras asignaturas de este mismo Área.

Subjects

Bóveda | Bóveda | Expresión Gráfica en la Ingeniería | Expresión Gráfica en la Ingeniería | Módulo | Módulo | 3D network | 3D network | Laboratorios Jorba | Laboratorios Jorba | Modelado | Modelado | Extrusión | Extrusión | Construcciones Arquitectónicas | Construcciones Arquitectónicas | Red plana | Red plana | Mezquita Mihrimah | Mezquita Mihrimah | Orden toscano | Orden toscano | Dome | Dome | Platonic solids | Platonic solids | Dibujo | Dibujo | Booleana | Booleana | Graphic | Graphic | Solid | Solid | Dibujo 3D | Dibujo 3D | Arquitecto | Arquitecto | Architect | Architect | Poliedros semirregulares | Poliedros semirregulares | Hiperboloides | Hiperboloides | Polyhedron | Polyhedron | NURBS | NURBS | Render | Render | Malla | Malla | Computing | Computing | Ermita de la Virgen del Puerto | Ermita de la Virgen del Puerto | Surface | Surface | Particiones | Particiones | Historia del Arte | Historia del Arte | Poliedros regulares | Poliedros regulares | Red 3D | Red 3D | Choisy | Choisy | Axonometric | Axonometric | Network | Network | Pattern | Pattern | Geométrico | Geométrico | Mosque | Mosque | Spatial | Spatial | Sinan | Sinan | Red | Red | 3D | 3D | Form | Form | Perspectiva | Perspectiva | Ordenador | Ordenador | Ismael Garcia Rios | Ismael Garcia Rios | Lacería | Lacería | Paraboloides | Paraboloides | Axonometría | Axonometría | Souto de Moura | Souto de Moura | Revolve | Revolve | Kingo Houses | Kingo Houses | Santa María del Naranco | Santa María del Naranco | Pedro de Ribera | Pedro de Ribera | Regular tessellations | Regular tessellations | Expresión Gráfica Arquitectónica | Expresión Gráfica Arquitectónica | Architecture | Architecture | Molina de Aragón | Molina de Aragón | Tessellations | Tessellations | Ponte de Lima | Ponte de Lima | Arabesco | Arabesco | compactación | compactación | Pantheon | Pantheon | Computer | Computer | Eduardo Torroja | Eduardo Torroja | Fernández del Amo | Fernández del Amo | Ribbed vault | Ribbed vault | Organic architecture | Organic architecture | Superficie cuádrica | Superficie cuádrica | Arquitectura | Arquitectura | Miguel Fisac | Miguel Fisac | Crecimiento orgánico | Crecimiento orgánico | Autocad | Autocad | Symmetrical polyhedra | Symmetrical polyhedra | Rhinoceros | Rhinoceros | Domical vault | Domical vault | Templete de los Evangelistas | Templete de los Evangelistas | Gráfica | Gráfica | Luigi Canina | Luigi Canina | MicroStation | MicroStation | Irregular tessellations | Irregular tessellations | Patio de los Evangelistas | Patio de los Evangelistas | Composición Arquitectónica | Composición Arquitectónica | Sabil | Sabil | Modelling | Modelling | quadric surface | quadric surface | Mesh | Mesh | Panteón | Panteón | Monasterio de El Escorial | Monasterio de El Escorial | Superficie | Superficie | Tunnel vault | Tunnel vault | Vegaviana | Vegaviana | Iglesia Santos Apóstoles | Iglesia Santos Apóstoles | Proyectos Arquitectónicos | Proyectos Arquitectónicos | Parabólicos | Parabólicos | Geometry | Geometry | Carmen Garcia Reig | Carmen Garcia Reig | Expression | Expression | Cuba hiperbólica | Cuba hiperbólica | Red espacial | Red espacial | Geometría | Geometría | skeletal polyhedra | skeletal polyhedra | Bóveda de rincón de claustro | Bóveda de rincón de claustro | Sólido | Sólido | Poliedros platónicos | Poliedros platónicos | Hiperbólicos | Hiperbólicos | Perspective | Perspective | Mimbar | Mimbar | Fedala | Fedala | Furniture | Furniture | Polygon | Polygon | Hyperbolic | Hyperbolic | Bóveda de arista | Bóveda de arista | Revolución | Revolución | Cúpula | Cúpula | Bóveda de cañón | Bóveda de cañón | Estructura | Estructura | Expresión | Expresión | Computer aided design | Computer aided design | Informática | Informática | Hollow faced polyhedra | Hollow faced polyhedra | Flexible | Flexible | Geometric | Geometric | CAD | CAD | Formas | Formas | Bóveda de crucería | Bóveda de crucería | Infografía | Infografía | Polyhedra | Polyhedra | Groin vault | Groin vault | Plane | Plane | Architectural | Architectural | Plano | Plano | Structure | Structure | Architectural drawing | Architectural drawing | Arquitectónica | Arquitectónica | Barrel vault | Barrel vault | Arabesque | Arabesque | Sombra | Sombra | Utzon | Utzon | Escher | Escher | Mueble | Mueble | Vault | Vault | Annular vault | Annular vault | Egipto | Egipto | Polígono | Polígono | Archimedean solids | Archimedean solids | Poliedros arquimedianos | Poliedros arquimedianos | Egypt | Egypt | Paraboloid | Paraboloid | Module | Module | Extrude | Extrude | Boolean | Boolean | Tuscan Order | Tuscan Order | Hyperboloid | Hyperboloid | Poliedros vacuus | Poliedros vacuus

License

Copyright 2009, by the Contributing Authors http://creativecommons.org/licenses/by-nc-sa/3.0/

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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.

Subjects

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

License

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

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

Description

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

Subjects

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

License

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

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6.849 Geometric Folding Algorithms: Linkages, Origami, Polyhedra (MIT) 6.849 Geometric Folding Algorithms: Linkages, Origami, Polyhedra (MIT)

Description

Includes audio/video content: AV lectures. This course focuses on the algorithms for analyzing and designing geometric foldings. Topics include reconfiguration of foldable structures, linkages made from one-dimensional rods connected by hinges, folding two-dimensional paper (origami), and unfolding and folding three-dimensional polyhedra. Applications to architecture, robotics, manufacturing, and biology are also covered in this course. Acknowledgments Thanks to videographers Martin Demaine and Jayson Lynch. Includes audio/video content: AV lectures. This course focuses on the algorithms for analyzing and designing geometric foldings. Topics include reconfiguration of foldable structures, linkages made from one-dimensional rods connected by hinges, folding two-dimensional paper (origami), and unfolding and folding three-dimensional polyhedra. Applications to architecture, robotics, manufacturing, and biology are also covered in this course. Acknowledgments Thanks to videographers Martin Demaine and Jayson Lynch.

Subjects

origami | origami | geometry | geometry | algorithm | algorithm | folding | folding | linkage | linkage | polyhedra | polyhedra | seam | seam | crease pattern | crease pattern | universal molecule | universal molecule | box pleating | box pleating | triangulation | triangulation | vertex | vertex | edge | edge | curved crease | curved crease | rigidity | rigidity | tensegrity | tensegrity | hinged dissection | hinged dissection | unfolding | unfolding | gluing | gluing | platonic solid | platonic solid | refolding | refolding | sculpture | sculpture | paper | paper | 3D chain | 3D chain | design | design

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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FeSi Single Crystal: Polyhedral Morphology

Description

CORE-Materials posted a photo: This photograph shows polyhedral morphology of a FeSi single crystal grown by dissolving arc-melted FeSi pieces in Sb or Sn flux at ˜1200°C, then slowly cooling down to ˜700°C. This material has a simple cubic structure and crystals grow as polyhedra or long bars. FeSi is a narrow-gap semiconductor, with a high density of states above and below the Fermi surface. Courtesy of Paul Canfield, Ames Laboratory, US Department of Energy.

Subjects

crystal growth semiconductor polyhedral fesi fermisurface arcmelting cubicstructure

License

https://creativecommons.org/licenses/by/2.0/deed.en

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ZrNiSn Single Crystal: Polyhedral Morphology

Description

CORE-Materials posted a photo: This photograph shows polyhedral morphology of a ZrNiSn single crystal grown by using the self-flux method (excess Sn), this half-Heusler compound, whose structure can be respresented as 4 interpenetrating cubic fcc sublattices, is part of a series of narrow-gap semiconductors with potential low and intermediate temperature thermoelectric applications. Courtesy of Paul Canfield, Ames Laboratory, US Department of Energy.

Subjects

crystal growth semiconductor polyhedral selffluxmethod heuslercompound fcclattice thermoelectricapplications

License

https://creativecommons.org/licenses/by/2.0/deed.en

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Spherulites in poly-3-hydroxy butyrate (PHB)

Description

Polymer melts often crystallise from heterogeneous nuclei to form ribbon-like lamellae, which have a folded chain molecular structure. They commonly radiate outwards from the nucleation point, to form spherical features called spherulites - which are often quite large. The characteristic Maltese cross pattern, seen when viewed between crossed polars, results from isoclinic fringes formed when one of the principal vibration directions is approximately parallel to the polariser. In some cases (including the Poly-3-hydroxy butyrate (PHB) viewed here), the lamella twist as they grow outwards, in phase with their neighbours. This gives rise to the observed ring pattern. When viewed with a full wave plate, alternate (fast and slow) rings are coloured blue and yellow.

Subjects

alignment | crystal | polyhedral | polyhydroxybutyrate (PHB) | polymer | spherulite | DoITPoMS | University of Cambridge | micrograph | corematerials | ukoer

License

http://creativecommons.org/licenses/by-nc-sa/2.0/uk/

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Spherulites in poly-3-hydroxy butyrate (PHB)

Description

Polymer melts often crystallise from heterogeneous nuclei to form ribbon-like lamellae, which have a folded chain molecular structure. They commonly radiate outwards from the nucleation point, to form spherical features called spherulites - which are often quite large. The characteristic Maltese cross pattern, seen when viewed between crossed polars, results from isoclinic fringes formed when one of the principal vibration directions is approximately parallel to the polariser. In some cases (including the Poly-3-hydroxy butyrate (PHB) viewed here), the lamella twist as they grow outwards, in phase with their neighbours. This gives rise to the observed ring pattern. When viewed with a full wave plate, alternate (fast and slow) rings are coloured blue and yellow.

Subjects

alignment | crystal | polyhedral | polyhydroxybutyrate (phb) | polymer | spherulite | doitpoms | university of cambridge | micrograph | corematerials | ukoer | Engineering | H000

License

Attribution-Noncommercial-Share Alike 2.0 UK: England & Wales Attribution-Noncommercial-Share Alike 2.0 UK: England & Wales http://creativecommons.org/licenses/by-nc-sa/2.0/uk/ http://creativecommons.org/licenses/by-nc-sa/2.0/uk/

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6.849 Geometric Folding Algorithms: Linkages, Origami, Polyhedra (MIT)

Description

This course focuses on the algorithms for analyzing and designing geometric foldings. Topics include reconfiguration of foldable structures, linkages made from one-dimensional rods connected by hinges, folding two-dimensional paper (origami), and unfolding and folding three-dimensional polyhedra. Applications to architecture, robotics, manufacturing, and biology are also covered in this course. Acknowledgments Thanks to videographers Martin Demaine and Jayson Lynch.

Subjects

origami | geometry | algorithm | folding | linkage | polyhedra | seam | crease pattern | universal molecule | box pleating | triangulation | vertex | edge | curved crease | rigidity | tensegrity | hinged dissection | unfolding | gluing | platonic solid | refolding | sculpture | paper | 3D chain | design

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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6.253 Convex Analysis and Optimization (MIT)

Description

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

Subjects

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

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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