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18.755 Introduction to Lie Groups (MIT) 18.755 Introduction to Lie Groups (MIT)

Description

This course is devoted to the theory of Lie Groups with emphasis on its connections with Differential Geometry. The text for this class is Differential Geometry, Lie Groups and Symmetric Spaces by Sigurdur Helgason (American Mathematical Society, 2001). Much of the course material is based on Chapter I (first half) and Chapter II of the text. The text however develops basic Riemannian Geometry, Complex Manifolds, as well as a detailed theory of Semisimple Lie Groups and Symmetric Spaces. This course is devoted to the theory of Lie Groups with emphasis on its connections with Differential Geometry. The text for this class is Differential Geometry, Lie Groups and Symmetric Spaces by Sigurdur Helgason (American Mathematical Society, 2001). Much of the course material is based on Chapter I (first half) and Chapter II of the text. The text however develops basic Riemannian Geometry, Complex Manifolds, as well as a detailed theory of Semisimple Lie Groups and Symmetric Spaces.

Subjects

Manifolds | Manifolds | Lie groups | Lie groups | exponential mapping | exponential mapping | Lie algebras | Lie algebras | Homogeneous spaces | Homogeneous spaces | transformation groups | transformation groups | Adjoint representation | Adjoint representation | Covering groups | Covering groups | Automorphism groups | Automorphism groups | Invariant differential forms | Invariant differential forms | cohomology of Lie groups | cohomology of Lie groups | homogeneous spaces. | homogeneous spaces. | Lie Groups | Lie Groups | Exponential Mapping | Exponential Mapping | Lie Algebras | Lie Algebras | Homogeneous Spaces | Homogeneous Spaces | Transformation Groups | Transformation Groups | Covering Groups | Covering Groups | Automorphism Groups | Automorphism Groups | Invariant Differential Forms | Invariant Differential Forms | Cohomology of Lie Groups | Cohomology of Lie Groups | Homogeneous Spaces. | Homogeneous Spaces.

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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18.701 Algebra I (MIT) 18.701 Algebra I (MIT)

Description

This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups. This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups.

Subjects

Group Theory | Group Theory | Linear Algebra | and Geometry | Linear Algebra | and Geometry | groups | groups | vector spaces | vector spaces | linear transformations | linear transformations | symmetry groups | symmetry groups | bilinear | bilinear | bilinear forms | and linear groups | bilinear forms | and linear groups

License

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18.701 Algebra I (MIT) 18.701 Algebra I (MIT)

Description

This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups. This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups.

Subjects

Group Theory | Group Theory | Linear Algebra | Linear Algebra | Geometry | Geometry | groups | groups | vector spaces | vector spaces | linear transformations | linear transformations | symmetry groups | symmetry groups | bilinear forms | bilinear forms | linear groups | linear groups

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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18.701 Algebra I (MIT) 18.701 Algebra I (MIT)

Description

The subjects to be covered include groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups. The subjects to be covered include groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups.

Subjects

Group Theory | Group Theory | Linear Algebra | and Geometry | Linear Algebra | and Geometry | groups | groups | vector spaces | vector spaces | linear transformations | linear transformations | symmetry groups | symmetry groups | bilinear | bilinear | bilinear forms | and linear groups | bilinear forms | and linear groups

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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18.755 Introduction to Lie Groups (MIT)

Description

This course is devoted to the theory of Lie Groups with emphasis on its connections with Differential Geometry. The text for this class is Differential Geometry, Lie Groups and Symmetric Spaces by Sigurdur Helgason (American Mathematical Society, 2001). Much of the course material is based on Chapter I (first half) and Chapter II of the text. The text however develops basic Riemannian Geometry, Complex Manifolds, as well as a detailed theory of Semisimple Lie Groups and Symmetric Spaces.

Subjects

Manifolds | Lie groups | exponential mapping | Lie algebras | Homogeneous spaces | transformation groups | Adjoint representation | Covering groups | Automorphism groups | Invariant differential forms | cohomology of Lie groups | homogeneous spaces. | Lie Groups | Exponential Mapping | Lie Algebras | Homogeneous Spaces | Transformation Groups | Covering Groups | Automorphism Groups | Invariant Differential Forms | Cohomology of Lie Groups | Homogeneous Spaces.

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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5.04 Principles of Inorganic Chemistry II (MIT) 5.04 Principles of Inorganic Chemistry II (MIT)

Description

This course provides a systematic presentation of the chemical applications of group theory with emphasis on the formal development of the subject and its applications to the physical methods of inorganic chemical compounds. Against the backdrop of electronic structure, the electronic, vibrational, and magnetic properties of transition metal complexes are presented and their investigation by the appropriate spectroscopy described. This course provides a systematic presentation of the chemical applications of group theory with emphasis on the formal development of the subject and its applications to the physical methods of inorganic chemical compounds. Against the backdrop of electronic structure, the electronic, vibrational, and magnetic properties of transition metal complexes are presented and their investigation by the appropriate spectroscopy described.

Subjects

inorganic chemistry | inorganic chemistry | group theory | group theory | electronic structure of molecules | electronic structure of molecules | transition metal complexes | transition metal complexes | spectroscopy | spectroscopy | symmetry elements | symmetry elements | mathematical groups | mathematical groups | character tables | character tables | molecular point groups | molecular point groups | Huckel Theory | Huckel Theory | N-Dimensional cyclic systems | N-Dimensional cyclic systems | solid state theory | solid state theory | band theory | band theory | frontier molecular orbitals | frontier molecular orbitals | similarity transformations | similarity transformations | complexes | complexes | organometallic complexes | organometallic complexes | two electron bond | two electron bond | vibrational spectroscopy | vibrational spectroscopy | symmetry | symmetry | overtones | overtones | normal coordinat analysis | normal coordinat analysis | AOM | AOM | single electron CFT | single electron CFT | tanabe-sugano diagram | tanabe-sugano diagram | ligand | ligand | crystal field theory | crystal field theory | LCAO | LCAO

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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8.321 Quantum Theory I (MIT) 8.321 Quantum Theory I (MIT)

Description

8.321 is the first semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: Hilbert spaces, observables, uncertainty relations, eigenvalue problems and methods for solution thereof, time-evolution in the Schrodinger, Heisenberg, and interaction pictures, connections between classical and quantum mechanics, path integrals, quantum mechanics in EM fields, angular momentum, time-independent perturbation theory, density operators, and quantum measurement. 8.321 is the first semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: Hilbert spaces, observables, uncertainty relations, eigenvalue problems and methods for solution thereof, time-evolution in the Schrodinger, Heisenberg, and interaction pictures, connections between classical and quantum mechanics, path integrals, quantum mechanics in EM fields, angular momentum, time-independent perturbation theory, density operators, and quantum measurement.

Subjects

eigenstates | eigenstates | uncertainty relation | uncertainty relation | observables | observables | eigenvalues | eigenvalues | probabilities of the results of measurement | probabilities of the results of measurement | transformation theory | transformation theory | equations of motion | equations of motion | constants of motion | constants of motion | Symmetry in quantum mechanics | Symmetry in quantum mechanics | representations of symmetry groups | representations of symmetry groups | Variational and perturbation approximations | Variational and perturbation approximations | Systems of identical particles and applications | Systems of identical particles and applications | Time-dependent perturbation theory | Time-dependent perturbation theory | Scattering theory: phase shifts | Scattering theory: phase shifts | Born approximation | Born approximation | The quantum theory of radiation | The quantum theory of radiation | Second quantization and many-body theory | Second quantization and many-body theory | Relativistic quantum mechanics of one electron | Relativistic quantum mechanics of one electron | probability | probability | measurement | measurement | motion equations | motion equations | motion constants | motion constants | symmetry groups | symmetry groups | quantum mechanics | quantum mechanics | variational approximations | variational approximations | perturbation approximations | perturbation approximations | identical particles | identical particles | time-dependent perturbation theory | time-dependent perturbation theory | scattering theory | scattering theory | phase shifts | phase shifts | quantum theory of radiation | quantum theory of radiation | second quantization | second quantization | many-body theory | many-body theory | relativistic quantum mechanics | relativistic quantum mechanics | one electron | one electron | Hilbert spaces | Hilbert spaces | time evolution | time evolution | Schrodinger picture | Schrodinger picture | Heisenberg picture | Heisenberg picture | interaction picture | interaction picture | classical mechanics | classical mechanics | path integrals | path integrals | EM fields | EM fields | electromagnetic fields | electromagnetic fields | angular momentum | angular momentum | density operators | density operators | quantum measurement | quantum measurement | quantum statistics | quantum statistics | quantum dynamics | quantum dynamics

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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8.322 Quantum Theory II (MIT) 8.322 Quantum Theory II (MIT)

Description

8.322 is the second semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: time-dependent perturbation theory and applications to radiation, quantization of EM radiation field, adiabatic theorem and Berry's phase, symmetries in QM, many-particle systems, scattering theory, relativistic quantum mechanics, and Dirac equation. 8.322 is the second semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: time-dependent perturbation theory and applications to radiation, quantization of EM radiation field, adiabatic theorem and Berry's phase, symmetries in QM, many-particle systems, scattering theory, relativistic quantum mechanics, and Dirac equation.

Subjects

uncertainty relation | uncertainty relation | observables | observables | eigenstates | eigenstates | eigenvalues | eigenvalues | probabilities of the results of measurement | probabilities of the results of measurement | transformation theory | transformation theory | equations of motion | equations of motion | constants of motion | constants of motion | Symmetry in quantum mechanics | Symmetry in quantum mechanics | representations of symmetry groups | representations of symmetry groups | Variational and perturbation approximations | Variational and perturbation approximations | Systems of identical particles and applications | Systems of identical particles and applications | Time-dependent perturbation theory | Time-dependent perturbation theory | Scattering theory: phase shifts | Scattering theory: phase shifts | Born approximation | Born approximation | The quantum theory of radiation | The quantum theory of radiation | Second quantization and many-body theory | Second quantization and many-body theory | Relativistic quantum mechanics of one electron | Relativistic quantum mechanics of one electron | probability | probability | measurement | measurement | motion equations | motion equations | motion constants | motion constants | symmetry groups | symmetry groups | quantum mechanics | quantum mechanics | variational approximations | variational approximations | perturbation approximations | perturbation approximations | identical particles | identical particles | time-dependent perturbation theory | time-dependent perturbation theory | scattering theory | scattering theory | phase shifts | phase shifts | quantum theory of radiation | quantum theory of radiation | second quantization | second quantization | many-body theory | many-body theory | relativistic quantum mechanics | relativistic quantum mechanics | one electron | one electron | quantization | quantization | EM radiation field | EM radiation field | electromagnetic radiation field | electromagnetic radiation field | adiabatic theorem | adiabatic theorem | Berry?s phase | Berry?s phase | many-particle systems | many-particle systems | Dirac equation | Dirac equation | Hilbert spaces | Hilbert spaces | time evolution | time evolution | Schrodinger picture | Schrodinger picture | Heisenberg picture | Heisenberg picture | interaction picture | interaction picture | classical mechanics | classical mechanics | path integrals | path integrals | EM fields | EM fields | electromagnetic fields | electromagnetic fields | angular momentum | angular momentum | density operators | density operators | quantum measurement | quantum measurement | quantum statistics | quantum statistics | quantum dynamics | quantum dynamics

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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9.916 Special Topics: Social Animals (MIT) 9.916 Special Topics: Social Animals (MIT)

Description

Humans are social animals; social demands, both cooperative and competitive, structure our development, our brain and our mind. This course covers social development, social behaviour, social cognition and social neuroscience, in both human and non-human social animals. Topics include altruism, empathy, communication, theory of mind, aggression, power, groups, mating, and morality. Methods include evolutionary biology, neuroscience, cognitive science, social psychology and anthropology. Humans are social animals; social demands, both cooperative and competitive, structure our development, our brain and our mind. This course covers social development, social behaviour, social cognition and social neuroscience, in both human and non-human social animals. Topics include altruism, empathy, communication, theory of mind, aggression, power, groups, mating, and morality. Methods include evolutionary biology, neuroscience, cognitive science, social psychology and anthropology.

Subjects

social animals | social animals | social | social | animals | animals | society | society | human society | human society | members | members | community | community | living together | living together | mutual benefit | mutual benefit | people | people | region | region | country | country | world | world | whole | whole | association | association | body | body | individuals | individuals | functional interdependence | functional interdependence | national or cultural identity | national or cultural identity | social solidarity | social solidarity | language or hierarchical organization | language or hierarchical organization | patterns of relationships between individuals sharing a distinctive culture and institutions | patterns of relationships between individuals sharing a distinctive culture and institutions | groups | groups | economic | economic | social or industrial infrastructure | social or industrial infrastructure | made up of a varied collection of individuals | made up of a varied collection of individuals | ethnic groups | ethnic groups | nation state | nation state | broader cultural group | broader cultural group | organized voluntary association of people for religious | organized voluntary association of people for religious | benevolent | benevolent | cultural | cultural | scientific | scientific | political | political | patriotic | patriotic | or other purposes. | or other purposes.

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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15.356 How to Develop "Breakthrough" Products and Services (MIT) 15.356 How to Develop "Breakthrough" Products and Services (MIT)

Description

Includes audio/video content: AV selected lectures. To prosper, firms must develop major product and service innovations. Often, though, they don't know how. Recent research into the innovation process has made it possible to develop "breakthroughs" systematically. 15.356 presents several practical concept development methods, such as the "Lead User Method," where manufacturers learn from innovative customers. Expert guest speakers present case studies that show the "art" required to implement a concept development method. 15.356 is a half-term subject. Includes audio/video content: AV selected lectures. To prosper, firms must develop major product and service innovations. Often, though, they don't know how. Recent research into the innovation process has made it possible to develop "breakthroughs" systematically. 15.356 presents several practical concept development methods, such as the "Lead User Method," where manufacturers learn from innovative customers. Expert guest speakers present case studies that show the "art" required to implement a concept development method. 15.356 is a half-term subject.

Subjects

lead user method; innovations; innovation process; idea generation; brainstorming; concept development methods; prototypes; solutions; problem solving; business breakthroughs; incremental improvements; market research; focus groups; MIT Media Lab; creativity | lead user method; innovations; innovation process; idea generation; brainstorming; concept development methods; prototypes; solutions; problem solving; business breakthroughs; incremental improvements; market research; focus groups; MIT Media Lab; creativity | lead user method | lead user method | innovations | innovations | innovation process | innovation process | idea generation | idea generation | brainstorming | brainstorming | concept development methods | concept development methods | prototypes | prototypes | solutions | solutions | problem solving | problem solving | business breakthroughs | business breakthroughs | incremental improvements | incremental improvements | market research | market research | focus groups | focus groups | MIT Media Lab | MIT Media Lab | creativity | creativity

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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Working in Groups - for Regional Regeneration section 1

Description

This JISC funded project has re-used and re-purposed learning content from external and internal sources to develop and pilot a level 4 module “Working in Groups” for multi-professional teams working in the field of Regional Regeneration. Permissions received from all external sources.

Subjects

studying online | ukoer | working in groups | group working | working groups | groups versus teams | group | team | groups vs teams | regional regeneration | non-verbal communication | action learning sets | group work reflection | Education | X000

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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Working in Groups - for Regional Regeneration section 2

Description

This JISC funded project has re-used and re-purposed learning content from external and internal sources to develop and pilot a level 4 module “Working in Groups” for multi-professional teams working in the field of Regional Regeneration. Permissions received from all external sources.

Subjects

ukoer | working in groups | group working | working groups | differences between groups and teams | group formation and planning | group | team | groups vs teams | benefits of group work | stages of group development | regional regeneration | non-verbal communication | action learning sets | legal actions | group work reflection | techniques to improve group output | equality and diversity | participation and involvement | team resources | resource needs of a group | issues of inclusion and exclusion with group work | studying online | Education | X000

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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Working in Groups - for Regional Regeneration section 3

Description

This JISC funded project has re-used and re-purposed learning content from external and internal sources to develop and pilot a level 4 module “Working in Groups” for multi-professional teams working in the field of Regional Regeneration. Permissions received from all external sources.

Subjects

working in groups | group working | working groups | group dynamics and behaviour | appreciative enquiry | transactional analysis | groups versus teams | group | team | groups vs teams | regional regeneration | non-verbal communication | action learning sets | group work reflection | studying online | ukoer | Social studies | Education | X000 | L000 | EDUCATION / TRAINING / TEACHING | G

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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Working in Groups - for Regional Regeneration section 4

Description

This JISC funded project has re-used and re-purposed learning content from external and internal sources to develop and pilot a level 4 module “Working in Groups” for multi-professional teams working in the field of Regional Regeneration. Permissions received from all external sources.

Subjects

ukoer | working in groups | group working | working groups | communicating effectively in groups | effective communication | data sharing | sharing intelligence | giving briefings | giving presentations | planning a presentation | conducting meetings | meetings | roles people play in meetings | managing meetings | meeting minutes | minute takers | groups and teams | regional regeneration | non-verbal communication | action learning sets | group work reflection | studying online | Social studies | Education | X000 | L000 | EDUCATION / TRAINING / TEACHING | G

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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18.701 Algebra I (MIT)

Description

This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups.

Subjects

Group Theory | Linear Algebra | Geometry | groups | vector spaces | linear transformations | symmetry groups | bilinear forms | linear groups

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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18.701 Algebra I (MIT)

Description

The subjects to be covered include groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups.

Subjects

Group Theory | Linear Algebra | and Geometry | groups | vector spaces | linear transformations | symmetry groups | bilinear | bilinear forms | and linear groups

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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18.701 Algebra I (MIT)

Description

This undergraduate level Algebra I course covers groups, vector spaces, linear transformations, symmetry groups, bilinear forms, and linear groups.

Subjects

Group Theory | Linear Algebra | and Geometry | groups | vector spaces | linear transformations | symmetry groups | bilinear | bilinear forms | and linear groups

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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Learning to Teach Inclusively OER Videos in Human Resources

Description

Project web-site – www.wlv.ac.uk/teachinclusively e-mail: oer@wlv.ac.uk Videos are in .mp4 format (H.264 compression) and are playable with most media players. If you have any problems with the playback we recommend free open source VLC media player – www.videolan.org or all of our published videos can be viewed online on www.vimeo.com/oer/videos in your browser. Please contact us if you need any help oer@wlv.ac.uk, or fill in Problem Report form - https://docs.google.com/spreadsheet/viewform?formkey=dEg4a1h6T0ZzenRkVnlIcW1iYXpadHc6MQ

Subjects

inclusive teaching | video resource | classroom | session | teaching activity | interview | connecting lives and theory | teacher | student | participation | human resources | sharing knowledge | experience | benefits | working | diverse groups | strategies | coordinating interaction | challenges | mixed | groups | diverse | ukoer | omac | oer | lti | administrative studies | N000

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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18.325 Topics in Applied Mathematics: Mathematical Methods in Nanophotonics (MIT) 18.325 Topics in Applied Mathematics: Mathematical Methods in Nanophotonics (MIT)

Description

This course covers algebraic approaches to electromagnetism and nano-photonics. Topics include photonic crystals, waveguides, perturbation theory, diffraction, computational methods, applications to integrated optical devices, and fiber-optic systems. Emphasis is placed on abstract algebraic approaches rather than detailed solutions of partial differential equations, the latter being done by computers. This course covers algebraic approaches to electromagnetism and nano-photonics. Topics include photonic crystals, waveguides, perturbation theory, diffraction, computational methods, applications to integrated optical devices, and fiber-optic systems. Emphasis is placed on abstract algebraic approaches rather than detailed solutions of partial differential equations, the latter being done by computers.

Subjects

linear algebra | linear algebra | eigensystems for Maxwell's equations | eigensystems for Maxwell's equations | symmetry groups | symmetry groups | representation theory | representation theory | Bloch's theorem | Bloch's theorem | numerical eigensolver methods | numerical eigensolver methods | time and frequency-domain computation | time and frequency-domain computation | perturbation theory | perturbation theory | coupled-mode theories | coupled-mode theories | waveguide theory | waveguide theory | adiabatic transitions | adiabatic transitions | Optical phenomena | Optical phenomena | photonic crystals | photonic crystals | band gaps | band gaps | anomalous diffraction | anomalous diffraction | mechanisms for optical confinement | mechanisms for optical confinement | optical fibers | optical fibers | integrated optical devices | integrated optical devices

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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18.702 Algebra II (MIT) 18.702 Algebra II (MIT)

Description

The course covers group theory and its representations, and focuses on the Sylow theorem, Schur's lemma, and proof of the orthogonality relations. It also analyzes the rings, the factorization processes, and the fields. Topics such as the formal construction of integers and polynomials, homomorphisms and ideals, the Gauss' lemma, quadratic imaginary integers, Gauss primes, and finite and function fields are discussed in detail. The course covers group theory and its representations, and focuses on the Sylow theorem, Schur's lemma, and proof of the orthogonality relations. It also analyzes the rings, the factorization processes, and the fields. Topics such as the formal construction of integers and polynomials, homomorphisms and ideals, the Gauss' lemma, quadratic imaginary integers, Gauss primes, and finite and function fields are discussed in detail.

Subjects

Sylow theorems | Sylow theorems | Group Representations | Group Representations | definitions | definitions | unitary representations | unitary representations | characters | characters | Schur's Lemma | Schur's Lemma | Rings: Basic Definitions | Rings: Basic Definitions | homomorphisms | homomorphisms | fractions | fractions | Factorization | Factorization | unique factorization | unique factorization | Gauss' Lemma | Gauss' Lemma | explicit factorization | explicit factorization | maximal ideals | maximal ideals | Quadratic Imaginary Integers | Quadratic Imaginary Integers | Gauss Primes | Gauss Primes | quadratic integers | quadratic integers | ideal factorization | ideal factorization | ideal classes | ideal classes | Linear Algebra over a Ring | Linear Algebra over a Ring | free modules | free modules | integer matrices | integer matrices | generators and relations | generators and relations | structure of abelian groups | structure of abelian groups | Rings: Abstract Constructions | Rings: Abstract Constructions | relations in a ring | relations in a ring | adjoining elements | adjoining elements | Fields: Field Extensions | Fields: Field Extensions | algebraic elements | algebraic elements | degree of field extension | degree of field extension | ruler and compass | ruler and compass | symbolic adjunction | symbolic adjunction | finite fields | finite fields | Fields: Galois Theory | Fields: Galois Theory | the main theorem | the main theorem | cubic equations | cubic equations | symmetric functions | symmetric functions | primitive elements | primitive elements | quartic equations | quartic equations | quintic equations | quintic equations

License

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17.588 Field Seminar in Comparative Politics (MIT) 17.588 Field Seminar in Comparative Politics (MIT)

Description

This course provides an introduction to the field of comparative politics. Readings include both classic and recent materials. Discussions include research design and research methods, in addition to topics such as political culture, social cleavages, the state, and democratic institutions. The emphasis on each issue depends in part on the interests of the students. This course provides an introduction to the field of comparative politics. Readings include both classic and recent materials. Discussions include research design and research methods, in addition to topics such as political culture, social cleavages, the state, and democratic institutions. The emphasis on each issue depends in part on the interests of the students.

Subjects

comparative politics | comparative politics | Aristotle | Aristotle | political research | political research | regimes | regimes | Marxist model | Marxist model | class alliances | class alliances | democracy | democracy | pluralism | pluralism | economic growth | economic growth | party formation | party formation | political elites | political elites | interest groups | interest groups | constitutional reform | constitutional reform | political system | political system | constitutional choice | constitutional choice | leadership | leadership | state formation | state formation | modernization | modernization | political institution | political institution | embedded autonomy | embedded autonomy | dead capital | dead capital | nationalism | nationalism | electoral behavior | electoral behavior | clientelism | clientelism | patronage politics | patronage politics | corruption | corruption | self-government | self-government

License

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9.916 Special Topics: Social Animals (MIT)

Description

Humans are social animals; social demands, both cooperative and competitive, structure our development, our brain and our mind. This course covers social development, social behaviour, social cognition and social neuroscience, in both human and non-human social animals. Topics include altruism, empathy, communication, theory of mind, aggression, power, groups, mating, and morality. Methods include evolutionary biology, neuroscience, cognitive science, social psychology and anthropology.

Subjects

social animals | social | animals | society | human society | members | community | living together | mutual benefit | people | region | country | world | whole | association | body | individuals | functional interdependence | national or cultural identity | social solidarity | language or hierarchical organization | patterns of relationships between individuals sharing a distinctive culture and institutions | groups | economic | social or industrial infrastructure | made up of a varied collection of individuals | ethnic groups | nation state | broader cultural group | organized voluntary association of people for religious | benevolent | cultural | scientific | political | patriotic | or other purposes.

License

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18.702 Algebra II (MIT) 18.702 Algebra II (MIT)

Description

This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory. This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

Subjects

Sylow theorems | Sylow theorems | Group Representations | Group Representations | definitions | definitions | unitary representations | unitary representations | characters | characters | Schur's Lemma | Schur's Lemma | Rings: Basic Definitions | Rings: Basic Definitions | homomorphisms | homomorphisms | fractions | fractions | Factorization | Factorization | unique factorization | unique factorization | Gauss' Lemma | Gauss' Lemma | explicit factorization | explicit factorization | maximal ideals | maximal ideals | Quadratic Imaginary Integers | Quadratic Imaginary Integers | Gauss Primes | Gauss Primes | quadratic integers | quadratic integers | ideal factorization | ideal factorization | ideal classes | ideal classes | Linear Algebra over a Ring | Linear Algebra over a Ring | free modules | free modules | integer matrices | integer matrices | generators and relations | generators and relations | structure of abelian groups | structure of abelian groups | Rings: Abstract Constructions | Rings: Abstract Constructions | relations in a ring | relations in a ring | adjoining elements | adjoining elements | Fields: Field Extensions | Fields: Field Extensions | algebraic elements | algebraic elements | degree of field extension | degree of field extension | ruler and compass | ruler and compass | symbolic adjunction | symbolic adjunction | finite fields | finite fields | Fields: Galois Theory | Fields: Galois Theory | the main theorem | the main theorem | cubic equations | cubic equations | symmetric functions | symmetric functions | primitive elements | primitive elements | quartic equations | quartic equations | quintic equations | quintic 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.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

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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18.703 Modern Algebra (MIT) 18.703 Modern Algebra (MIT)

Description

This undergraduate course focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics. This undergraduate course focuses on traditional algebra topics that have found greatest application in science and engineering as well as in mathematics.

Subjects

algebra | algebra | group theory | group theory | finite groups | finite groups | ring theory | ring theory | unique factorization | unique factorization | Euclidean rings | Euclidean rings | field theory | field theory | finite fields | finite fields

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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