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18.304 Undergraduate Seminar in Discrete Mathematics (MIT) 18.304 Undergraduate Seminar in Discrete Mathematics (MIT)

Description

This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic. This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic.

Subjects

discrete math; discrete mathematics; discrete; math; mathematics; seminar; presentations; student presentations; oral; communication; stable marriage; dych; emergency; response vehicles; ambulance; game theory; congruences; color theorem; four color; cake cutting; algorithm; RSA; encryption; numberical integration; sorting; post correspondence problem; PCP; ramsey; van der waals; fibonacci; recursion; domino; tiling; towers; hanoi; pigeonhole; principle; matrix; hamming; code; hat game; juggling; zero-knowledge; proof; repeated games; lewis carroll; determinants; infinitude of primes; bridges; konigsberg; koenigsberg; time series analysis; GARCH; rational; recurrence; relations; digital; image; compression; quantum computing | discrete math; discrete mathematics; discrete; math; mathematics; seminar; presentations; student presentations; oral; communication; stable marriage; dych; emergency; response vehicles; ambulance; game theory; congruences; color theorem; four color; cake cutting; algorithm; RSA; encryption; numberical integration; sorting; post correspondence problem; PCP; ramsey; van der waals; fibonacci; recursion; domino; tiling; towers; hanoi; pigeonhole; principle; matrix; hamming; code; hat game; juggling; zero-knowledge; proof; repeated games; lewis carroll; determinants; infinitude of primes; bridges; konigsberg; koenigsberg; time series analysis; GARCH; rational; recurrence; relations; digital; image; compression; quantum computing | discrete math | discrete math | discrete mathematics | discrete mathematics | discrete | discrete | math | math | mathematics | mathematics | seminar | seminar | presentations | presentations | student presentations | student presentations | oral | oral | communication | communication | stable marriage | stable marriage | dych | dych | emergency | emergency | response vehicles | response vehicles | ambulance | ambulance | game theory | game theory | congruences | congruences | color theorem | color theorem | four color | four color | cake cutting | cake cutting | algorithm | algorithm | RSA | RSA | encryption | encryption | numberical integration | numberical integration | sorting | sorting | post correspondence problem | post correspondence problem | PCP | PCP | ramsey | ramsey | van der waals | van der waals | fibonacci | fibonacci | recursion | recursion | domino | domino | tiling | tiling | towers | towers | hanoi | hanoi | pigeonhole | pigeonhole | principle | principle | matrix | matrix | hamming | hamming | code | code | hat game | hat game | juggling | juggling | zero-knowledge | zero-knowledge | proof | proof | repeated games | repeated games | lewis carroll | lewis carroll | determinants | determinants | infinitude of primes | infinitude of primes | bridges | bridges | konigsberg | konigsberg | koenigsberg | koenigsberg | time series analysis | time series analysis | GARCH | GARCH | rational | rational | recurrence | recurrence | relations | relations | digital | digital | image | image | compression | compression | quantum computing | quantum computing

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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18.304 Undergraduate Seminar in Discrete Mathematics (MIT) 18.304 Undergraduate Seminar in Discrete Mathematics (MIT)

Description

This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic. This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic.

Subjects

discrete math; discrete mathematics; discrete; math; mathematics; seminar; presentations; student presentations; oral; communication; stable marriage; dych; emergency; response vehicles; ambulance; game theory; congruences; color theorem; four color; cake cutting; algorithm; RSA; encryption; numberical integration; sorting; post correspondence problem; PCP; ramsey; van der waals; fibonacci; recursion; domino; tiling; towers; hanoi; pigeonhole; principle; matrix; hamming; code; hat game; juggling; zero-knowledge; proof; repeated games; lewis carroll; determinants; infinitude of primes; bridges; konigsberg; koenigsberg; time series analysis; GARCH; rational; recurrence; relations; digital; image; compression; quantum computing | discrete math; discrete mathematics; discrete; math; mathematics; seminar; presentations; student presentations; oral; communication; stable marriage; dych; emergency; response vehicles; ambulance; game theory; congruences; color theorem; four color; cake cutting; algorithm; RSA; encryption; numberical integration; sorting; post correspondence problem; PCP; ramsey; van der waals; fibonacci; recursion; domino; tiling; towers; hanoi; pigeonhole; principle; matrix; hamming; code; hat game; juggling; zero-knowledge; proof; repeated games; lewis carroll; determinants; infinitude of primes; bridges; konigsberg; koenigsberg; time series analysis; GARCH; rational; recurrence; relations; digital; image; compression; quantum computing | discrete math | discrete math | discrete mathematics | discrete mathematics | discrete | discrete | math | math | mathematics | mathematics | seminar | seminar | presentations | presentations | student presentations | student presentations | oral | oral | communication | communication | stable marriage | stable marriage | dych | dych | emergency | emergency | response vehicles | response vehicles | ambulance | ambulance | game theory | game theory | congruences | congruences | color theorem | color theorem | four color | four color | cake cutting | cake cutting | algorithm | algorithm | RSA | RSA | encryption | encryption | numberical integration | numberical integration | sorting | sorting | post correspondence problem | post correspondence problem | PCP | PCP | ramsey | ramsey | van der waals | van der waals | fibonacci | fibonacci | recursion | recursion | domino | domino | tiling | tiling | towers | towers | hanoi | hanoi | pigeonhole | pigeonhole | principle | principle | matrix | matrix | hamming | hamming | code | code | hat game | hat game | juggling | juggling | zero-knowledge | zero-knowledge | proof | proof | repeated games | repeated games | lewis carroll | lewis carroll | determinants | determinants | infinitude of primes | infinitude of primes | bridges | bridges | konigsberg | konigsberg | koenigsberg | koenigsberg | time series analysis | time series analysis | GARCH | GARCH | rational | rational | recurrence | recurrence | relations | relations | digital | digital | image | image | compression | compression | quantum computing | quantum computing

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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6.003 Signals and Systems (MIT) 6.003 Signals and Systems (MIT)

Description

6.003 covers the fundamentals of signal and system analysis, focusing on representations of discrete-time and continuous-time signals (singularity functions, complex exponentials and geometrics, Fourier representations, Laplace and Z transforms, sampling) and representations of linear, time-invariant systems (difference and differential equations, block diagrams, system functions, poles and zeros, convolution, impulse and step responses, frequency responses). Applications are drawn broadly from engineering and physics, including feedback and control, communications, and signal processing. 6.003 covers the fundamentals of signal and system analysis, focusing on representations of discrete-time and continuous-time signals (singularity functions, complex exponentials and geometrics, Fourier representations, Laplace and Z transforms, sampling) and representations of linear, time-invariant systems (difference and differential equations, block diagrams, system functions, poles and zeros, convolution, impulse and step responses, frequency responses). Applications are drawn broadly from engineering and physics, including feedback and control, communications, and signal processing.

Subjects

signal and system analysis | signal and system analysis | representations of discrete-time and continuous-time signals | representations of discrete-time and continuous-time signals | representations of linear time-invariant systems | representations of linear time-invariant systems | Fourier representations | Fourier representations | Laplace and Z transforms | Laplace and Z transforms | sampling | sampling | difference and differential equations | difference and differential equations | feedback and control | feedback and control | communications | communications | signal processing | signal processing

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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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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18.702 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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18.712 Introduction to Representation Theory (MIT) 18.712 Introduction to Representation Theory (MIT)

Description

The goal of this course is to give an undergraduate-level introduction to representation theory (of groups, Lie algebras, and associative algebras). Representation theory is an area of mathematics which, roughly speaking, studies symmetry in linear spaces. The goal of this course is to give an undergraduate-level introduction to representation theory (of groups, Lie algebras, and associative algebras). Representation theory is an area of mathematics which, roughly speaking, studies symmetry in linear spaces.

Subjects

finite dimensional algebras | finite dimensional algebras | Quiver Representations | Quiver Representations | series Representations | series Representations | finite groups | finite groups | representation theory | representation theory | Lie algebras | Lie algebras | Tensor products | Tensor products | density theorem | density theorem | Jordan-H?older theorem | Jordan-H?older theorem | Krull-Schmidt theorem | Krull-Schmidt theorem | Maschke?s Theorem | Maschke?s Theorem | Frobenius-Schur indicator | Frobenius-Schur indicator | Frobenius divisibility | Frobenius divisibility | Burnside?s Theorem | Burnside?s Theorem

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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18.304 Undergraduate Seminar in Discrete Mathematics (MIT)

Description

This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic.

Subjects

discrete math; discrete mathematics; discrete; math; mathematics; seminar; presentations; student presentations; oral; communication; stable marriage; dych; emergency; response vehicles; ambulance; game theory; congruences; color theorem; four color; cake cutting; algorithm; RSA; encryption; numberical integration; sorting; post correspondence problem; PCP; ramsey; van der waals; fibonacci; recursion; domino; tiling; towers; hanoi; pigeonhole; principle; matrix; hamming; code; hat game; juggling; zero-knowledge; proof; repeated games; lewis carroll; determinants; infinitude of primes; bridges; konigsberg; koenigsberg; time series analysis; GARCH; rational; recurrence; relations; digital; image; compression; quantum computing | discrete math | discrete mathematics | discrete | math | mathematics | seminar | presentations | student presentations | oral | communication | stable marriage | dych | emergency | response vehicles | ambulance | game theory | congruences | color theorem | four color | cake cutting | algorithm | RSA | encryption | numberical integration | sorting | post correspondence problem | PCP | ramsey | van der waals | fibonacci | recursion | domino | tiling | towers | hanoi | pigeonhole | principle | matrix | hamming | code | hat game | juggling | zero-knowledge | proof | repeated games | lewis carroll | determinants | infinitude of primes | bridges | konigsberg | koenigsberg | time series analysis | GARCH | rational | recurrence | relations | digital | image | compression | quantum computing

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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9.916 The Neural Basis of Visual Object Recognition in Monkeys and Humans (MIT) 9.916 The Neural Basis of Visual Object Recognition in Monkeys and Humans (MIT)

Description

Understanding the brain's remarkable ability for visual object recognition is one of the greatest challenges of brain research. The goal of this course is to provide an overview of key issues of object representation and to survey data from primate physiology and human fMRI that bear on those issues. Topics include the computational problems of object representation, the nature of object representations in the brain, the tolerance and selectivity of those representations, and the effects of attention and learning. Understanding the brain's remarkable ability for visual object recognition is one of the greatest challenges of brain research. The goal of this course is to provide an overview of key issues of object representation and to survey data from primate physiology and human fMRI that bear on those issues. Topics include the computational problems of object representation, the nature of object representations in the brain, the tolerance and selectivity of those representations, and the effects of attention and learning.

Subjects

vision | vision | object recognition | object recognition | monkey versus human | monkey versus human | object representations | object representations | fMRI | fMRI | temporal lobe | temporal lobe | visual cortex | visual cortex | neuronal representations | neuronal representations | neurophysiology | neurophysiology | retinal image | retinal image | pattern recognition | pattern recognition | perceptual awareness | perceptual awareness

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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18.304 Undergraduate Seminar in Discrete Mathematics (MIT) 18.304 Undergraduate Seminar in Discrete Mathematics (MIT)

Description

This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic. This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic.

Subjects

discrete math | discrete math | discrete mathematics | discrete mathematics | presentations | presentations | student presentations | student presentations | oral communication | oral communication | combinatorics | combinatorics | graph theory | graph theory | Proofs from the Book | Proofs from the Book | mathematics communication | mathematics communication

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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18.712 Introduction to Representation Theory (MIT) 18.712 Introduction to Representation Theory (MIT)

Description

This is a new course, whose goal is to give an undergraduate-level introduction to representation theory (of groups, Lie algebras, and associative algebras). Representation theory is an area of mathematics which, roughly speaking, studies symmetry in linear spaces. This is a new course, whose goal is to give an undergraduate-level introduction to representation theory (of groups, Lie algebras, and associative algebras). Representation theory is an area of mathematics which, roughly speaking, studies symmetry in linear spaces.

Subjects

finite dimensional algebras | finite dimensional algebras | Quiver Representations | Quiver Representations | series Representations | series Representations | finite groups | finite groups | representation theory | representation theory | Lie algebras | Lie algebras | Tensor products | Tensor products | density theorem | density theorem | Jordan-H?older theorem | Jordan-H?older theorem | Krull-Schmidt theorem | Krull-Schmidt theorem | Maschke?s Theorem | Maschke?s Theorem | Frobenius-Schur indicator | Frobenius-Schur indicator | Frobenius divisibility | Frobenius divisibility | Burnside?s Theorem | Burnside?s Theorem

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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6.003 Signals and Systems (MIT)

Description

6.003 covers the fundamentals of signal and system analysis, focusing on representations of discrete-time and continuous-time signals (singularity functions, complex exponentials and geometrics, Fourier representations, Laplace and Z transforms, sampling) and representations of linear, time-invariant systems (difference and differential equations, block diagrams, system functions, poles and zeros, convolution, impulse and step responses, frequency responses). Applications are drawn broadly from engineering and physics, including feedback and control, communications, and signal processing.

Subjects

signal and system analysis | representations of discrete-time and continuous-time signals | representations of linear time-invariant systems | Fourier representations | Laplace and Z transforms | sampling | difference and differential equations | feedback and control | communications | signal processing

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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24.901 Language and its Structure I: Phonology (MIT) 24.901 Language and its Structure I: Phonology (MIT)

Description

24.901 is designed to give you a preliminary understanding of how the sound systems of different languages are structured, how and why they may differ from each other. The course also aims to provide you with analytical tools in phonology, enough to allow you to sketch the analysis of an entire phonological system by the end of the term. On a non-linguistic level, the couse aims to teach you by example the virtues of formulating precise and explicit descriptive statements; and to develop your skills in making and evaluating arguments. 24.901 is designed to give you a preliminary understanding of how the sound systems of different languages are structured, how and why they may differ from each other. The course also aims to provide you with analytical tools in phonology, enough to allow you to sketch the analysis of an entire phonological system by the end of the term. On a non-linguistic level, the couse aims to teach you by example the virtues of formulating precise and explicit descriptive statements; and to develop your skills in making and evaluating arguments.

Subjects

fundamental concepts | fundamental concepts | phonological theory | phonological theory | philosophy | philosophy | cognitive psychology | cognitive psychology | articulatory phonetics | articulatory phonetics | acoustic phonetics | acoustic phonetics | feature systems | feature systems | underlying representations | underlying representations | underspecification | underspecification | phonological rules | phonological rules | phonological derivations | phonological derivations | syllable structure | syllable structure | accentual systems | accentual systems | morphology-phonology interface | morphology-phonology interface

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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18.304 Undergraduate Seminar in Discrete Mathematics (MIT)

Description

This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic.

Subjects

discrete math | discrete mathematics | presentations | student presentations | oral communication | combinatorics | graph theory | Proofs from the Book | mathematics communication

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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7.18 Topics in Experimental Biology (MIT) 7.18 Topics in Experimental Biology (MIT)

Description

This independent experimental study course is designed to allow students with a strong interest in independent research to fulfill the project laboratory requirement for the Biology Department Program in the context of a research laboratory at MIT. The research should be a continuation of a previous project under the direction of a member of the Biology Department faculty. This course provides instruction and practice in written and oral communication. Journal club discussions are used to help students evaluate and write scientific papers. This independent experimental study course is designed to allow students with a strong interest in independent research to fulfill the project laboratory requirement for the Biology Department Program in the context of a research laboratory at MIT. The research should be a continuation of a previous project under the direction of a member of the Biology Department faculty. This course provides instruction and practice in written and oral communication. Journal club discussions are used to help students evaluate and write scientific papers.

Subjects

experimental biology | experimental biology | journal club | journal club | primary literature | primary literature | scientific research | scientific research | oral presentations | oral presentations | communication | communication | abstracts | abstracts | materials and methods | materials and methods | discussion | discussion | IMRAD | IMRAD | research report | research report | laboratory research | laboratory research | results section | results section

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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4.301 Introduction to the Visual Arts (MIT) 4.301 Introduction to the Visual Arts (MIT)

Description

This class will introduce students to a variety of contemporary art practices and ideas. The class will begin with a brief overview of 'visual language' by looking at a variety of artworks and discussing basic concepts revolving around artistic practice. The rest of the class will focus on notions of the real/unreal as explored with various mediums and practices. The class will work in video, sculpture and in public space. This class will introduce students to a variety of contemporary art practices and ideas. The class will begin with a brief overview of 'visual language' by looking at a variety of artworks and discussing basic concepts revolving around artistic practice. The rest of the class will focus on notions of the real/unreal as explored with various mediums and practices. The class will work in video, sculpture and in public space.

Subjects

visual art practice | visual art practice | critical analysis | critical analysis | long-range artistic development | long-range artistic development | two-dimensional | two-dimensional | three-dimensional | three-dimensional | time-based media | time-based media | installations | installations | performance and video | performance and video | visiting artist presentations | visiting artist presentations | field trips | field trips | studio practice | studio practice | aesthetic analyses | aesthetic analyses | modern art | modern art | art history | art history | body | body | phenomenology | phenomenology | personal space | personal space | installation | installation

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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3.042 Materials Project Laboratory (MIT) 3.042 Materials Project Laboratory (MIT)

Description

As its name implies, the 3.042 Materials Project Laboratory involves working with such operations as investment casting of metals, injection molding of polymers, and sintering of ceramics. After all the abstraction and theory in the lecture part of the DMSE curriculum, many students have found this hands-on experience with materials to be very fun stuff - several have said that 3.042/3.082 was their favorite DMSE subject. The lab is more than operating processing equipment, however. It is intended also to emulate professional practice in materials engineering project management, with aspects of design, analysis, teamwork, literature and patent searching, Web creation and oral presentation, and more. As its name implies, the 3.042 Materials Project Laboratory involves working with such operations as investment casting of metals, injection molding of polymers, and sintering of ceramics. After all the abstraction and theory in the lecture part of the DMSE curriculum, many students have found this hands-on experience with materials to be very fun stuff - several have said that 3.042/3.082 was their favorite DMSE subject. The lab is more than operating processing equipment, however. It is intended also to emulate professional practice in materials engineering project management, with aspects of design, analysis, teamwork, literature and patent searching, Web creation and oral presentation, and more.

Subjects

Student project teams design and fabricate a materials engineering prototype using processing technologies (injection molding | Student project teams design and fabricate a materials engineering prototype using processing technologies (injection molding | thermoforming | thermoforming | investment casting | investment casting | powder processing | powder processing | three-dimensional printing | three-dimensional printing | physical vapor deposition | physical vapor deposition | etc.) appropriate for the materials and device of interest. Goals include using MSE fundamentals in a practical application; understanding trade-offs between design | etc.) appropriate for the materials and device of interest. Goals include using MSE fundamentals in a practical application; understanding trade-offs between design | processing and performance; and fabrication of a deliverable prototype. Emphasis on teamwork | processing and performance; and fabrication of a deliverable prototype. Emphasis on teamwork | project management | project management | communications and computer skills | communications and computer skills | and hands-on work using student and MIT laboratory shops. Teams document their progress and final results by means of web pages and weekly oral presentations. Instruction and practice in oral communication provided. | and hands-on work using student and MIT laboratory shops. Teams document their progress and final results by means of web pages and weekly oral presentations. Instruction and practice in oral communication provided.

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4.303 The Production of Space: Art, Architecture and Urbanism in Dialogue (MIT) 4.303 The Production of Space: Art, Architecture and Urbanism in Dialogue (MIT)

Description

Includes audio/video content: AV selected lectures. This seminar engages in the notion of space from various points of departure. The goal is first of all to engage in the term and secondly to examine possibilities of art, architecture within urban settings in order to produce what is your interpretation of space. Includes audio/video content: AV selected lectures. This seminar engages in the notion of space from various points of departure. The goal is first of all to engage in the term and secondly to examine possibilities of art, architecture within urban settings in order to produce what is your interpretation of space.

Subjects

architecture | architecture | urbanisml gender | urbanisml gender | space | space | visual art practice | visual art practice | critical analysis | critical analysis | long-range artistic development | long-range artistic development | two-dimensional | two-dimensional | three-dimensional | three-dimensional | time-based media | time-based media | installations | installations | performance and video | performance and video | visiting artist presentations | visiting artist presentations | field trips | field trips | studio practice | studio practice | aesthetic analyses | aesthetic analyses | modern art | modern art | art history | art history | body | body | phenomenology | phenomenology | personal space | personal space | installation | installation

License

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4.367 Studio Seminar in Public Art (MIT) 4.367 Studio Seminar in Public Art (MIT)

Description

Includes audio/video content: AV special element video. How do we define Public Art? This course focuses on the production of projects for public places. Public Art is a concept that is in constant discussion and revision, as much as the evolution and transformation of public spaces and cities are. Monuments are repositories of memory and historical presences with the expectation of being permanent. Public interventions are created not to impose and be temporary, but as forms intended to activate discourse and discussion. Considering the concept of a museum as a public device and how they are searching for new ways of avoiding generic identities, we will deal with the concept of the personal imaginary museum. It should be considered as a point of departure to propose a personal individual Includes audio/video content: AV special element video. How do we define Public Art? This course focuses on the production of projects for public places. Public Art is a concept that is in constant discussion and revision, as much as the evolution and transformation of public spaces and cities are. Monuments are repositories of memory and historical presences with the expectation of being permanent. Public interventions are created not to impose and be temporary, but as forms intended to activate discourse and discussion. Considering the concept of a museum as a public device and how they are searching for new ways of avoiding generic identities, we will deal with the concept of the personal imaginary museum. It should be considered as a point of departure to propose a personal individual

Subjects

cities | cities | urbanism | urbanism | artists | artists | architects | architects | collaboration | collaboration | translation | translation | revitalization | revitalization | urban space | urban space | redistricting | redistricting | planned cities | planned cities | development | development | ground zero | ground zero | blank slate | blank slate | interventions | interventions | visual art practice | visual art practice | critical analysis | critical analysis | long-range artistic development | long-range artistic development | two-dimensional | two-dimensional | three-dimensional | three-dimensional | time-based media | time-based media | installations | installations | performance and video | performance and video | visiting artist presentations | visiting artist presentations | field trips | field trips | studio practice | studio practice | aesthetic analyses | aesthetic analyses | modern art | modern art | art history | art history | body | body | phenomenology | phenomenology | personal space | personal space | installation | installation

License

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4.370 Interrogative Design Workshop (MIT) 4.370 Interrogative Design Workshop (MIT)

Description

Includes audio/video content: AV selected lectures. "Parrhesia" was an Athenian right to frank and open speaking, the right that, like the First Amendment, demands a "fearless speaker" who must challenge political powers with criticism and unsolicited advice. Can designer and artist respond today to such a democratic call and demand? Is it possible to do so despite the (increasing) restrictions imposed on our liberties today? Can the designer or public artist operate as a proactive "parrhesiatic" agent and contribute to the protection, development and dissemination of "fearless speaking" in Public Space? Includes audio/video content: AV selected lectures. "Parrhesia" was an Athenian right to frank and open speaking, the right that, like the First Amendment, demands a "fearless speaker" who must challenge political powers with criticism and unsolicited advice. Can designer and artist respond today to such a democratic call and demand? Is it possible to do so despite the (increasing) restrictions imposed on our liberties today? Can the designer or public artist operate as a proactive "parrhesiatic" agent and contribute to the protection, development and dissemination of "fearless speaking" in Public Space?

Subjects

visual art practice | visual art practice | critical analysis | critical analysis | long-range artistic development | long-range artistic development | two-dimensional | two-dimensional | three-dimensional | three-dimensional | time-based media | time-based media | installations | installations | performance and video | performance and video | visiting artist presentations | visiting artist presentations | field trips | field trips | studio practice | studio practice | aesthetic analyses | aesthetic analyses | modern art | modern art | art history | art history | body | body | phenomenology | phenomenology | personal space | personal space | installation | installation

License

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SP.240 Composing Your Life: Exploration of Self through Visual Arts and Writing (MIT) SP.240 Composing Your Life: Exploration of Self through Visual Arts and Writing (MIT)

Description

In this interdisciplinary seminar, we explore a variety of visual and written tools for self exploration and self expression. Through discussion, written assignments, and directed exercises, students practice utilizing a variety of media to explore and express who they are. In this interdisciplinary seminar, we explore a variety of visual and written tools for self exploration and self expression. Through discussion, written assignments, and directed exercises, students practice utilizing a variety of media to explore and express who they are.

Subjects

self-exploration | self-exploration | self-expression | self-expression | photography | photography | representations of self | representations of self | family history | family history | race | race | gender | gender | personal values | personal values | letters | letters | emails | emails | blogs | blogs | journals | journals | poetry | poetry | memoirs | memoirs | autobiographies | autobiographies | self-portraiture | self-portraiture | narrative | narrative | ESG.SP240 | ESG.SP240

License

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8.05 Quantum Physics II (MIT) 8.05 Quantum Physics II (MIT)

Description

This course, along with the next course in this sequence (8.06, Quantum Physics III) in a two-course sequence covering quantum physics with applications drawn from modern physics. General formalism of quantum mechanics: states, operators, Dirac notation, representations, measurement theory. Harmonic oscillator: operator algebra, states. Quantum mechanics in three-dimensions: central potentials and the radial equation, bound and scattering states, qualitative analysis of wavefunctions. Angular momentum: operators, commutator algebra, eigenvalues and eigenstates, spherical harmonics. Spin: Stern-Gerlach devices and measurements, nuclear magnetic resonance, spin and statistics. Addition of angular momentum: Clebsch-Gordan series and coefficients, spin systems, and allotropic forms of hydrogen This course, along with the next course in this sequence (8.06, Quantum Physics III) in a two-course sequence covering quantum physics with applications drawn from modern physics. General formalism of quantum mechanics: states, operators, Dirac notation, representations, measurement theory. Harmonic oscillator: operator algebra, states. Quantum mechanics in three-dimensions: central potentials and the radial equation, bound and scattering states, qualitative analysis of wavefunctions. Angular momentum: operators, commutator algebra, eigenvalues and eigenstates, spherical harmonics. Spin: Stern-Gerlach devices and measurements, nuclear magnetic resonance, spin and statistics. Addition of angular momentum: Clebsch-Gordan series and coefficients, spin systems, and allotropic forms of hydrogen

Subjects

General formalism of quantum mechanics: states | General formalism of quantum mechanics: states | operators | operators | Dirac notation | Dirac notation | representations | representations | measurement theory | measurement theory | Harmonic oscillator: operator algebra | Harmonic oscillator: operator algebra | states | states | Quantum mechanics in three-dimensions: central potentials and the radial equation | Quantum mechanics in three-dimensions: central potentials and the radial equation | bound and scattering states | bound and scattering states | qualitative analysis of wavefunctions | qualitative analysis of wavefunctions | Angular momentum: operators | Angular momentum: operators | commutator algebra | commutator algebra | eigenvalues and eigenstates | eigenvalues and eigenstates | spherical harmonics | spherical harmonics | Spin: Stern-Gerlach devices and measurements | Spin: Stern-Gerlach devices and measurements | nuclear magnetic resonance | nuclear magnetic resonance | spin and statistics | spin and statistics | Addition of angular momentum: Clebsch-Gordan series and coefficients | Addition of angular momentum: Clebsch-Gordan series and coefficients | spin systems | spin systems | allotropic forms of hydrogen | allotropic forms of hydrogen | Angular momentum | Angular momentum | Harmonic oscillator | Harmonic oscillator | operator algebra | operator algebra | Spin | Spin | Stern-Gerlach devices and measurements | Stern-Gerlach devices and measurements | central potentials and the radial equation | central potentials and the radial equation | Clebsch-Gordan series and coefficients | Clebsch-Gordan series and coefficients | quantum physics | quantum physics

License

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17.869 Political Science Scope and Methods (MIT) 17.869 Political Science Scope and Methods (MIT)

Description

This course is designed to provide an introduction to a variety of empirical research methods used by political scientists. The primary aims of the course are to make you a more sophisticated consumer of diverse empirical research and to allow you to conduct sophisticated independent work in your junior and senior years. This is not a course in data analysis. Rather, it is a course on how to approach political science research. This course is designed to provide an introduction to a variety of empirical research methods used by political scientists. The primary aims of the course are to make you a more sophisticated consumer of diverse empirical research and to allow you to conduct sophisticated independent work in your junior and senior years. This is not a course in data analysis. Rather, it is a course on how to approach political science research.

Subjects

political science | political science | empirical research | empirical research | scientific method | scientific method | research design | research design | models | models | samping | samping | statistical analysis | statistical analysis | measurement | measurement | ethics | ethics | empirical | empirical | research | research | scientific | scientific | methods | methods | statistics | statistics | statistical | statistical | analysis | analysis | political | political | politics | politics | science | science | design | design | sampling | sampling | theoretical | theoretical | observation | observation | data | data | case studies | case studies | cases | cases | empirical research methods | empirical research methods | political scientists | political scientists | empirical analysis | empirical analysis | theoretical analysis | theoretical analysis | research projects | research projects | department faculty | department faculty | inference | inference | writing | writing | revision | revision | oral presentations | oral presentations | experimental method | experimental method | theories | theories | political implications | political implications

License

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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.

Subjects

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

License

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18.712 Introduction to Representation Theory (MIT)

Description

This is a new course, whose goal is to give an undergraduate-level introduction to representation theory (of groups, Lie algebras, and associative algebras). Representation theory is an area of mathematics which, roughly speaking, studies symmetry in linear spaces.

Subjects

finite dimensional algebras | Quiver Representations | series Representations | finite groups | representation theory | Lie algebras | Tensor products | density theorem | Jordan-H?older theorem | Krull-Schmidt theorem | Maschke?s Theorem | Frobenius-Schur indicator | Frobenius divisibility | Burnside?s Theorem

License

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