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

Description

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

Subjects

mathematics | mathematics | geometry | geometry | history | history | philosophy | philosophy | Greek philosophy | Greek philosophy | Plato | Plato | Euclid | Euclid | Aristotle | Aristotle | Rene Descartes | Rene Descartes | Nicomachus | Nicomachus | Francis Bacon | Francis Bacon | number | number | irrational number | irrational number | ratio | ratio | ethics | ethics | logos | logos | logic | logic | ancient knowing | ancient knowing | modern knowing | modern knowing | Greek conception of number | Greek conception of number | idea of number | idea of number | courage | courage | justice | justice | pursuit of truth | pursuit of truth | truth as a surd | truth as a surd

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

Description

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

Subjects

mathematics | mathematics | geometry | geometry | history | history | philosophy | philosophy | Greek philosophy | Greek philosophy | Plato | Plato | Euclid | Euclid | Aristotle | Aristotle | Rene Descartes | Rene Descartes | Nicomachus | Nicomachus | Francis Bacon | Francis Bacon | number | number | irrational number | irrational number | ratio | ratio | ethics | ethics | logos | logos | logic | logic | ancient knowing | ancient knowing | modern knowing | modern knowing | Greek conception of number | Greek conception of number | idea of number | idea of number | courage | courage | justice | justice | pursuit of truth | pursuit of truth | truth as a surd | truth as a surd

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

Description

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

Subjects

mathematics | mathematics | geometry | geometry | history | history | philosophy | philosophy | Greek philosophy | Greek philosophy | Plato | Plato | Euclid | Euclid | Aristotle | Aristotle | Rene Descartes | Rene Descartes | Nicomachus | Nicomachus | Francis Bacon | Francis Bacon | number | number | irrational number | irrational number | ratio | ratio | ethics | ethics | logos | logos | logic | logic | ancient knowing | ancient knowing | modern knowing | modern knowing | Greek conception of number | Greek conception of number | idea of number | idea of number | courage | courage | justice | justice | pursuit of truth | pursuit of truth | truth as a surd | truth as a surd

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

Description

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

Subjects

mathematics | mathematics | geometry | geometry | history | history | philosophy | philosophy | Greek philosophy | Greek philosophy | Plato | Plato | Euclid | Euclid | Aristotle | Aristotle | Rene Descartes | Rene Descartes | Nicomachus | Nicomachus | Francis Bacon | Francis Bacon | number | number | irrational number | irrational number | ratio | ratio | ethics | ethics | logos | logos | logic | logic | ancient knowing | ancient knowing | modern knowing | modern knowing | Greek conception of number | Greek conception of number | idea of number | idea of number | courage | courage | justice | justice | pursuit of truth | pursuit of truth | truth as a surd | truth as a surd

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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18.785 Analytic Number Theory (MIT) 18.785 Analytic Number Theory (MIT)

Description

This course is an introduction to analytic number theory, including the use of zeta functions and L-functions to prove distribution results concerning prime numbers (e.g., the prime number theorem in arithmetic progressions). This course is an introduction to analytic number theory, including the use of zeta functions and L-functions to prove distribution results concerning prime numbers (e.g., the prime number theorem in arithmetic progressions).

Subjects

analytic number theory | analytic number theory | Riemann zeta function | Riemann zeta function | L-functions | L-functions | prime number theorem | prime number theorem | Dirichlet's theorem | Dirichlet's theorem | Riemann Hypothesis | Riemann Hypothesis | Sieving methods | Sieving methods | Linnik | Linnik | Linnik's large sieve | Linnik's large sieve | Selberg | Selberg | Selberg's sieve | Selberg's sieve | distribution of prime numbers | distribution of prime numbers

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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18.786 Topics in Algebraic Number Theory (MIT) 18.786 Topics in Algebraic Number Theory (MIT)

Description

This course is a first course in algebraic number theory. Topics to be covered include number fields, class numbers, Dirichlet's units theorem, cyclotomic fields, local fields, valuations, decomposition and inertia groups, ramification, basic analytic methods, and basic class field theory. An additional theme running throughout the course will be the use of computer algebra to investigate number-theoretic questions; this theme will appear primarily in the problem sets. This course is a first course in algebraic number theory. Topics to be covered include number fields, class numbers, Dirichlet's units theorem, cyclotomic fields, local fields, valuations, decomposition and inertia groups, ramification, basic analytic methods, and basic class field theory. An additional theme running throughout the course will be the use of computer algebra to investigate number-theoretic questions; this theme will appear primarily in the problem sets.

Subjects

algebraic number theory | algebraic number theory | number fields | number fields | class numbers | class numbers | Dirichlet's units theorem | Dirichlet's units theorem | cyclotomic fields | cyclotomic fields | local fields | local fields | valuations | valuations | decomposition and inertia groups | decomposition and inertia groups | ramification | ramification | basic analytic methods | basic analytic methods | basic class field theory | basic class field theory

License

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

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16.100 Aerodynamics (MIT) 16.100 Aerodynamics (MIT)

Description

This course extends fluid mechanic concepts from Unified Engineering to the aerodynamic performance of wings and bodies in sub/supersonic regimes. 16.100 generally has four components: subsonic potential flows, including source/vortex panel methods; viscous flows, including laminar and turbulent boundary layers; aerodynamics of airfoils and wings, including thin airfoil theory, lifting line theory, and panel method/interacting boundary layer methods; and supersonic and hypersonic airfoil theory. Course material varies each year depending upon the focus of the design problem. Technical RequirementsFile decompression software, such as Winzip® or StuffIt®, is required to open the .tar files found on this course site. MATLAB&#1 This course extends fluid mechanic concepts from Unified Engineering to the aerodynamic performance of wings and bodies in sub/supersonic regimes. 16.100 generally has four components: subsonic potential flows, including source/vortex panel methods; viscous flows, including laminar and turbulent boundary layers; aerodynamics of airfoils and wings, including thin airfoil theory, lifting line theory, and panel method/interacting boundary layer methods; and supersonic and hypersonic airfoil theory. Course material varies each year depending upon the focus of the design problem. Technical RequirementsFile decompression software, such as Winzip® or StuffIt®, is required to open the .tar files found on this course site. MATLAB&#1

Subjects

aerodynamics | aerodynamics | airflow | airflow | air | air | body | body | aircraft | aircraft | aerodynamic modes | aerodynamic modes | aero | aero | forces | forces | flow | flow | computational | computational | CFD | CFD | aerodynamic analysis | aerodynamic analysis | lift | lift | drag | drag | potential flows | potential flows | imcompressible | imcompressible | supersonic | supersonic | subsonic | subsonic | panel method | panel method | vortex lattice method | vortex lattice method | boudary layer | boudary layer | transition | transition | turbulence | turbulence | inviscid | inviscid | viscous | viscous | euler | euler | navier-stokes | navier-stokes | wind tunnel | wind tunnel | flow similarity | flow similarity | non-dimensional | non-dimensional | mach number | mach number | reynolds number | reynolds number | integral momentum | integral momentum | airfoil | airfoil | wing | wing | stall | stall | friction drag | friction drag | induced drag | induced drag | wave drag | wave drag | pressure drag | pressure drag | fluid element | fluid element | shear strain | shear strain | normal strain | normal strain | vorticity | vorticity | divergence | divergence | substantial derviative | substantial derviative | laminar | laminar | displacement thickness | displacement thickness | momentum thickness | momentum thickness | skin friction | skin friction | separation | separation | velocity profile | velocity profile | 2-d panel | 2-d panel | 3-d vortex | 3-d vortex | thin airfoil | thin airfoil | lifting line | lifting line | aspect ratio | aspect ratio | twist | twist | camber | camber | wing loading | wing loading | roll moments | roll moments | finite volume approximation | finite volume approximation | shocks | shocks | expansion fans | expansion fans | shock-expansion theory | shock-expansion theory | transonic | transonic | critical mach number | critical mach number | wing sweep | wing sweep | Kutta condition | Kutta condition | team project | team project | blended-wing-body | blended-wing-body | computational fluid dynamics | computational fluid dynamics | Incompressible | Incompressible

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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16.100 Aerodynamics (MIT) 16.100 Aerodynamics (MIT)

Description

This course extends fluid mechanic concepts from Unified Engineering to the aerodynamic performance of wings and bodies in sub/supersonic regimes. 16.100 generally has four components: subsonic potential flows, including source/vortex panel methods; viscous flows, including laminar and turbulent boundary layers; aerodynamics of airfoils and wings, including thin airfoil theory, lifting line theory, and panel method/interacting boundary layer methods; and supersonic and hypersonic airfoil theory. Course material varies each year depending upon the focus of the design problem. This course extends fluid mechanic concepts from Unified Engineering to the aerodynamic performance of wings and bodies in sub/supersonic regimes. 16.100 generally has four components: subsonic potential flows, including source/vortex panel methods; viscous flows, including laminar and turbulent boundary layers; aerodynamics of airfoils and wings, including thin airfoil theory, lifting line theory, and panel method/interacting boundary layer methods; and supersonic and hypersonic airfoil theory. Course material varies each year depending upon the focus of the design problem.

Subjects

aerodynamics | aerodynamics | airflow | airflow | air | air | body | body | aircraft | aircraft | aerodynamic modes | aerodynamic modes | aero | aero | forces | forces | flow | flow | computational | computational | CFD | CFD | aerodynamic analysis | aerodynamic analysis | lift | lift | drag | drag | potential flows | potential flows | imcompressible | imcompressible | supersonic | supersonic | subsonic | subsonic | panel method | panel method | vortex lattice method | vortex lattice method | boudary layer | boudary layer | transition | transition | turbulence | turbulence | inviscid | inviscid | viscous | viscous | euler | euler | navier-stokes | navier-stokes | wind tunnel | wind tunnel | flow similarity | flow similarity | non-dimensional | non-dimensional | mach number | mach number | reynolds number | reynolds number | integral momentum | integral momentum | airfoil | airfoil | wing | wing | stall | stall | friction drag | friction drag | induced drag | induced drag | wave drag | wave drag | pressure drag | pressure drag | fluid element | fluid element | shear strain | shear strain | normal strain | normal strain | vorticity | vorticity | divergence | divergence | substantial derivative | substantial derivative | laminar | laminar | displacement thickness | displacement thickness | momentum thickness | momentum thickness | skin friction | skin friction | separation | separation | velocity profile | velocity profile | 2-d panel | 2-d panel | 3-d vortex | 3-d vortex | thin airfoil | thin airfoil | lifting line | lifting line | aspect ratio | aspect ratio | twist | twist | camber | camber | wing loading | wing loading | roll moments | roll moments | finite volume approximation | finite volume approximation | shocks | shocks | expansion fans | expansion fans | shock-expansion theory | shock-expansion theory | transonic | transonic | critical mach number | critical mach number | wing sweep | wing sweep | Kutta condition | Kutta condition | team project | team project | blended-wing-body | blended-wing-body | computational fluid dynamics | computational fluid 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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Collecting scrap at Hetton Station Goods Yard Collecting scrap at Hetton Station Goods Yard

Description

Subjects

road | road | roof | roof | sky | sky | abstract | abstract | blur | blur | industry | industry | wheel | wheel | metal | metal | stone | stone | wall | wall | shirt | shirt | yard | yard | standing | standing | fence | fence | buildings | buildings | 1974 | 1974 | interesting | interesting | workers | workers | industrial | industrial | carriage | carriage | unitedkingdom | unitedkingdom | path | path | timber | timber | mark | mark | coat | coat | debris | debris | caps | caps | grain | grain | plate | plate | ground | ground | social | social | number | number | soil | soil | cap | cap | transportation | transportation | signage | signage | bolt | bolt | archives | archives | land | land | letter | letter | vehicle | vehicle | trousers | trousers | unusual | unusual | telegraphpole | telegraphpole | scrap | scrap | railways | railways | crease | crease | flap | flap | attentive | attentive | slope | slope | collecting | collecting | numberplate | numberplate | fascinating | fascinating | digitalimage | digitalimage | sunderland | sunderland | scrapmetal | scrapmetal | citycouncil | citycouncil | 1895 | 1895 | blackandwhitephotograph | blackandwhitephotograph | northeastofengland | northeastofengland | goodsyard | goodsyard | moorsley | moorsley | mid20thcentury | mid20thcentury | eastrainton | eastrainton | hettonlehole | hettonlehole | easingtonlane | easingtonlane | hettondowns | hettondowns | hettonurbandistrictcouncil | hettonurbandistrictcouncil | hettonstationgoodsyard | hettonstationgoodsyard | hettonleholeurbandistrict | hettonleholeurbandistrict | sunderlandmetropolitanborough | sunderlandmetropolitanborough | localgovernmentact1894 | localgovernmentact1894

License

No known copyright restrictions

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ES.2H3 Ancient Philosophy and Mathematics (MIT)

Description

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

Subjects

mathematics | geometry | history | philosophy | Greek philosophy | Plato | Euclid | Aristotle | Rene Descartes | Nicomachus | Francis Bacon | number | irrational number | ratio | ethics | logos | logic | ancient knowing | modern knowing | Greek conception of number | idea of number | courage | justice | pursuit of truth | truth as a surd

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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Tom had been warned about drinking and driving Tom had been warned about drinking and driving

Description

Subjects

show | show | road | road | door | door | roof | roof | shadow | shadow | sky | sky | horses | horses | people | people | abstract | abstract | brick | brick | industry | industry | window | window | car | car | wall | wall | shirt | shirt | buildings | buildings | hair | hair | daylight | daylight | bottle | bottle | interesting | interesting | construction | construction | stair | stair | industrial | industrial | driving | driving | carriage | carriage | display | display | label | label | flag | flag | bald | bald | tie | tie | social | social | rope | rope | parade | parade | number | number | celebration | celebration | event | event | frame | frame | gathering | gathering | archives | archives | vehicle | vehicle | driver | driver | 1960s | 1960s | unusual | unusual | striking | striking | wacky | wacky | occasion | occasion | spectator | spectator | tyneside | tyneside | crease | crease | development | development | crowds | crowds | numberplate | numberplate | bizzare | bizzare | newcastleupontyne | newcastleupontyne | fascinating | fascinating | digitalimage | digitalimage | factories | factories | beerbottle | beerbottle | rivertyne | rivertyne | manufacturing | manufacturing | showpiece | showpiece | northeastengland | northeastengland | grandparade | grandparade | blackandwhitephotograph | blackandwhitephotograph | scotswoodroad | scotswoodroad | coachwork | coachwork | lordarmstrong | lordarmstrong | vickersarmstrong | vickersarmstrong | elswickworks | elswickworks | williamgeorgearmstrong | williamgeorgearmstrong | workshopoftheworld | workshopoftheworld | blaydonraces | blaydonraces | scotswoodworks | scotswoodworks | centenaryoftheblaydonraces | centenaryoftheblaydonraces | 9june1962 | 9june1962

License

No known copyright restrictions

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Prime Numbers

Description

Dr Richard Earl of the Mathematical Institute, Oxford presents a talk about prime numbers. What they are and their role in internet security. Wales; http://creativecommons.org/licenses/by-nc-sa/2.0/uk/

Subjects

maths | numbers | prime numbers | Internet security | maths | numbers | prime numbers | Internet security | 2013-12-09

License

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

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Prime Numbers

Description

Dr Richard Earl of the Mathematical Institute, Oxford presents a talk about prime numbers. What they are and their role in internet security. Wales; http://creativecommons.org/licenses/by-nc-sa/2.0/uk/

Subjects

maths | numbers | prime numbers | Internet security | maths | numbers | prime numbers | Internet security | 2013-12-09

License

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

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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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18.104 Seminar in Analysis: Applications to Number Theory (MIT) 18.104 Seminar in Analysis: Applications to Number Theory (MIT)

Description

18.104 is an undergraduate level seminar for mathematics majors. Students present and discuss subject matter taken from current journals or books. Instruction and practice in written and oral communication is provided. The topics vary from year to year. The topic for this term is Applications to Number Theory. 18.104 is an undergraduate level seminar for mathematics majors. Students present and discuss subject matter taken from current journals or books. Instruction and practice in written and oral communication is provided. The topics vary from year to year. The topic for this term is Applications to Number Theory.

Subjects

Infinitude of the primes | Infinitude of the primes | Summing powers of integers | Summing powers of integers | Bernoulli polynomials | Bernoulli polynomials | sine product formula | sine product formula | $\zeta(2n)$ | $\zeta(2n)$ | Fermat's Little Theorem | Fermat's Little Theorem | Fermat's Great Theorem | Fermat's Great Theorem | Averages of arithmetic functions | Averages of arithmetic functions | arithmetic-geometric mean | arithmetic-geometric mean | Gauss' theorem | Gauss' theorem | Wallis's formula | Wallis's formula | Stirling's formula | Stirling's formula | prime number theorem | prime number theorem | Riemann's hypothesis | Riemann's hypothesis | Euler's proof of infinitude of primes | Euler's proof of infinitude of primes | Density of prime numbers | Density of prime numbers | Euclidean algorithm | Euclidean algorithm | Golden Ratio | Golden Ratio

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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18.409 Behavior of Algorithms (MIT) 18.409 Behavior of Algorithms (MIT)

Description

This course is a study of Behavior of Algorithms and covers an area of current interest in theoretical computer science. The topics vary from term to term. During this term, we discuss rigorous approaches to explaining the typical performance of algorithms with a focus on the following approaches: smoothed analysis, condition numbers/parametric analysis, and subclassing inputs. This course is a study of Behavior of Algorithms and covers an area of current interest in theoretical computer science. The topics vary from term to term. During this term, we discuss rigorous approaches to explaining the typical performance of algorithms with a focus on the following approaches: smoothed analysis, condition numbers/parametric analysis, and subclassing inputs.

Subjects

Condition number | Condition number | largest singluar value of a matrix | largest singluar value of a matrix | Smoothed analysis | Smoothed analysis | Gaussian elimination | Gaussian elimination | Growth factors of partial and complete pivoting | Growth factors of partial and complete pivoting | GE of graphs with low bandwidth or small separators | GE of graphs with low bandwidth or small separators | Spectral Partitioning of planar graphs | Spectral Partitioning of planar graphs | spectral paritioning of well-shaped meshes | spectral paritioning of well-shaped meshes | spectral paritioning of nearest neighbor graphs | spectral paritioning of nearest neighbor graphs | Turner's theorem | Turner's theorem | bandwidth of semi-random graphs. | bandwidth of semi-random graphs. | McSherry's spectral bisection algorithm | McSherry's spectral bisection algorithm | Linear Programming | Linear Programming | von Neumann's algorithm | von Neumann's algorithm | primal and dual simplex methods | and duality Strong duality theorem | primal and dual simplex methods | and duality Strong duality theorem | Renegar's condition numbers | Renegar's condition numbers

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18.238 Geometry and Quantum Field Theory (MIT) 18.238 Geometry and Quantum Field Theory (MIT)

Description

Geometry and Quantum Field Theory, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals. It covers the basics of classical field theory, free quantum theories and Feynman diagrams. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and String Theory. Geometry and Quantum Field Theory, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals. It covers the basics of classical field theory, free quantum theories and Feynman diagrams. The goal is to discuss, using mathematical language, a number of basic notions and results of QFT that are necessary to understand talks and papers in QFT and String Theory.

Subjects

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

License

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18.781 Theory of Numbers (MIT) 18.781 Theory of Numbers (MIT)

Description

This course provides an elementary introduction to number theory with no algebraic prerequisites. Topics include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions and elliptic curves. This course provides an elementary introduction to number theory with no algebraic prerequisites. Topics include primes, congruences, quadratic reciprocity, diophantine equations, irrational numbers, continued fractions and elliptic curves.

Subjects

number theory with no algebraic prerequisites | number theory with no algebraic prerequisites | primes | congruences | primes | congruences | quadratic reciprocity | quadratic reciprocity | diophantine equations | diophantine equations | irrational numbers | irrational numbers | continued fractions | continued fractions | partitions | partitions

License

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6 Stages of Diagnostics, Part 4 - Stage 3, Evaluate

Description

(Total video duration:05:09)

Subjects

Electronic Control Unit | Scanner | Right hand front wheel speed sensor | Read fault codes | Fault Codes | VIN number | ILRforSkills | Evaluate | Stage 3 | Motor Vehicle | ABS Fault diagnostics | motor vehicle apprenticeship | Motor Vehicle Apprentice | Electrical Diagnosis | 6 Stages of Diagnostics | UKOER | vehicle identification number | LV08 | Level 3 Diploma in Light Vehicle Maintenance and Repair Principles | registration number | Processing information | Cars | Diagnostic fault | Equipment | Automotive | Evaluating Information | ECU information | ECU | EOBD diagnostic connector | 16 pin diagnostic socket | 16 pin diagnostic connector | permanent fault | Model number | manufacturer's specification | Diagnostic equipment | VIN

License

Attribution 4.0 International Attribution 4.0 International http://creativecommons.org/licenses/by/4.0/ http://creativecommons.org/licenses/by/4.0/

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18.03 Differential Equations (MIT) 18.03 Differential Equations (MIT)

Description

Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE's) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and

Subjects

Ordinary Differential Equations | Ordinary Differential Equations | ODE | ODE | modeling physical systems | modeling physical systems | first-order ODE's | first-order ODE's | Linear ODE's | Linear ODE's | second order ODE's | second order ODE's | second order ODE's with constant coefficients | second order ODE's with constant coefficients | Undetermined coefficients | Undetermined coefficients | variation of parameters | variation of parameters | Sinusoidal signals | Sinusoidal signals | exponential signals | exponential signals | oscillations | oscillations | damping | damping | resonance | resonance | Complex numbers and exponentials | Complex numbers and exponentials | Fourier series | Fourier series | periodic solutions | periodic solutions | Delta functions | Delta functions | convolution | convolution | Laplace transform methods | Laplace transform methods | Matrix systems | Matrix systems | first order linear systems | first order linear systems | eigenvalues and eigenvectors | eigenvalues and eigenvectors | Non-linear autonomous systems | Non-linear autonomous systems | critical point analysis | critical point analysis | phase plane diagrams | phase plane diagrams | constant coefficients | constant coefficients | complex numbers | complex numbers | exponentials | exponentials | eigenvalues | eigenvalues | eigenvectors | eigenvectors

License

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3.044 Materials Processing (MIT) 3.044 Materials Processing (MIT)

Description

The goal of 3.044 is to teach cost-effective and sustainable production of solid material with a desired geometry, structure or distribution of structures, and production volume. Toward this end, it is organized around different types of phase transformations which determine the structure in various processes for making materials, in roughly increasing order of entropy change during those transformations: solid heat treatment, liquid-solid processing, fluid behavior, deformation processing, and vapor-solid processing. The course ends with several lectures that place the subject in the context of society at large. The goal of 3.044 is to teach cost-effective and sustainable production of solid material with a desired geometry, structure or distribution of structures, and production volume. Toward this end, it is organized around different types of phase transformations which determine the structure in various processes for making materials, in roughly increasing order of entropy change during those transformations: solid heat treatment, liquid-solid processing, fluid behavior, deformation processing, and vapor-solid processing. The course ends with several lectures that place the subject in the context of society at large.

Subjects

diffusion | diffusion | chemical reaction | chemical reaction | phase transformation | phase transformation | heat transport | heat transport | mass transport | mass transport | fluid | fluid | fluid flow | fluid flow | recycling | recycling | cost modeling | cost modeling | multilayer | multilayer | biot number | biot number | radiation | radiation | convection | convection | titanium | titanium | moving bodies | moving bodies | Reynolds number | Reynolds number | turbulence | turbulence | reactor | reactor | deformation | deformation | polymer | polymer | vapor transport | vapor transport | nanotechnology | nanotechnology

License

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15.099 Readings in Optimization (MIT) 15.099 Readings in Optimization (MIT)

Description

In keeping with the tradition of the last twenty-some years, the Readings in Optimization seminar will focus on an advanced topic of interest to a portion of the MIT optimization community: randomized methods for deterministic optimization. In contrast to conventional optimization algorithms whose iterates are computed and analyzed deterministically, randomized methods rely on stochastic processes and random number/vector generation as part of the algorithm and/or its analysis. In the seminar, we will study some very recent papers on this topic, many by MIT faculty, as well as some older papers from the existing literature that are only now receiving attention. In keeping with the tradition of the last twenty-some years, the Readings in Optimization seminar will focus on an advanced topic of interest to a portion of the MIT optimization community: randomized methods for deterministic optimization. In contrast to conventional optimization algorithms whose iterates are computed and analyzed deterministically, randomized methods rely on stochastic processes and random number/vector generation as part of the algorithm and/or its analysis. In the seminar, we will study some very recent papers on this topic, many by MIT faculty, as well as some older papers from the existing literature that are only now receiving attention.

Subjects

deterministic optimization; algorithms; stochastic processes; random number generation; simplex method; nonlinear; convex; complexity analysis; semidefinite programming; heuristic; global optimization; Las Vegas algorithm; randomized algorithm; linear programming; search techniques; hit and run; NP-hard; approximation | deterministic optimization; algorithms; stochastic processes; random number generation; simplex method; nonlinear; convex; complexity analysis; semidefinite programming; heuristic; global optimization; Las Vegas algorithm; randomized algorithm; linear programming; search techniques; hit and run; NP-hard; approximation | deterministic optimization | deterministic optimization | algorithms | algorithms | stochastic processes | stochastic processes | random number generation | random number generation | simplex method | simplex method | nonlinear | nonlinear | convex | convex | complexity analysis | complexity analysis | semidefinite programming | semidefinite programming | heuristic | heuristic | global optimization | global optimization | Las Vegas algorithm | Las Vegas algorithm | randomized algorithm | randomized algorithm | linear programming | linear programming | search techniques | search techniques | hit and run | hit and run | NP-hard | NP-hard | approximation | approximation

License

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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

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Falling Numbers

Description

Games aimed at those with learning difficulties or disabilities - games - Falling Numbers - Using a fraction as an operator on whole numbers.

Subjects

number games | falling numbers | whole numbers | fractions | SCIENCES and MATHEMATICS | R

License

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

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18.785 Analytic Number Theory (MIT)

Description

This course is an introduction to analytic number theory, including the use of zeta functions and L-functions to prove distribution results concerning prime numbers (e.g., the prime number theorem in arithmetic progressions).

Subjects

analytic number theory | Riemann zeta function | L-functions | prime number theorem | Dirichlet's theorem | Riemann Hypothesis | Sieving methods | Linnik | Linnik's large sieve | Selberg | Selberg's sieve | distribution of prime numbers

License

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