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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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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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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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18.781 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

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 provides an introduction to algebraic number theory. Topics covered include dedekind domains, unique factorization of prime ideals, number fields, splitting of primes, class group, lattice methods, finiteness of the class number, Dirichlet's units theorem, local fields, ramification, discriminants. This course provides an introduction to algebraic number theory. Topics covered include dedekind domains, unique factorization of prime ideals, number fields, splitting of primes, class group, lattice methods, finiteness of the class number, Dirichlet's units theorem, local fields, ramification, discriminants.

Subjects

number fields | number fields | dedekind domain | dedekind domain | prime ideal | prime ideal | class group | class group | lattice method | lattice method

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.S66 The Art of Counting (MIT) 18.S66 The Art of Counting (MIT)

Description

The subject of enumerative combinatorics deals with counting the number of elements of a finite set. For instance, the number of ways to write a positive integer n as a sum of positive integers, taking order into account, is 2n-1. We will be concerned primarily with bijective proofs, i.e., showing that two sets have the same number of elements by exhibiting a bijection (one-to-one correspondence) between them. This is a subject which requires little mathematical background to reach the frontiers of current research. Students will therefore have the opportunity to do original research. It might be necessary to limit enrollment. The subject of enumerative combinatorics deals with counting the number of elements of a finite set. For instance, the number of ways to write a positive integer n as a sum of positive integers, taking order into account, is 2n-1. We will be concerned primarily with bijective proofs, i.e., showing that two sets have the same number of elements by exhibiting a bijection (one-to-one correspondence) between them. This is a subject which requires little mathematical background to reach the frontiers of current research. Students will therefore have the opportunity to do original research. It might be necessary to limit enrollment.

Subjects

enumerative combinatorics | enumerative combinatorics | finite set | finite set | sum of positive integers | sum of positive integers | bijective proofs | bijective proofs | bijection (one-to-one correspondence) | bijection (one-to-one correspondence) | permutations | permutations | partitions | partitions | Catalan numbers | Catalan numbers | Young tableaux | Young tableaux | lattice paths and tilings | lattice paths and tilings

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)

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.

Subjects

algebraic number theory | number fields | class numbers | Dirichlet's units theorem | cyclotomic fields | local fields | valuations | decomposition and inertia groups | ramification | basic analytic methods | 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 https://ocw.mit.edu/terms/index.htm

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Praštevila in sestavljena števila Prime and composite numbers

Description

PowerPointova diaprojekcija s predstavitvijo in delitvijo števil na praštevila in sestavljena števila vsebuje tudi Eratostenovo rešeto. Presentation of prime and composite numbers, divisibility of prime numbers, searching prime numbers with sieve of Eratosthenes.

Subjects

znanstvene vede | sciences | matematika | mathematics | praštevilo | prime number | sestavljeno število | composite number | deljivost | divisibility | eratostenovo rešeto | sieve of eratosthenes

License

http://creativecommons.org/licenses/by-nc-sa/2.5/si/ http://creativecommons.org/licenses/by-nc-sa/2.5/si/

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

Description

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

Subjects

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

License

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

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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

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

Description

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

Subjects

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

License

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

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15.099 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

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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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

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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18.238 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

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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Number systems

Description

Number systems and the rules for combining numbers can be daunting. This unit will help you to understand the detail of rational and real numbers, complex numbers and integers. You will also be introduced to modular arithmetic and the concept of a relation between elements of a set.

Subjects

linear_equations | number_systems | quadratic_equations | real_numbers | mathematics and statistics | complex_numbers | Education | X000

License

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

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Complex numbers

Description

You may have met complex numbers before, but not had experience in manipulating them. This unit gives an accessible introduction to complex numbers, which are very important in science and technology, as well as mathematics. The unit includes definitions, concepts and techniques which will be very helpful and interesting to a wide variety of people with a reasonable background in algebra and trigonometry.

Subjects

imaginary_numbers | real_numbers | mathematics and statistics | complex_numbers | Education | X000

License

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

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Sustainability: the business perspective Sustainability: the business perspective

Description

The primary aim of this unit is to capture this transition and define what businesses are doing to adopt a more sustainable approach. Looking at a number of case studies, the unit will attempt to demonstrate how individual businesses are attempting to align their activities to address global sustainability challenges such as climate change and carbon reduction, energy and water scarcity and poverty reduction. The primary aim of this unit is to capture this transition and define what businesses are doing to adopt a more sustainable approach. Looking at a number of case studies, the unit will attempt to demonstrate how individual businesses are attempting to align their activities to address global sustainability challenges such as climate change and carbon reduction, energy and water scarcity and poverty reduction. There is growing recognition across business that the reductionist ‘mind set’ founded on unlimited economic growth impervious to the social and environmental impacts of commercial activities will not resolve the converging environmental, social and economic crises now faced by the global community. Ever greater numbers of Boards and CEOs are grappling with a notion of sustainability and attempting to define precisely what it means for their business. The primary aim of this unit is to capture this transition and define what businesses are doing to adopt a more sustainable approach. Looking at a number of case studies, the unit will attempt to demonstrate how individual businesses are attempting to align their activities to address global sustainability challenges such as climate chang There is growing recognition across business that the reductionist ‘mind set’ founded on unlimited economic growth impervious to the social and environmental impacts of commercial activities will not resolve the converging environmental, social and economic crises now faced by the global community. Ever greater numbers of Boards and CEOs are grappling with a notion of sustainability and attempting to define precisely what it means for their business. The primary aim of this unit is to capture this transition and define what businesses are doing to adopt a more sustainable approach. Looking at a number of case studies, the unit will attempt to demonstrate how individual businesses are attempting to align their activities to address global sustainability challenges such as climate chang

Subjects

UNow | UNow | UKOER | UKOER | Nottingham | Nottingham | Sustainability | Sustainability | Business | Business

License

Except for third party materials (materials owned by someone other than The University of Nottingham) and where otherwise indicated, the copyright in the content provided in this resource is owned by The University of Nottingham and licensed under a Creative Commons Attribution-NonCommercial-ShareAlike UK 2.0 Licence (BY-NC-SA) Except for third party materials (materials owned by someone other than The University of Nottingham) and where otherwise indicated, the copyright in the content provided in this resource is owned by The University of Nottingham and licensed under a Creative Commons Attribution-NonCommercial-ShareAlike UK 2.0 Licence (BY-NC-SA)

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Numbers, units and arithmetic Numbers, units and arithmetic

Description

Do fractions and decimals make you apprehensive about maths? Do you lack confidence in dealing with numbers? If so, then this free course, Numbers, courses and arithmetic, is for you. The course will explain the basics of working with positive and negative numbers and how to multiply and divide with fractions and decimals. First published on Mon, 27 Jul 2015 as Numbers, units and arithmetic. To find out more visit The Open University's Openlearn website. Creative-Commons 2015 Do fractions and decimals make you apprehensive about maths? Do you lack confidence in dealing with numbers? If so, then this free course, Numbers, courses and arithmetic, is for you. The course will explain the basics of working with positive and negative numbers and how to multiply and divide with fractions and decimals. First published on Mon, 27 Jul 2015 as Numbers, units and arithmetic. To find out more visit The Open University's Openlearn website. Creative-Commons 2015

Subjects

Mathematics Education | Mathematics Education | Arit | Arit | basic math | basic math | arithmetic | arithmetic | expotential notation | expotential notation | m363 | m363 | Maths | Maths | mu123 Learning Club | mu123 Learning Club | numbers | numbers | pre course work | pre course work | Units | Units | using maths | using maths | MU120_4M1 | MU120_4M1

License

Except for third party materials and otherwise stated (see http://www.open.ac.uk/conditions terms and conditions), this content is made available under a http://creativecommons.org/licenses/by-nc-sa/2.0/uk/ Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence Licensed under a Creative Commons Attribution - NonCommercial-ShareAlike 2.0 Licence - see http://creativecommons.org/licenses/by-nc-sa/2.0/uk/ - Original copyright The Open University

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Using numbers and handling data Using numbers and handling data

Description

You might not realise it, but maths is an essential component of healthcare. In fact, sloppy calculations can have fatal consequences. This free course, Using numbers and handling data, is designed for those contemplating a future in the health services industry. First published on Wed, 16 Mar 2016 as Using numbers and handling data. To find out more visit The Open University's Openlearn website. Creative-Commons 2016 You might not realise it, but maths is an essential component of healthcare. In fact, sloppy calculations can have fatal consequences. This free course, Using numbers and handling data, is designed for those contemplating a future in the health services industry. First published on Wed, 16 Mar 2016 as Using numbers and handling data. To find out more visit The Open University's Openlearn website. Creative-Commons 2016

Subjects

Public Health | Public Health | medication | medication | S110_1 | S110_1 | health | health | handling data | handling data

License

Except for third party materials and otherwise stated (see http://www.open.ac.uk/conditions terms and conditions), this content is made available under a http://creativecommons.org/licenses/by-nc-sa/2.0/uk/ Creative Commons Attribution-NonCommercial-ShareAlike 2.0 Licence Licensed under a Creative Commons Attribution - NonCommercial-ShareAlike 2.0 Licence - see http://creativecommons.org/licenses/by-nc-sa/2.0/uk/ - Original copyright The Open University

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18.785 Number Theory I (MIT) 18.785 Number Theory I (MIT)

Description

This is the first semester of a one year graduate course in number theory covering standard topics in algebraic and analytic number theory. At various points in the course, we will make reference to material from other branches of mathematics, including topology, complex analysis, representation theory, and algebraic geometry. This is the first semester of a one year graduate course in number theory covering standard topics in algebraic and analytic number theory. At various points in the course, we will make reference to material from other branches of mathematics, including topology, complex analysis, representation theory, and algebraic geometry.

Subjects

number theory | number theory | Dedekind domains | Dedekind domains | decomposition of prime ideals | decomposition of prime ideals | local field | local field | ideal class groups | ideal class groups | Dirichlet's unit theorm | Dirichlet's unit theorm | ring of adeles | ring of adeles | group of ideles | group of ideles | zeta functions | zeta functions | L-functions | L-functions | Chebotarev density theorem | Chebotarev density theorem | Sato-Tate theorem | Sato-Tate 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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