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6.896 covers mathematical foundations of parallel hardware, from computer arithmetic to physical design, focusing on algorithmic underpinnings. Topics covered include: arithmetic circuits, parallel prefix, systolic arrays, retiming, clocking methodologies, boolean logic, sorting networks, interconnection networks, hypercubic networks, P-completeness, VLSI layout theory, reconfigurable wiring, fat-trees, and area-time complexity. This course was also taught as part of the Singapore-MIT Alliance (SMA) programme as course number SMA 5511 (Theory of Parallel Hardware). 6.896 covers mathematical foundations of parallel hardware, from computer arithmetic to physical design, focusing on algorithmic underpinnings. Topics covered include: arithmetic circuits, parallel prefix, systolic arrays, retiming, clocking methodologies, boolean logic, sorting networks, interconnection networks, hypercubic networks, P-completeness, VLSI layout theory, reconfigurable wiring, fat-trees, and area-time complexity. This course was also taught as part of the Singapore-MIT Alliance (SMA) programme as course number SMA 5511 (Theory of Parallel Hardware).

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

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 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.

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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This course begins with an introduction to the theory of computability, then proceeds to a detailed study of its most illustrious result: Kurt Gödel's theorem that, for any system of true arithmetical statements we might propose as an axiomatic basis for proving truths of arithmetic, there will be some arithmetical statements that we can recognize as true even though they don't follow from the system of axioms. In my opinion, which is widely shared, this is the most important single result in the entire history of logic, important not only on its own right but for the many applications of the technique by which it's proved. We'll discuss some of these applications, among them: Church's theorem that there is no algorithm for deciding when a formula is valid in the predicate calculus; This course begins with an introduction to the theory of computability, then proceeds to a detailed study of its most illustrious result: Kurt Gödel's theorem that, for any system of true arithmetical statements we might propose as an axiomatic basis for proving truths of arithmetic, there will be some arithmetical statements that we can recognize as true even though they don't follow from the system of axioms. In my opinion, which is widely shared, this is the most important single result in the entire history of logic, important not only on its own right but for the many applications of the technique by which it's proved. We'll discuss some of these applications, among them: Church's theorem that there is no algorithm for deciding when a formula is valid in the predicate calculus;

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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This task guide forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This lecture forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This reading material forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This zip file contains a visual aid which forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This zip file contains reading material which forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This zip file contains a visual aid which forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This zip file contains a visual aid which forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This reading material which forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This reading material forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This reading material forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This visual aid forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This reading material forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This reading material forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This reading material forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This visual aid forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This reading material forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This visual aid forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This visual aid forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This visual aid forms part of the "Classes and arithmetic" topic in the Introduction to OO Programming in Java module.

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This is an introductory course in Discrete Mathematics oriented toward Computer Science and Engineering. The course divides roughly into thirds: Fundamental concepts of Mathematics: definitions, proofs, sets, functions, relations. Discrete structures: modular arithmetic, graphs, state machines, counting. Discrete probability theory. This course was also taught as part of the Singapore-MIT Alliance (SMA) programme as course number SMA 5512 (Mathematics for Computer Science). Contributors Srinivas Devadas Lars Engebretsen David Karger Eric Lehman Thomson Leighton Charles Leiserson Nancy Lynch Santosh Vempala This is an introductory course in Discrete Mathematics oriented toward Computer Science and Engineering. The course divides roughly into thirds: Fundamental concepts of Mathematics: definitions, proofs, sets, functions, relations. Discrete structures: modular arithmetic, graphs, state machines, counting. Discrete probability theory. This course was also taught as part of the Singapore-MIT Alliance (SMA) programme as course number SMA 5512 (Mathematics for Computer Science). Contributors Srinivas Devadas Lars Engebretsen David Karger Eric Lehman Thomson Leighton Charles Leiserson Nancy Lynch Santosh Vempala

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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