This section contains video lectures. http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures 2013-01-12T04:33:08+05:00 MIT OpenCourseWare http://ocw.mit.edu en-US 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 <p><strong>Description:</strong> Introduction to mathematical proofs using axioms and propositions. Covers basics of truth tables and implications, as well as some famous hypotheses and conjectures.</p> <p><strong>Speaker:</strong> Tom Leighton</p>Keywords: proof, axiom, proposition, lemma, predicate, factoring, coloring, truth table, implication<br><br>Thumbnail - <a href= http://img.youtube.com/vi/L3LMbpZIKhQ/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec01_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-1-introduction-proofs/id503873536?i=110644965>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/L3LMbpZIKhQ>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-1-introduction-and-proofs Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US proof axiom proposition lemma predicate factoring coloring truth table implication MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> An introduction to proof techniques, covering proof by contradiction and induction, with an emphasis on the inductive techniques used in proof by induction.</p> <p><strong>Speaker:</strong> Tom Leighton</p>Keywords: proof, induction, proof by contradiction, hypothesis, false proof, base case, inductive step<br><br>Thumbnail - <a href= http://img.youtube.com/vi/z8HKWUWS-lA/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec02_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-2-induction/id503873536?i=110644980>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/z8HKWUWS-lA>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-2-induction Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US proof induction proof by contradiction hypothesis false proof base case inductive step MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Covers strong induction as a tool for proofs. Introduction to invariants with different games, including the n&ndash;block game and grid puzzles.</p> <p><strong>Speaker:</strong> Tom Leighton</p>Keywords: strong induction, proof, Fermat's last theorem, proof technique, invariant, n-block game<br><br>Thumbnail - <a href= http://img.youtube.com/vi/NuGDkmwEObM/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec03_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-3-strong-induction/id503873536?i=110644967>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/NuGDkmwEObM>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-3-strong-induction Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US strong induction proof Fermat's last theorem proof technique invariant, n-block game MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Explores the basics of number theory with state machines, linear combinations, and algorithms for computation with integers.</p> <p><strong>Speaker:</strong> Marten van Dijk</p>Keywords: number theory, divisibility, state machine, induction, linear combination, Euclid's algorithm, pulverizer, greatest common divsor<br><br>Thumbnail - <a href= http://img.youtube.com/vi/NuY7szYSXSw/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec04_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-4-number-theory-i/id503873536?i=110644966>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/NuY7szYSXSw>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-4-number-theory-i Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US number theory divisibility state machine induction linear combination Euclid's algorithm pulverizer greatest common divsor MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Delves deeper into number theory, covering the basics of encryption and decryption using modular arithmetic.</p> <p><strong>Speaker:</strong> Marten van Dijk</p>Keywords: number theory, encryption, Turing's code, modular arithmetic, totient function, Euler's theorem, Fermat's little theorem, RSA<br><br>Thumbnail - <a href= http://img.youtube.com/vi/XX7ePR21Ook/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec05_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-5-number-theory-ii/id503873536?i=110644960>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/XX7ePR21Ook>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-5-number-theory-ii Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US number theory encryption Turing's code modular arithmetic totient function Euler's theorem Fermat's little theorem RSA MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> An introduction to graph theory basics and intuition with applications to scheduling, coloring, and even sexual promiscuity.</p> <p><strong>Speaker:</strong> Tom Leighton</p>Keywords: simple graphs, nodes, edges, vertices, sexual promiscuity, scheduling, coloring, chromatic number, NP completeness, bipartite<br><br>Thumbnail - <a href= http://img.youtube.com/vi/h9wxtqoa1jY/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec06_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-6-graph-theory-coloring/id503873536?i=110644981>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/h9wxtqoa1jY>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-6-graph-theory-and-coloring Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US simple graphs nodes edges vertices sexual promiscuity scheduling coloring chromatic number NP completeness bipartite MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Introduces the concept of matching. Discusses the mating algorithm, its fairness, and relation to practical applications.</p> <p><strong>Speaker:</strong> Tom Leighton</p>Keywords: matching, min-weight matching, rouge couple, mating algorithm, fairness<br><br>Thumbnail - <a href= http://img.youtube.com/vi/5RSMLgy06Ew/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec07_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-7-matching-problems/id503873536?i=110644963>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/5RSMLgy06Ew>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-7-matching-problems Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US matching min-weight matching rouge couple mating algorithm fairness MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Explores the various measures of connectivity of graphs and how these can be used to categorize and analyze graphs.</p> <p><strong>Speaker:</strong> Marten van Dijk</p>Keywords: walks, paths, connectivity, cycles, closed walk, acyclic, tree, spanning trees, minimum weight spanning trees<br><br>Thumbnail - <a href= http://img.youtube.com/vi/GJpt_3ie4WU/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec08_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-8-graph-theory-ii/id503873536?i=110644969>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/GJpt_3ie4WU>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-8-graph-theory-ii-minimum-spanning-trees Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US walks paths connectivity cycles closed walk acyclic tree spanning trees minimum weight spanning trees MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Covers the application of graph theory to communication networks, surveying their configuration, topology, and optimization.</p> <p><strong>Speaker:</strong> Marten van Dijk</p>Keywords: networks, switch, diameter, congestion, switch size, butterfly network, binary tree, benes, latency<br><br>Thumbnail - <a href= http://img.youtube.com/vi/bTyxpoi2dmM/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec09_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-9-communication-networks/id503873536?i=110644973>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/bTyxpoi2dmM>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-9-communication-networks Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US networks switch diameter congestion switch size butterfly network binary tree benes latency MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Builds upon previous lectures to cover additional graph classifications and criteria, including tournament graphs and directed acyclic graphs. Also covers Euler Tours, Hamiltonian paths, and adjacency matrices.</p> <p><strong>Speaker:</strong> Marten van Dijk</p>Keywords: Euler tour, walks, directed graphs, strong connectivity, directed acyclic graphs, DAG, tournament graphs, Hamiltonian paths, adjacency matrices<br><br>Thumbnail - <a href= http://img.youtube.com/vi/DOIp5D7VMS4/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec10_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-10-graph-theory-iii/id503873536?i=110644959>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/DOIp5D7VMS4>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-10-graph-theory-iii Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US Euler tour walks directed graphs strong connectivity directed acyclic graphs DAG tournament graphs Hamiltonian paths adjacency matrices MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Covers definitions and examples of basic relations, equivalence classes, Hasse diagrams and topological sorts, as well as other topics.</p> <p><strong>Speaker:</strong> Marten van Dijk</p> <p>The last 30 minutes of this video are not available.</p>Keywords: relations, partial orders, poset, Hasse diagram, total order, topological sort, parallel task scheduling, Dilworth's lemma, reflexive, antisymmetric, transitive, symmetric<br><br>Thumbnail - <a href= http://img.youtube.com/vi/1nScXLQAQ9A/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec11_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-11-relations-partial/id503873536?i=110644975>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/1nScXLQAQ9A>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-11-relations-partial-orders-and-scheduling Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US relations partial orders poset Hasse diagram total order topological sort parallel task scheduling Dilworth's lemma reflexive antisymmetric transitive symmetric MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> An introduction to sums through examination of real&ndash;world problems like annuities. Covers finding closed form solutions and bounds with the perturbation, derivative, and integral methods.</p> <p><strong>Speaker:</strong> Tom Leighton</p>Keywords: sum, annuities, interest rate, perturbation method, closed form sums, derivative method<br><br>Thumbnail - <a href= http://img.youtube.com/vi/fAeShezAGLE/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec12_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-12-sums/id503873536?i=110644979>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/fAeShezAGLE>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-12-sums Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US sum annuities interest rate perturbation method closed form sums derivative method MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Analysis of sums, formulation of asymptotic bounds using various techniques, and introduction to asymptotic notation.</p> <p><strong>Speaker:</strong> Tom Leighton</p>Keywords: block stacking, greedy algorithm, center of mass, sum, recursion, harmonic numbers, closed form expression, Sterling's Formula, bounds<br><br>Thumbnail - <a href= http://img.youtube.com/vi/X9eErxRjQEI/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec13_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-13-sums-asymptotics/id503873536?i=110644972>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/X9eErxRjQEI>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-13-sums-and-asymptotics Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US block stacking greedy algorithm center of mass sum recursion harmonic numbers closed form expression Sterling's Formula bounds MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Introduces the concept of recursion applied to various recurrence problems, such as the Towers of Hanoi and the Merge Sort algorithm, as well as their asymptotic analysis using the Akra&ndash;Bazzi method.</p> <p><strong>Speaker:</strong> Tom Leighton</p>Keywords: recurrences, Towers of Hanoi, Akra-Bazzi, recursion, merge sort, asymptotic runtime<br><br>Thumbnail - <a href= http://img.youtube.com/vi/Kqf0uO0oV6s/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec14_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-14-divide-conquer/id503873536?i=110644962>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/Kqf0uO0oV6s>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-14-divide-and-conquer-recurrences Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US recurrences Towers of Hanoi Akra-Bazzi recursion merge sort asymptotic runtime MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Covers the mechanics of solving general linear recurrences as well as applications to the graduate student job problem and Fibonacci modeling of populations.</p> <p><strong>Speaker:</strong> Tom Leighton</p>Keywords: linear recurrence, graduate student job problem, Fibonacci, order, boundary conditions, homogeneous equation, characteristic equation, inhomogeneous, particular solution, repeated roots<br><br>Thumbnail - <a href= http://img.youtube.com/vi/TWBB-JlmYUc/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec15_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-15-linear-recurrences/id503873536?i=110644968>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/TWBB-JlmYUc>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-15-linear-recurrences Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US linear recurrence graduate student job problem Fibonacci order boundary conditions homogeneous equation characteristic equation inhomogeneous particular solution repeated roots MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Introduces and defines relationships between sets and covers how they are used to reason about counting.</p> <p><strong>Speaker:</strong> Marten van Dijk</p>Keywords: set, sequence, permutation, surjective, injective, bijective, mapping rule, pigeonhole principle, product rule, sum rule, division rule<br><br>Thumbnail - <a href= http://img.youtube.com/vi/pNt5Ll6hGqo/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec16_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-16-counting-rules-i/id503873536?i=110644982>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/pNt5Ll6hGqo>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-16-counting-rules-i Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US set sequence permutation surjective injective bijective mapping rule pigeonhole principle product rule sum rule division rule MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Covers computing cardinality of sets with inclusion&ndash;exclusion, the bookkeeper rule, the subset rule, and poker hands with applications to probability and counting.</p> <p><strong>Speaker:</strong> Marten van Dijk</p>Keywords: inclusion-exclusion, intersection, union, permutation, bookkeeper rule, subset rule, binomial theorem, combinatorial proofs<br><br>Thumbnail - <a href= http://img.youtube.com/vi/09yIb3VHhMI/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec17_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-17-counting-rules-ii/id503873536?i=110644964>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/09yIb3VHhMI>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-17-counting-rules-ii Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US inclusion-exclusion intersection union permutation bookkeeper rule subset rule binomial theorem combinatorial proofs MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Gives an overview of probability, including basic definitions, the Monty Hall problem, and strange dice games.</p> <p><strong>Speaker:</strong> Tom Leighton</p>Keywords: probability, Monty Hall problem, sample space, outcome, tree method, probability space, dice game<br><br>Thumbnail - <a href= http://img.youtube.com/vi/SmFwFdESMHI/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec18_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-18-probability-introduction/id503873536?i=110644977>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/SmFwFdESMHI>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-18-probability-introduction Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US probability Monty Hall problem sample space outcome tree method probability space dice game MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Covers conditional probability and its applications to examples including medical testing, gambling, and court cases.</p> <p><strong>Speaker:</strong> Tom Leighton</p> <p><strong>Instructor's Note:</strong> The actual details of the Berkeley sex discrimination case may have been different than what was stated in the lecture, so it is best to consider the description given in lecture as fictional but illustrative of the mathematical point being made.</p>Keywords: conditional probability, a postieri, medical testing, false, positive, negative, carnival dice, inclusion-exclusion, gender bias<br><br>Thumbnail - <a href= http://img.youtube.com/vi/E6FbvM-FGZ8/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec19_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-19-conditional-probability/id503873536?i=110644974>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/E6FbvM-FGZ8>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-19-conditional-probability Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US conditional probability a postieri medical testing false positive negative carnival dice inclusion-exclusion gender bias MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Differentiates between independent and dependent events as it pertains to probability, covering applications like coin flips, the distribution of birthdays, hashing, and cryptography.</p> <p><strong>Speaker:</strong> Tom Leighton</p>Keywords: independence, probability, event, pairwise independence, mutual independence, the birthday problem, blizzard babies, hashing, cryptography<br><br>Thumbnail - <a href= http://img.youtube.com/vi/l1BCv3qqW4A/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec20_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-20-independence/id503873536?i=110644970>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/l1BCv3qqW4A>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-20-independence Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US independence probability event pairwise independence mutual independence the birthday problem blizzard babies hashing cryptography MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Introduces partitioning of the probabilistic sample space using random variables. Distribution functions, notably, the binomial distribution, are discussed.</p> <p><strong>Speaker:</strong> Tom Leighton</p>Keywords: random variable, sample space, indicator, Bernoulli, characteristic, independence, dependent, probability distribution function, cumulative distribution function, uniform, binomial distribution function<br><br>Thumbnail - <a href= http://img.youtube.com/vi/MOfhhFaQdjw/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec21_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-21-random-variables/id503873536?i=110644961>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/MOfhhFaQdjw>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-21-random-variables Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US random variable sample space indicator Bernoulli characteristic independence dependent probability distribution function cumulative distribution function uniform binomial distribution function MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Covers expected value as it relates to random variables, discussing coin games, network latency, and the hat check problem.</p> <p><strong>Speaker:</strong> Tom Leighton</p>Keywords: expected value, gambling, summation, linearity of expectation, hat check problem<br><br>Thumbnail - <a href= http://img.youtube.com/vi/gGlMSe7uEkA/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec22_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-22-expectation-i/id503873536?i=110644978>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/gGlMSe7uEkA>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-22-expectation-i Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US expected value gambling summation linearity of expectation hat check problem MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Continues exploring expectation with a discussion of likelihood in cases of card games, bit transmission errors, and algorithms, and concludes with definitions of variance and standard deviation for random variables.</p> <p><strong>Speaker:</strong> Tom Leighton</p>Keywords: random variable, expected value, linearity of expectation, Murphy's Law, mutual independence, distribution, variance, standard deviation<br><br>Thumbnail - <a href= http://img.youtube.com/vi/oI9fMUqgfxY/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec23_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-23-expectation-ii/id503873536?i=110644971>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/oI9fMUqgfxY>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-23-expectation-ii Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US random variable expected value linearity of expectation Murphy's Law mutual independence distribution variance standard deviation MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Covers large deviation. Like expectation, it gives three other notions in solving bounds and many frequently experienced problems in computer science, such as determining the probability a random variable will deviate from its expectation.</p> <p><strong>Speaker:</strong> Tom Leighton</p>Keywords: random variable, variance, expectation, Markov's theorem, Chebyshev's theorem, Chernoff bound, load balancing<br><br>Thumbnail - <a href= http://img.youtube.com/vi/q4mwO2qS2z4/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec24_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-24-large-deviations/id503873536?i=110644976>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/q4mwO2qS2z4>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-24-large-deviations Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US random variable variance expectation Markov's theorem Chebyshev's theorem Chernoff bound load balancing MIT OpenCourseWare http://ocw.mit.edu 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 <p><strong>Description:</strong> Discusses random walks and their non&ndash;intuitive effect on systems, such as gambling at roulette and gambler's ruin.</p> <p><strong>Speaker:</strong> Tom Leighton</p>Keywords: random walks, gambler's ruin, roulette, martingale, biased, unbiased, drift<br><br>Thumbnail - <a href= http://img.youtube.com/vi/56iFMY8QW2k/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MIT6.042JF10/MIT6_042JF10_lec25_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/lecture-25-random-walks/id503873536?i=110644958>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/56iFMY8QW2k>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/video-lectures/lecture-25-random-walks Leighton, Tom Dijk, Marten van 2011-06-23T12:46:03+05:00 en-US random walks gambler's ruin roulette martingale biased unbiased drift MIT OpenCourseWare http://ocw.mit.edu 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