- published: 20 Aug 2012
- views: 12254
- author: Jaap Weel
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- Algebra
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Bletchley Park
Coordinates: 51°59′47″N 0°44′34″W / 51.99651°N 0.74276°W / 51.99651; -0.74276
Bletchley Park is an estate located in the town of Bletchley, in Buckinghamshire, England, which currently houses the National Codes Centre and the National Museum of Computing. During the Second World War, Bletchley Park was the site of the United Kingdom's main decryption establishment, the Government Code and Cypher School (GC&CS), where ciphers and codes of several Axis countries were decrypted, most importantly the ciphers generated by the German Enigma and Lorenz machines. It also housed Station X, a secret radio intercept station. Although interception was soon moved to a location with better reception, the name persisted for the Bletchley Park wartime activities.
The high-level intelligence produced at Bletchley Park, codenamed Ultra, provided crucial assistance to the Allied war effort. Sir Harry Hinsley, a Bletchley veteran and the official historian of British Intelligence during the Second World War, said that Ultra shortened the war by two to four years and that the outcome of the war would have been uncertain without it.
wn.com/Bletchley_Park
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Andrew Clive Simpson
Dr Andrew Clive Simpson is a British Computer Scientist. He is Director of Studies, Software Engineering Programme at University of Oxford. He is Governing Body Fellow of Kellogg College.
He obtained first class BSc (Hons) in Computer Science from University of Wales, Swansea (1989–1992); followed by MSc in Computation (1992–1993) and DPhil in Computation (1993–1996) from University of Oxford.
Before his current post he was research officer in Oxford University Computing Laboratory (1996–1999) and Principal Lecturer in Computing at Oxford Brookes University (1999–2001).
Andrew Simpson's publications covers a wide range of topics covering Software Engineering, Computational Biology, Security, and Formal Methods.
wn.com/Andrew_Clive_Simpson -
Bojan Mohar
Bojan Mohar is a Slovenian and Canadian mathematician, specializing in graph theory. He is a professor of mathematics at the University of Ljubljana and the holder of a Canada Research Chair in graph theory at Simon Fraser University in Vancouver, Canada.
Mohar received his Ph.D. from the University of Ljubljana in 1986, under the supervision of Tomo Pisanski.
Mohar's research concerns topological graph theory, algebraic graph theory, graph minors, and graph coloring.
With Carsten Thomassen he is the co-author of the book Graphs on Surfaces (Johns Hopkins University Press, 2001).
Mohar was a Fulbright visiting scholar at Ohio State University in 1988, and won the Boris Kidrič prize of the Socialist Republic of Slovenia in 1990. He has been a member of the Slovenian Academy of Engineering since 1999.
wn.com/Bojan_Mohar -
Carsten Thomassen
Carsten Thomassen (born August 22, 1948 in Grindsted) is a Danish mathematician. He has been a Professor of Mathematics at the Technical University of Denmark since 1981, and since 1990 a member of the Royal Danish Academy of Sciences and Letters. His research concerns discrete mathematics and more specifically graph theory.
Thomassen received his Ph.D. in 1976 from the University of Waterloo.
He is editor-in-chief of the Journal of Graph Theory and the Electronic Journal of Combinatorics, and editor of Combinatorica, the Journal of Combinatorial Theory, Ser B, Discrete Mathematics, and the European Journal of Combinatorics.
He was awarded the Dedicatory Award of the 6th International Conference on the Theory and Applications of Graphs by the Western Michigan University in May 1988, the Lester R. Ford Award by the Mathematical Association of America in 1993, and the Faculty of Mathematics Alumni Achievement Medal by the University of Waterloo in 2005. He was included on the ISI Web of Knowledge list of the 250 most cited mathematicians.
wn.com/Carsten_Thomassen -
David Hilbert
David Hilbert, FRS (German pronunciation: [ˈdaːvɪt ˈhɪlbɐt]; January 23, 1862 – February 14, 1943) was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis.
Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set the course for much of the mathematical research of the 20th century.
Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.
wn.com/David_Hilbert -
Donald Knuth
Donald Ervin Knuth (
/kəˈnuːθ/kə-NOOTH; born January 10, 1938) is a computer scientist and Professor Emeritus at Stanford University.
He is the author of the seminal multi-volume work The Art of Computer Programming. Knuth has been called the "father" of the analysis of algorithms. He contributed to the development of the rigorous analysis of the computational complexity of algorithms and systematized formal mathematical techniques for it. In the process he also popularized the asymptotic notation.
In addition to fundamental contributions in several branches of theoretical computer science, Knuth is the creator of the TeX computer typesetting system, the related METAFONT font definition language and rendering system, and the Computer Modern family of typefaces.
As a writer and scholar, Knuth created the WEB/CWEB computer programming systems designed to encourage and facilitate literate programming, and designed the MIX/MMIX instruction set architectures.
Knuth was born in Milwaukee, Wisconsin, where his father owned a small printing business and taught bookkeeping at Milwaukee Lutheran High School, where he enrolled, earning achievement awards. He applied his intelligence in unconventional ways, winning a contest when he was in eighth grade by finding over 4,500 words that could be formed from the letters in "Ziegler's Giant Bar." The judges had only about 2,500 words on their master list. This won him a television set for his school and a candy bar for everyone in his class.
wn.com/Donald_Knuth -
Georg Cantor
Georg Ferdinand Ludwig Philipp Cantor ( /ˈkæntɔr/ KAN-tor; German: [ɡeˈɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfiːlɪp ˈkantɔʁ]; March 3 [O.S. February 19] 1845 – January 6, 1918) was a German mathematician, best known as the inventor of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between the members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's method of proof of this theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.
wn.com/Georg_Cantor -
Jaroslav Nešetřil
Jaroslav Nešetřil (Czech pronunciation: [ˈjaroslaf ˈnɛʃɛtr̝̊ɪl]; born March 13, 1946 in Brno) is a Czech mathematician, working at Charles University in Prague. His research areas include combinatorics (structural combinatorics, Ramsey theory), graph theory (coloring problems, sparse structures), algebra (representation of structures, categories, homomorphisms), posets (diagram and dimension problems), computer science (complexity, NP-completeness).
Nešetřil received his Ph.D. from Charles University in 1973. He is responsible for more than 300 publications, was awarded the state prize (1985 jointly with Vojtěch Rödl) for a collection of papers in Ramsey theory, is a corresponding member of the German Academy of Sciences since 1996 and has been declared Doctor Honoris Causa of the University of Alaska (Fairbanks) in 2002. Since 2006, he is chairman of the Committee of Mathematics of Czech Republic (the Czech partner of IMU). He has also been declared Doctor Honoris Causa of the University of Bordeaux 1 in 2009; the speech he made in French at this occasion attracted a great deal of attention. He received in 2010 the Medal of Merit of Czech Republic.
wn.com/Jaroslav_Nešetřil -
Kenneth Appel
Kenneth Ira Appel (born October 8, 1932 in Brooklyn, New York) is a mathematician who in 1976, with colleague Wolfgang Haken at the University of Illinois at Urbana-Champaign, solved one of the most famous problems in mathematics, the four-color theorem. They proved that any two-dimensional map, with certain limitations, can be filled in with four colors without any adjacent "countries" sharing the same color.
The proof has been one of the most controversial of modern mathematics because of its heavy dependence on computer "number-crunching" to sort through possibilities, which drew criticism from many in the mathematical community for its inelegance: "a good mathematical proof is like a poem — this is a telephone directory!" Appel and Haken agreed in a 1977 interview that it was not "elegant, concise and completely comprehensible by a human mathematical mind".
However, the proof was the start of a change in mathematicians' attitudes toward computers – which they had largely disdained as a tool for engineers rather than for theoreticians - leading to the creation of what is sometimes called "experimental mathematics".
wn.com/Kenneth_Appel -
Oren Patashnik
Oren Patashnik (born 1954) is a computer scientist. He is notable for co-creating BibTeX, and co-writing Concrete Mathematics: A Foundation for Computer Science. He is a researcher at the Center for Communications Research, La Jolla, and lives nearby in San Diego. Oren and his wife Amy have three children, Josh, Ariel, and Jeremy.
Oren Patashnik graduated from Yale University in 1976, and later became a doctoral student in computer science at Stanford University, where his research was supervised by Donald Knuth.
While working at Bell Labs in 1980, Patashnik proved that Qubic can always be won by the first player. Using 1500 hours of computer time, Patashnik's proof is a notable example of a computer-assisted proof.
In 1985, Patashnik created the bibliography-system, BibTeX, in collaboration with Leslie Lamport, the creator of LaTeX. LaTeX is a system and programming language for formatting documents, which is especially designed for mathematical documents. BibTeX is a widely used bibliography-formatting tool for LaTeX.
wn.com/Oren_Patashnik -
Ralph Grimaldi
Ralph Peter Grimaldi (born January 1943) is an American mathematician specializing in discrete mathematics who is a full professor at Rose-Hulman Institute of Technology. He is known for his textbook Discrete and Combinatorial Mathematics: An Applied Introduction [1], first published in 1985 and now in its fifth edition, and his numerous research papers.
He was born and raised in New York City and graduated from what is now the State University of New York at Albany in 1964 (B.S.) and 1965 (M.S.), then earned his Ph.D. in Mathematical Sciences in 1972 at New Mexico State University under Ray Mines. He previously taught at the State University of New York at Oswego and has held sabbatical appointments at Clemson University and New Mexico State University, as well as in industry.
wn.com/Ralph_Grimaldi -
Richard Johnsonbaugh
Richard Johnsonbaugh is a notable author and mathematician. His interests include discrete mathematics and the history of mathematics. He is the successful author of several books including Discrete Mathematics, 7th ed, Macmillan. He holds degrees in computer science and mathematics from the University of Oregon, Yale University, and the University of Illinois at Chicago. He is currently Emeritus Professor at De Paul University.
wn.com/Richard_Johnsonbaugh -
Ronald Graham
Ronald Lewis Graham (born October 31, 1935) is a mathematician credited by the American Mathematical Society as being "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years". He has done important work in scheduling theory, computational geometry, Ramsey theory, and quasi-randomness.
He is currently the Chief Scientist at the California Institute for Telecommunications and Information Technology (also known as Cal-(IT)2) and the Irwin and Joan Jacobs Professor in Computer Science and Engineering at the University of California, San Diego (UCSD).
Graham was born in Taft, California. In 1962, he received his Ph.D. in mathematics from the University of California, Berkeley.
His 1977 paper considered a problem in Ramsey theory, and gave a "large number" as an upper bound for its solution. This number has since become well known as the largest number ever used in a mathematical proof (is listed as such in the Guinness Book of Records), and is now known as Graham's number.
wn.com/Ronald_Graham -
Wolfgang Haken
Wolfgang Haken (born June 21, 1928 in Berlin, Germany) is a mathematician who specializes in topology, in particular 3-manifolds.
In 1976 together with colleague Kenneth Appel at the University of Illinois at Urbana-Champaign, Haken solved one of the most famous problems in mathematics, the four-color theorem. They proved that any two-dimensional map, with certain limitations, can be filled in with four colors without any adjacent "countries" sharing the same color.
Haken has introduced several important ideas, including Haken manifolds, Kneser-Haken finiteness, and an expansion of the work of Kneser into a theory of normal surfaces. Much of his work has an algorithmic aspect, and he is one of the influential figures in algorithmic topology. One of his key contributions to this field is an algorithm to detect if a knot is unknotted.
Haken is the father of six children. His eldest son Armin Haken proved that there exist propositional tautologies that require resolution proofs of exponential size. Lippold Haken, the inventor of the Continuum fingerboard, is also his son.
wn.com/Wolfgang_Haken -
Yuri Matiyasevich
Yuri Vladimirovich Matiyasevich, (Russian: Ю́рий Влади́мирович Матиясе́вич; born March 2, 1947 in Leningrad) is a Russian mathematician and computer scientist. He is best known for his negative solution of Hilbert's tenth problem, presented in his doctoral thesis, at LOMI (the Leningrad Department of the Steklov Institute of Mathematics).
wn.com/Yuri_Matiyasevich
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Arsdigita 02 (Discrete Mathematics) Lecture 1/20
- Order: Reorder
- Duration: 79:02
- Updated: 05 Jan 2013
Course 02: Discrete Mathematics (Arsdigita University) NOTE: I will delete off-topic comments, especially offensive ones related to the lecturer's religious or purportedly implied political beliefs, so please stop bothering already. This course covers the mathematical topics most directly related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis is placed on providing a context for the application of the mathematics within computer science. Instructor: Shai Simonson Text: Discrete Mathematics and its Applications, Rosen. 01 What kinds of problems are solved in discrete math? 02 Boolean Algebra and formal logic 03 More logic: quantifiers and predicates 04 Sets 05 Diagonalization, functions and sums review 06 Basic arithmetic and geometric sums, closed forms. 07 Chinese rings puzzle 08 Solving recurrence equations 09 Solving recurrence equations (cont.) 10 Mathematical induction 11 Combinations and permutations 12 Counting Problems 13 Counting problems 14 Counting problems using combinations, distributions 15 Counting problems using combinations, distributions 16 The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick. 17 Equivalence Relations and Partial Orders 18 Euclid's Algorithm 19 ...
http://wn.com/Arsdigita_02_Discrete_Mathematics_Lecture_1_20
Course 02: Discrete Mathematics (Arsdigita University) NOTE: I will delete off-topic comments, especially offensive ones related to the lecturer's religious or purportedly implied political beliefs, so please stop bothering already. This course covers the mathematical topics most directly related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis is placed on providing a context for the application of the mathematics within computer science. Instructor: Shai Simonson Text: Discrete Mathematics and its Applications, Rosen. 01 What kinds of problems are solved in discrete math? 02 Boolean Algebra and formal logic 03 More logic: quantifiers and predicates 04 Sets 05 Diagonalization, functions and sums review 06 Basic arithmetic and geometric sums, closed forms. 07 Chinese rings puzzle 08 Solving recurrence equations 09 Solving recurrence equations (cont.) 10 Mathematical induction 11 Combinations and permutations 12 Counting Problems 13 Counting problems 14 Counting problems using combinations, distributions 15 Counting problems using combinations, distributions 16 The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick. 17 Equivalence Relations and Partial Orders 18 Euclid's Algorithm 19 ...
Lecture 1 - Propositional Logic
- Order: Reorder
- Duration: 56:47
- Updated: 06 Jan 2013
Discrete Mathematical Structures
http://wn.com/Lecture_1_Propositional_Logic
Arsdigita 02 (Discrete Mathematics) Lecture 20/20
- Order: Reorder
- Duration: 76:52
- Updated: 07 Aug 2012
Course 02: Discrete Mathematics (Arsdigita University) This course covers the mathematical topics most directly related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis is placed on providing a context for the application of the mathematics within computer science. Instructor: Shai Simonson Text: Discrete Mathematics and its Applications, Rosen. 01 What kinds of problems are solved in discrete math? 02 Boolean Algebra and formal logic 03 More logic: quantifiers and predicates 04 Sets 05 Diagonalization, functions and sums review 06 Basic arithmetic and geometric sums, closed forms. 07 Chinese rings puzzle 08 Solving recurrence equations 09 Solving recurrence equations (cont.) 10 Mathematical induction 11 Combinations and permutations 12 Counting Problems 13 Counting problems 14 Counting problems using combinations, distributions 15 Counting problems using combinations, distributions 16 The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick. 17 Equivalence Relations and Partial Orders 18 Euclid's Algorithm 19 Recitation -- a combinatorial card trick 20 Cryptography More information about this course: www.archive.org www.aduni.org Licensed under Creative Commons Attribution ShareAlike ...
http://wn.com/Arsdigita_02_Discrete_Mathematics_Lecture_20_20
Course 02: Discrete Mathematics (Arsdigita University) This course covers the mathematical topics most directly related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis is placed on providing a context for the application of the mathematics within computer science. Instructor: Shai Simonson Text: Discrete Mathematics and its Applications, Rosen. 01 What kinds of problems are solved in discrete math? 02 Boolean Algebra and formal logic 03 More logic: quantifiers and predicates 04 Sets 05 Diagonalization, functions and sums review 06 Basic arithmetic and geometric sums, closed forms. 07 Chinese rings puzzle 08 Solving recurrence equations 09 Solving recurrence equations (cont.) 10 Mathematical induction 11 Combinations and permutations 12 Counting Problems 13 Counting problems 14 Counting problems using combinations, distributions 15 Counting problems using combinations, distributions 16 The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick. 17 Equivalence Relations and Partial Orders 18 Euclid's Algorithm 19 Recitation -- a combinatorial card trick 20 Cryptography More information about this course: www.archive.org www.aduni.org Licensed under Creative Commons Attribution ShareAlike ...
- published: 07 Jun 2012
- views: 707
- author: Jaap Weel
DISCRETE MATHS -- INDUCTION AND RECURSION -. LECTURE 1.
- Order: Reorder
- Duration: 39:01
- Updated: 03 Jan 2013
DISCRETE MATHS -- INDUCTION AND RECURSION -. LECTURE 1. - April. 5th - ESCUELA UNIVERSITARIA DE INFORMÁTICA - PACO GÓMEZ Properties depending on natural numbers. Inductive proofs: basis step and inductive step. Examples: summation and inequalities
http://wn.com/DISCRETE_MATHS_-_INDUCTION_AND_RECURSION_LECTURE_1
DISCRETE MATHS -- INDUCTION AND RECURSION -. LECTURE 1. - April. 5th - ESCUELA UNIVERSITARIA DE INFORMÁTICA - PACO GÓMEZ Properties depending on natural numbers. Inductive proofs: basis step and inductive step. Examples: summation and inequalities
- published: 12 Apr 2011
- views: 10019
- author: UPM
Labyrinth Discrete Math
- Order: Reorder
- Duration: 1:21
- Updated: 30 Dec 2012
Labyrinth Discrete Math
http://wn.com/Labyrinth_Discrete_Math
Video Lesson - Sets (Discrete Mathematics)
- Order: Reorder
- Duration: 13:44
- Updated: 24 Dec 2012
Here's a second project for my discrete math class.
http://wn.com/Video_Lesson_Sets_Discrete_Mathematics
Here's a second project for my discrete math class.
- published: 13 Feb 2012
- views: 2520
- author: 4ce0fSp4des
Discrete Math 2 - Tutorial 16 - Generating Functions
- Order: Reorder
- Duration: 10:17
- Updated: 10 Dec 2012
First look at Generating Functions; another way of counting objects. Please comment rate and subscribe. for latest updates : follow us on twitter : twitter.com Become a fan on facebook : www.facebook.com Add us on Google+ : plus.google.com
http://wn.com/Discrete_Math_2_Tutorial_16_Generating_Functions
First look at Generating Functions; another way of counting objects. Please comment rate and subscribe. for latest updates : follow us on twitter : twitter.com Become a fan on facebook : www.facebook.com Add us on Google+ : plus.google.com
- published: 31 Dec 2011
- views: 4152
- author: coursehack
Discrete Math 2-Tutorial 14 - Example 2: Inclusion/Exlusion
- Order: Reorder
- Duration: 11:23
- Updated: 20 Nov 2012
2nd example in Inclusion and Exclusion Principle. If you have anymore questions; please leave me a comment. Please rate, comment and subscribe.
http://wn.com/Discrete_Math_2-Tutorial_14_Example_2_Inclusion_Exlusion
2nd example in Inclusion and Exclusion Principle. If you have anymore questions; please leave me a comment. Please rate, comment and subscribe.
- published: 27 Dec 2011
- views: 1228
- author: coursehack
Discrete Math 2-Tutorial 12-Inclusion/Exclusion Principle
- Order: Reorder
- Duration: 9:50
- Updated: 28 Dec 2012
The very first look at Inclusion and Exclusion principle. This video demonstrates why we need the principle. Dont forget to Rate comment and subscribe
http://wn.com/Discrete_Math_2-Tutorial_12-Inclusion_Exclusion_Principle
The very first look at Inclusion and Exclusion principle. This video demonstrates why we need the principle. Dont forget to Rate comment and subscribe
- published: 27 Dec 2011
- views: 4629
- author: coursehack
Discrete Mathematics-Selected Topics Lecture 1
- Order: Reorder
- Duration: 82:08
- Updated: 04 Jan 2013
This Video is the First Lecture in Discrete Mathematics-Selected Topics Videos By Dr.Omar Hamdy omar.hamdy@fci.helwan.edu.eg Lecture Notes: hci.fcih.net Topics: 1- Propositional Logic 2- Logical Connectives 3- Truth Table 4- Translation
http://wn.com/Discrete_Mathematics-Selected_Topics_Lecture_1
This Video is the First Lecture in Discrete Mathematics-Selected Topics Videos By Dr.Omar Hamdy omar.hamdy@fci.helwan.edu.eg Lecture Notes: hci.fcih.net Topics: 1- Propositional Logic 2- Logical Connectives 3- Truth Table 4- Translation
- published: 24 Dec 2011
- views: 2921
- author: FCI HU
Lecture 38-Finite State Automaton
- Order: Reorder
- Duration: 57:17
- Updated: 04 Jan 2013
Discrete Mathematical Structures
http://wn.com/Lecture_38-Finite_State_Automaton
Praxis 2 (0061) Discrete Math 1
- Order: Reorder
- Duration: 9:53
- Updated: 24 Dec 2012
sites.google.com A review of some basic discrete math problems
http://wn.com/Praxis_2_0061_Discrete_Math_1
sites.google.com A review of some basic discrete math problems
- published: 13 Mar 2011
- views: 4883
- author: shaunteaches
Lecture 10 - Sets
- Order: Reorder
- Duration: 55:13
- Updated: 25 Dec 2012
Discrete mathematical Structures Lecture 10 - Prof. Kamala Krithivasan
http://wn.com/Lecture_10_Sets
Discrete mathematical Structures Lecture 10 - Prof. Kamala Krithivasan
- published: 05 Dec 2007
- views: 67965
- author: nptelhrd
Base Conversion (Discrete Math)
- Order: Reorder
- Duration: 7:35
- Updated: 06 Dec 2012
In this video I explain the basics between different number systems and how to convert to and from base 10. Link: goo.gl - or - wartex8.com
http://wn.com/Base_Conversion_Discrete_Math
In this video I explain the basics between different number systems and how to convert to and from base 10. Link: goo.gl - or - wartex8.com
- published: 10 Oct 2010
- views: 5334
- author: wartex8
Discrete Mathematics
- Order: Reorder
- Duration: 3:16
- Updated: 07 Dec 2012
Discrete Mathematics - Spring Semester - Mr. Duane Jones
http://wn.com/Discrete_Mathematics
Discrete Mathematics - Spring Semester - Mr. Duane Jones
- published: 20 Sep 2010
- views: 1162
- author: GHAMASCourses
Arsdigita 02 (Discrete Mathematics) Lecture 6/20
- Order: Reorder
- Duration: 88:04
- Updated: 08 Dec 2012
Course 02: Discrete Mathematics (Arsdigita University) This course covers the mathematical topics most directly related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis is placed on providing a context for the application of the mathematics within computer science. Instructor: Shai Simonson Text: Discrete Mathematics and its Applications, Rosen. 01 What kinds of problems are solved in discrete math? 02 Boolean Algebra and formal logic 03 More logic: quantifiers and predicates 04 Sets 05 Diagonalization, functions and sums review 06 Basic arithmetic and geometric sums, closed forms. 07 Chinese rings puzzle 08 Solving recurrence equations 09 Solving recurrence equations (cont.) 10 Mathematical induction 11 Combinations and permutations 12 Counting Problems 13 Counting problems 14 Counting problems using combinations, distributions 15 Counting problems using combinations, distributions 16 The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick. 17 Equivalence Relations and Partial Orders 18 Euclid's Algorithm 19 Recitation -- a combinatorial card trick 20 Cryptography More information about this course: www.archive.org www.aduni.org Licensed under Creative Commons Attribution ShareAlike ...
http://wn.com/Arsdigita_02_Discrete_Mathematics_Lecture_6_20
Course 02: Discrete Mathematics (Arsdigita University) This course covers the mathematical topics most directly related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis is placed on providing a context for the application of the mathematics within computer science. Instructor: Shai Simonson Text: Discrete Mathematics and its Applications, Rosen. 01 What kinds of problems are solved in discrete math? 02 Boolean Algebra and formal logic 03 More logic: quantifiers and predicates 04 Sets 05 Diagonalization, functions and sums review 06 Basic arithmetic and geometric sums, closed forms. 07 Chinese rings puzzle 08 Solving recurrence equations 09 Solving recurrence equations (cont.) 10 Mathematical induction 11 Combinations and permutations 12 Counting Problems 13 Counting problems 14 Counting problems using combinations, distributions 15 Counting problems using combinations, distributions 16 The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick. 17 Equivalence Relations and Partial Orders 18 Euclid's Algorithm 19 Recitation -- a combinatorial card trick 20 Cryptography More information about this course: www.archive.org www.aduni.org Licensed under Creative Commons Attribution ShareAlike ...
- published: 07 Jun 2012
- views: 3376
- author: Jaap Weel
Arsdigita 02 (Discrete Mathematics) Lecture 7/20
- Order: Reorder
- Duration: 82:05
- Updated: 27 Dec 2012
Course 02: Discrete Mathematics (Arsdigita University) This course covers the mathematical topics most directly related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis is placed on providing a context for the application of the mathematics within computer science. Instructor: Shai Simonson Text: Discrete Mathematics and its Applications, Rosen. 01 What kinds of problems are solved in discrete math? 02 Boolean Algebra and formal logic 03 More logic: quantifiers and predicates 04 Sets 05 Diagonalization, functions and sums review 06 Basic arithmetic and geometric sums, closed forms. 07 Chinese rings puzzle 08 Solving recurrence equations 09 Solving recurrence equations (cont.) 10 Mathematical induction 11 Combinations and permutations 12 Counting Problems 13 Counting problems 14 Counting problems using combinations, distributions 15 Counting problems using combinations, distributions 16 The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick. 17 Equivalence Relations and Partial Orders 18 Euclid's Algorithm 19 Recitation -- a combinatorial card trick 20 Cryptography More information about this course: www.archive.org www.aduni.org Licensed under Creative Commons Attribution ShareAlike ...
http://wn.com/Arsdigita_02_Discrete_Mathematics_Lecture_7_20
Course 02: Discrete Mathematics (Arsdigita University) This course covers the mathematical topics most directly related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis is placed on providing a context for the application of the mathematics within computer science. Instructor: Shai Simonson Text: Discrete Mathematics and its Applications, Rosen. 01 What kinds of problems are solved in discrete math? 02 Boolean Algebra and formal logic 03 More logic: quantifiers and predicates 04 Sets 05 Diagonalization, functions and sums review 06 Basic arithmetic and geometric sums, closed forms. 07 Chinese rings puzzle 08 Solving recurrence equations 09 Solving recurrence equations (cont.) 10 Mathematical induction 11 Combinations and permutations 12 Counting Problems 13 Counting problems 14 Counting problems using combinations, distributions 15 Counting problems using combinations, distributions 16 The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick. 17 Equivalence Relations and Partial Orders 18 Euclid's Algorithm 19 Recitation -- a combinatorial card trick 20 Cryptography More information about this course: www.archive.org www.aduni.org Licensed under Creative Commons Attribution ShareAlike ...
- published: 07 Jun 2012
- views: 1430
- author: Jaap Weel
DISCRETE MATHS -- INDUCTION AND RECURSION -. LECTURE 9.
- Order: Reorder
- Duration: 37:30
- Updated: 25 Mar 2012
DISCRETE MATHS -- INDUCTION AND RECURSION -. LECTURE 9. - April. 27th - ESCUELA UNIVERSITARIA DE INFORMÁTICA - PACO GÓMEZ Descripción: Lecture 8. Problem set: exercises 9 to 11. Review of lists: definition of a list, length of a list, rest of a list, head of a list, concatenation of lists. Structural tree of a list. Depth of a list
http://wn.com/DISCRETE_MATHS_-_INDUCTION_AND_RECURSION_LECTURE_9
DISCRETE MATHS -- INDUCTION AND RECURSION -. LECTURE 9. - April. 27th - ESCUELA UNIVERSITARIA DE INFORMÁTICA - PACO GÓMEZ Descripción: Lecture 8. Problem set: exercises 9 to 11. Review of lists: definition of a list, length of a list, rest of a list, head of a list, concatenation of lists. Structural tree of a list. Depth of a list
- published: 13 May 2011
- views: 782
- author: UPM
Arsdigita 02 (Discrete Mathematics) Lecture 13/20
- Order: Reorder
- Duration: 69:14
- Updated: 04 Jan 2013
Course 02: Discrete Mathematics (Arsdigita University) This course covers the mathematical topics most directly related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis is placed on providing a context for the application of the mathematics within computer science. Instructor: Shai Simonson Text: Discrete Mathematics and its Applications, Rosen. 01 What kinds of problems are solved in discrete math? 02 Boolean Algebra and formal logic 03 More logic: quantifiers and predicates 04 Sets 05 Diagonalization, functions and sums review 06 Basic arithmetic and geometric sums, closed forms. 07 Chinese rings puzzle 08 Solving recurrence equations 09 Solving recurrence equations (cont.) 10 Mathematical induction 11 Combinations and permutations 12 Counting Problems 13 Counting problems 14 Counting problems using combinations, distributions 15 Counting problems using combinations, distributions 16 The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick. 17 Equivalence Relations and Partial Orders 18 Euclid's Algorithm 19 Recitation -- a combinatorial card trick 20 Cryptography More information about this course: www.archive.org www.aduni.org Licensed under Creative Commons Attribution ShareAlike ...
http://wn.com/Arsdigita_02_Discrete_Mathematics_Lecture_13_20
Course 02: Discrete Mathematics (Arsdigita University) This course covers the mathematical topics most directly related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis is placed on providing a context for the application of the mathematics within computer science. Instructor: Shai Simonson Text: Discrete Mathematics and its Applications, Rosen. 01 What kinds of problems are solved in discrete math? 02 Boolean Algebra and formal logic 03 More logic: quantifiers and predicates 04 Sets 05 Diagonalization, functions and sums review 06 Basic arithmetic and geometric sums, closed forms. 07 Chinese rings puzzle 08 Solving recurrence equations 09 Solving recurrence equations (cont.) 10 Mathematical induction 11 Combinations and permutations 12 Counting Problems 13 Counting problems 14 Counting problems using combinations, distributions 15 Counting problems using combinations, distributions 16 The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick. 17 Equivalence Relations and Partial Orders 18 Euclid's Algorithm 19 Recitation -- a combinatorial card trick 20 Cryptography More information about this course: www.archive.org www.aduni.org Licensed under Creative Commons Attribution ShareAlike ...
- published: 07 Jun 2012
- views: 567
- author: Jaap Weel
Discrete Math 2 - Tutorial 1 - Rule of Sum and Product
- Order: Reorder
- Duration: 6:01
- Updated: 20 Nov 2012
First video for Discrete math 2. Introduction to counting. Rule of Sum and Product. Please rate, comment and subscribe.
http://wn.com/Discrete_Math_2_Tutorial_1_Rule_of_Sum_and_Product
First video for Discrete math 2. Introduction to counting. Rule of Sum and Product. Please rate, comment and subscribe.
- published: 16 Dec 2011
- views: 2741
- author: coursehack
Arsdigita 02 (Discrete Mathematics) Lecture 17/20
- Order: Reorder
- Duration: 65:57
- Updated: 08 Dec 2012
Course 02: Discrete Mathematics (Arsdigita University) This course covers the mathematical topics most directly related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis is placed on providing a context for the application of the mathematics within computer science. Instructor: Shai Simonson Text: Discrete Mathematics and its Applications, Rosen. 01 What kinds of problems are solved in discrete math? 02 Boolean Algebra and formal logic 03 More logic: quantifiers and predicates 04 Sets 05 Diagonalization, functions and sums review 06 Basic arithmetic and geometric sums, closed forms. 07 Chinese rings puzzle 08 Solving recurrence equations 09 Solving recurrence equations (cont.) 10 Mathematical induction 11 Combinations and permutations 12 Counting Problems 13 Counting problems 14 Counting problems using combinations, distributions 15 Counting problems using combinations, distributions 16 The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick. 17 Equivalence Relations and Partial Orders 18 Euclid's Algorithm 19 Recitation -- a combinatorial card trick 20 Cryptography More information about this course: www.archive.org www.aduni.org Licensed under Creative Commons Attribution ShareAlike ...
http://wn.com/Arsdigita_02_Discrete_Mathematics_Lecture_17_20
Course 02: Discrete Mathematics (Arsdigita University) This course covers the mathematical topics most directly related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis is placed on providing a context for the application of the mathematics within computer science. Instructor: Shai Simonson Text: Discrete Mathematics and its Applications, Rosen. 01 What kinds of problems are solved in discrete math? 02 Boolean Algebra and formal logic 03 More logic: quantifiers and predicates 04 Sets 05 Diagonalization, functions and sums review 06 Basic arithmetic and geometric sums, closed forms. 07 Chinese rings puzzle 08 Solving recurrence equations 09 Solving recurrence equations (cont.) 10 Mathematical induction 11 Combinations and permutations 12 Counting Problems 13 Counting problems 14 Counting problems using combinations, distributions 15 Counting problems using combinations, distributions 16 The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick. 17 Equivalence Relations and Partial Orders 18 Euclid's Algorithm 19 Recitation -- a combinatorial card trick 20 Cryptography More information about this course: www.archive.org www.aduni.org Licensed under Creative Commons Attribution ShareAlike ...
- published: 07 Jun 2012
- views: 892
- author: Jaap Weel
Arsdigita 02 (Discrete Mathematics) Lecture 8/20
- Order: Reorder
- Duration: 70:22
- Updated: 02 Dec 2012
Course 02: Discrete Mathematics (Arsdigita University) This course covers the mathematical topics most directly related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis is placed on providing a context for the application of the mathematics within computer science. Instructor: Shai Simonson Text: Discrete Mathematics and its Applications, Rosen. 01 What kinds of problems are solved in discrete math? 02 Boolean Algebra and formal logic 03 More logic: quantifiers and predicates 04 Sets 05 Diagonalization, functions and sums review 06 Basic arithmetic and geometric sums, closed forms. 07 Chinese rings puzzle 08 Solving recurrence equations 09 Solving recurrence equations (cont.) 10 Mathematical induction 11 Combinations and permutations 12 Counting Problems 13 Counting problems 14 Counting problems using combinations, distributions 15 Counting problems using combinations, distributions 16 The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick. 17 Equivalence Relations and Partial Orders 18 Euclid's Algorithm 19 Recitation -- a combinatorial card trick 20 Cryptography More information about this course: www.archive.org www.aduni.org Licensed under Creative Commons Attribution ShareAlike ...
http://wn.com/Arsdigita_02_Discrete_Mathematics_Lecture_8_20
Course 02: Discrete Mathematics (Arsdigita University) This course covers the mathematical topics most directly related to computer science. Topics include: logic, relations, functions, basic set theory, countability and counting arguments, proof techniques, mathematical induction, graph theory, combinatorics, discrete probability, recursion, recurrence relations, and number theory. Emphasis is placed on providing a context for the application of the mathematics within computer science. Instructor: Shai Simonson Text: Discrete Mathematics and its Applications, Rosen. 01 What kinds of problems are solved in discrete math? 02 Boolean Algebra and formal logic 03 More logic: quantifiers and predicates 04 Sets 05 Diagonalization, functions and sums review 06 Basic arithmetic and geometric sums, closed forms. 07 Chinese rings puzzle 08 Solving recurrence equations 09 Solving recurrence equations (cont.) 10 Mathematical induction 11 Combinations and permutations 12 Counting Problems 13 Counting problems 14 Counting problems using combinations, distributions 15 Counting problems using combinations, distributions 16 The pigeonhole principle and examples. The inclusion/exclusion theorem and advanced examples. A combinatorial card trick. 17 Equivalence Relations and Partial Orders 18 Euclid's Algorithm 19 Recitation -- a combinatorial card trick 20 Cryptography More information about this course: www.archive.org www.aduni.org Licensed under Creative Commons Attribution ShareAlike ...
- published: 07 Jun 2012
- views: 1068
- author: Jaap Weel
Praxis 2 (0061) Discrete Math 2
- Order: Reorder
- Duration: 11:57
- Updated: 03 Nov 2012
sites.google.com More discrete math review problems for the praxis test
http://wn.com/Praxis_2_0061_Discrete_Math_2
sites.google.com More discrete math review problems for the praxis test
- published: 13 Mar 2011
- views: 2461
- author: shaunteaches
photo: AP / Juan Karita
04 Jan 2013
Argentine President Cristina Fernandez de Kirchner has released a public letter urging the United Kingdom to relinquish its control over the disputed Falkland Islands. ... The veterans delivered a letter addressed to the British Ambassador in Argentina, John Freeman, criticising the British government for refusing to discuss the sovereignty of the islands with the Argentine government ... 508 Source.. Al Jazeera and agencies. Share....(size: 19.6Kb)
photo: WN / Imran Nissar
06 Jan 2013
ISLAMABAD - The Pakistani military is accusing forces from neighboring India of crossing the boundary between the two sides' forces in Kashmir and attacking a Pakistani border post, killing a soldier. The military's public relations office said in a statement Sunday that another Pakistani soldier was critically wounded in the incident early Sunday. They said troops are still exchanging gunfire in the area ... All rights reserved ... Parker ... ....(size: 4.3Kb)
photo: UN / Evan Schneider
06 Jan 2013
LONDON -- They have barely 3,000 inhabitants and far more sheep than people. They are more than 8,700 miles from London. For much of the year, they are bitterly cold and wind-swept, with only open sea and ice between them and Antarctica ... But the Falkland Islands, the focus of a short war between Britain and Argentina in 1982, have been in the headlines again recently ... Their arguments suggest what has long been evident ... Ms ... Mr ... Cameron said....(size: 5.1Kb)
photo: AP / SANA
05 Jan 2013
Syrian government forces have bombarded several rebel strongholds in the suburbs of Damascus to quell the fighters battling the regime of President Bashar al-Assad from reaching the capital. Activists on Saturday said troops leveled their bombardment campaign on the districts of Hajar al-Aswad and Qaboun in southern Damascus while they continued to shell the suburbs of Daraya, Beit Hahm, Zibdeen, Maadamiya and Jisteen ... 445 Source. ... ....(size: 19.0Kb)
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05 Jan 2013
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noodls
16 Nov 2012
Two individuals are inducted each year ... This year's inductees are. ... She has made significant research contributions in several areas of mathematics and computer science, including functional analysis, discrete mathematics, theoretical computer science, human-computer interaction, gender issues in information technology and interactive multimedia for mathematics education ... Klawe's current research focuses on discrete mathematics....(size: 2.9Kb)
noodls
14 Nov 2012
"In other words," Torquato said, "what appear to be abstract mathematical problems can be related to the cooling of special liquids that undergo phase transitions to crystal states at a temperature of absolute zero ... The covering and quantizer problems are major problems in discrete mathematics and information theory, according to Henry Cohn, a mathematician with Microsoft Research New England in Cambridge, Mass....(size: 9.4Kb)
PR Newswire
14 Nov 2012
Google ... Faculty Scholar Program contributes to the VEF mission of bringing and closer together through educational exchanges related to science, engineering, mathematics, medicine, and technology ... Rosenfeld introduced discrete mathematics and discrete optimization into the Vietnamese curriculum ... Rosenfeld's fields of expertise are Mathematics and Computer Science and he will be teaching a course, entitled Discrete Optimization ... ....(size: 13.3Kb)
Jamaica Observer
16 Sep 2012
Packer is also urging the student-teachers to utilise calculators along with information and communication technology to improve learning in mathematics and other subject areas ... Stating that the use of calculators was not encouraged enough at the primary or lower-secondary level, he added that "calculator or computer utilisation should be an integral part of any discrete mathematics or statistics course" ... ....(size: 2.4Kb)
Huffington Post
01 Aug 2012
Hacker argues that the teaching of mathematics takes a toll beginning with students in junior high or middle school ... Why pick on mathematics? ... Detailed calculus-based graphing techniques (which can now be done better by computer) could give way to a greater emphasis on "discrete mathematics," including aspects of probability and statistics, which might better be termed "reasoning and reckoning." ... Mathematics for minorities....(size: 9.4Kb)
Kashmir Observer
31 Jul 2012
Dr Pirzada has published more than 80 research papers in various international journals and is the member of American Mathematical Society, Indian Mathematical Society, Research Group in Mathematical Inequalities and Applications (RIGMA) Australia, Allahabad Mathematical Society, Korean Society of Industrial and Applied Mathematics, Advances in Discrete Mathematics and Applications, World Turkic Math....(size: 1.9Kb)
The Examiner
05 Jul 2012
Rousseau, a retired University of Memphis mathematics professor, has been named one of two winners of the mathematics world’s highest honor, the Paul Erdös Award ... Rousseau’s reputation is that of an avid creator and solver of mathematical problems ... He got his PhD degree in the field of Discrete Mathematics from Stockholm University and, since 1974, he has been teaching at this institution....(size: 7.4Kb)
Business Wire
19 Jun 2012
DUBLIN--(BUSINESS WIRE)--Research and Markets (http.//www.researchandmarkets.com/research/xdvtq6/fibonacci_and_cata) has announced the addition of John Wiley and Sons Ltd's new book "Fibonacci and Catalan Numbers. An Introduction" to their offering ... ... Author ... GRIMALDI, PhD, is Professor of Mathematics at Rose-Hulman Institute of Technology ... Grimaldi has published numerous articles in discrete mathematics, combinatorics, and graph theory....(size: 4.3Kb)
The State
10 Jun 2012
At first blush he seems like any other 17-year-old. He's wearing a T-shirt and jeans and flashes a big, though halfway shy smile ... "He's quite a guy," says Don Jordan, a mathematics professor at the University of South Carolina.. Video from around the world ... An early interest ... So far Ben has taken six classes at USC – classes such as discrete mathematics and numerical linear algebra – and plans to take four more this summer ... ....(size: 6.7Kb)
For the mathematics journal, see Discrete Mathematics (journal).
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic[1] – do not vary smoothly in this way, but have distinct, separated values.[2] Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets[3] (sets that have the same cardinality as subsets of the natural numbers, including rational numbers but not real numbers). However, there is no exact, universally agreed, definition of the term "discrete mathematics."[4] Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.
The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business.
Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming languages, cryptography, automated theorem proving, and software development. Conversely, computer implementations are significant in applying ideas from discrete mathematics to real-world problems, such as in operations research.
Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well.
Grand challenges, past and present[link]
The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field. In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved until 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computer assistance).[5]
In logic, the second problem on David Hilbert's list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent. Gödel's second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself. Hilbert's tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. In 1970, Yuri Matiyasevich proved that this could not be done.
The need to break German codes in World War II led to advances in cryptography and theoretical computer science, with the first programmable digital electronic computer being developed at England's Bletchley Park. At the same time, military requirements motivated advances in operations research. The Cold War meant that cryptography remained important, with fundamental advances such as public-key cryptography being developed in the following decades. Operations research remained important as a tool in business and project management, with the critical path method being developed in the 1950s. The telecommunication industry has also motivated advances in discrete mathematics, particularly in graph theory and information theory. Formal verification of statements in logic has been necessary for software development of safety-critical systems, and advances in automated theorem proving have been driven by this need.
Computational geometry has been an important part of the computer graphics incorporated into modern video games and computer-aided design tools.
Several fields of discrete mathematics, particularly theoretical computer science, graph theory, and combinatorics, are important in addressing the challenging bioinformatics problems associated with understanding the tree of life.[6]
Currently, one of the most famous open problems in theoretical computer science is the P = NP problem, which involves the relationship between the complexity classes P and NP. The Clay Mathematics Institute has offered a $1 million USD prize for the first correct proof, along with prizes for six other mathematical problems.[7]
Topics in discrete mathematics[link]
Theoretical computer science[link]
Main article: Theoretical computer science
Theoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on graph theory and logic. Included within theoretical computer science is the study of algorithms for computing mathematical results. Computability studies what can be computed in principle, and has close ties to logic, while complexity studies the time taken by computations. Automata theory and formal language theory are closely related to computability. Petri nets and process algebras are used to model computer systems, and methods from discrete mathematics are used in analyzing VLSI electronic circuits. Computational geometry applies algorithms to geometrical problems, while computer image analysis applies them to representations of images. Theoretical computer science also includes the study of various continuous computational topics.
Information theory[link]
Main article: Information theory
Information theory involves the quantification of information. Closely related is coding theory which is used to design efficient and reliable data transmission and storage methods. Information theory also includes continuous topics such as: analog signals, analog coding, analog encryption.
Logic[link]
Main article: Mathematical logic
Logic is the study of the principles of valid reasoning and inference, as well as of consistency, soundness, and completeness. For example, in most systems of logic (but not in intuitionistic logic) Peirce's law (((P→Q)→P)→P) is a theorem. For classical logic, it can be easily verified with a truth table. The study of mathematical proof is particularly important in logic, and has applications to automated theorem proving and formal verification of software.
Logical formulas are discrete structures, as are proofs, which form finite trees[8] or, more generally, directed acyclic graph structures[9][10] (with each inference step combining one or more premise branches to give a single conclusion). The truth values of logical formulas usually form a finite set, generally restricted to two values: true and false, but logic can also be continuous-valued, e.g., fuzzy logic. Concepts such as infinite proof trees or infinite derivation trees have also been studied,[11] e.g. infinitary logic.
Set theory[link]
Main article: Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red} or the (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications in several areas.
In discrete mathematics, countable sets (including finite sets) are the main focus. The beginning of set theory as a branch of mathematics is usually marked by Georg Cantor's work distinguishing between different kinds of infinite set, motivated by the study of trigonometric series, and further development of the theory of infinite sets is outside the scope of discrete mathematics. Indeed, contemporary work in descriptive set theory makes extensive use of traditional continuous mathematics.
Combinatorics[link]
Main article: Combinatorics
Combinatorics studies the way in which discrete structures can be combined or arranged. Enumerative combinatorics concentrates on counting the number of certain combinatorial objects - e.g. the twelvefold way provides a unified framework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumeration (i.e., determining the number) of combinatorial structures using tools from complex analysis and probability theory. In contrast with enumerative combinatorics which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Design theory is a study of combinatorial designs, which are collections of subsets with certain intersection properties. Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, special functions and orthogonal polynomials. Originally a part of number theory and analysis, partition theory is now considered a part of combinatorics or an independent field. Order theory is the study of partially ordered sets, both finite and infinite.
Graph theory[link]
Main article: Graph theory
Graph theory, the study of graphs and networks, is often considered part of combinatorics, but has grown large enough and distinct enough, with its own kind of problems, to be regarded as a subject in its own right.[12] Graphs are one of the prime objects of study in discrete mathematics. They are among the most ubiquitous models of both natural and human-made structures. They can model many types of relations and process dynamics in physical, biological and social systems. In computer science, they can represent networks of communication, data organization, computational devices, the flow of computation, etc. In mathematics, they are useful in geometry and certain parts of topology, e.g. knot theory. Algebraic graph theory has close links with group theory. There are also continuous graphs, however for the most part research in graph theory falls within the domain of discrete mathematics.
Probability[link]
Main article: Discrete probability theory
Discrete probability theory deals with events that occur in countable sample spaces. For example, count observations such as the numbers of birds in flocks comprise only natural number values {0, 1, 2, ...}. On the other hand, continuous observations such as the weights of birds comprise real number values and would typically be modeled by a continuous probability distribution such as the normal. Discrete probability distributions can be used to approximate continuous ones and vice versa. For highly constrained situations such as throwing dice or experiments with decks of cards, calculating the probability of events is basically enumerative combinatorics.
Number theory[link]
Main article: Number theory
Number theory is concerned with the properties of numbers in general, particularly integers. It has applications to cryptography, cryptanalysis, and cryptology, particularly with regard to modular arithmetic, diophantine equations, linear and quadratic congruences, prime numbers and primality testing. Other discrete aspects of number theory include geometry of numbers. In analytic number theory, techniques from continuous mathematics are also used. Topics that go beyond discrete objects include transcendental numbers, diophantine approximation, p-adic analysis and function fields.
Algebra[link]
Main article: Abstract algebra
Algebraic structures occur as both discrete examples and continuous examples. Discrete algebras include: boolean algebra used in logic gates and programming; relational algebra used in databases; discrete and finite versions of groups, rings and fields are important in algebraic coding theory; discrete semigroups and monoids appear in the theory of formal languages.
Calculus of finite differences, discrete calculus or discrete analysis[link]
Main article: finite difference
A function defined on an interval of the integers is usually called a sequence. A sequence could be a finite sequence from some data source or an infinite sequence from a discrete dynamical system. Such a discrete function could be defined explicitly by a list (if its domain is finite), or by a formula for its general term, or it could be given implicitly by a recurrence relation or difference equation. Difference equations are similar to a differential equations, but replace differentiation by taking the difference between adjacent terms; they can be used to approximate differential equations or (more often) studied in their own right. Many questions and methods concerning differential equations have counterparts for difference equations. For instance where there are integral transforms in harmonic analysis for studying continuous functions or analog signals, there are discrete transforms for discrete functions or digital signals. As well as the discrete metric there are more general discrete or finite metric spaces and finite topological spaces.
Geometry[link]
Main articles: discrete geometry and computational geometry
Discrete geometry and combinatorial geometry are about combinatorial properties of discrete collections of geometrical objects. A long-standing topic in discrete geometry is tiling of the plane. Computational geometry applies algorithms to geometrical problems.
Topology[link]
Although topology is the field of mathematics that formalizes and generalizes the intuitive notion of "continuous deformation" of objects, it gives rise to many discrete topics; this can be attributed in part to the focus on topological invariants, which themselves usually take discrete values. See combinatorial topology, topological graph theory, topological combinatorics, computational topology, discrete topological space, finite topological space, topology (chemistry).
Operations research[link]
Main article: Operations research
Operations research provides techniques for solving practical problems in business and other fields — problems such as allocating resources to maximize profit, or scheduling project activities to minimize risk. Operations research techniques include linear programming and other areas of optimization, queuing theory, scheduling theory, network theory. Operations research also includes continuous topics such as continuous-time Markov process, continuous-time martingales, process optimization, and continuous and hybrid control theory.
Game theory, decision theory, utility theory, social choice theory[link]
| Cooperate | Defect | |
| Cooperate | -1, -1 | -10, 0 |
| Defect | 0, -10 | -5, -5 |
| Payoff matrix for the Prisoner's dilemma, a common example in game theory. One player chooses a row, the other a column; the resulting pair gives their payoffs | ||
Decision theory is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision.
Utility theory is about measures of the relative economic satisfaction from, or desirability of, consumption of various goods and services.
Social choice theory is about voting. A more puzzle-based approach to voting is ballot theory.
Game theory deals with situations where success depends on the choices of others, which makes choosing the best course of action more complex. There are even continuous games, see differential game. Topics include auction theory and fair division.
Discretization[link]
Main article: Discretization
Discretization concerns the process of transferring continuous models and equations into discrete counterparts, often for the purposes of making calculations easier by using approximations. Numerical analysis provides an important example.
Discrete analogues of continuous mathematics[link]
There are many concepts in continuous mathematics which have discrete versions, such as discrete calculus, discrete probability distributions, discrete Fourier transforms, discrete geometry, discrete logarithms, discrete differential geometry, discrete exterior calculus, discrete Morse theory, difference equations, discrete dynamical systems, and discrete vector measures.
In applied mathematics, discrete modelling is the discrete analogue of continuous modelling. In discrete modelling, discrete formulae are fit to data. A common method in this form of modelling is to use recurrence relations.
Hybrid discrete and continuous mathematics[link]
The time scale calculus is a unification of the theory of difference equations with that of differential equations, which has applications to fields requiring simultaneous modelling of discrete and continuous
See also[link]
- Outline of discrete mathematics
- CyberChase, a show that teaches Discrete Mathematics to children
References[link]
- ^ Richard Johnsonbaugh, Discrete Mathematics, Prentice Hall, 2008.
- ^ Weisstein, Eric W., "Discrete mathematics" from MathWorld.
- ^ Norman L. Biggs, Discrete mathematics, Oxford University Press, 2002.
- ^ Brian Hopkins, Resources for Teaching Discrete Mathematics, Mathematical Association of America, 2008.
- ^ a b Wilson, Robin (2002). Four Colors Suffice. London: Penguin Books. ISBN 0-691-11533-8.
- ^ Trevor R. Hodkinson and John A. N. Parnell, Reconstructing the Tree of Life: Taxonomy and systematics of species rich taxa, CRC Press, 2007, ISBN 0-8493-9579-8, p. 97.
- ^ "Millennium Prize Problems". 2000-05-24. http://www.claymath.org/millennium/. Retrieved 2008-01-12.
- ^ Sjerp Troelstra, Helmut Schwichtenberg, Basic Proof Theory, Cambridge University Press, 2000, ISBN 0-521-77911-1, p. 186.
- ^ Samuel R. Buss, Handbook of Proof Theory (Volume 137 of Studies in logic and the foundations of mathematics), Elsevier, 1998. ISBN 0-444-89840-9, p 13.
- ^ Stephan Schulz, "Learning Search Control Knowledge for Equational Theorem Proving," in KI 2001: Advances in Artificial Intelligence : Joint German/Austrian Conference on AI, Vienna, Austria, September 19-21, 2001 : Proceedings (Volume 2174 of Lecture notes in Artificial Intelligence), Franz Baader, Gerhard Brewka, and Thomas Eiter, eds., Springer, 2001, ISBN 3-540-42612-4, p. 325.
- ^ Cyclic proofs of program termination in separation logic, J Brotherston, R Bornat, C Calcagno, ACM SIGPLAN Notices, Volume 43 , Issue 1 (January 2008)
- ^ Graphs on Surfaces, Bojan Mohar and Carsten Thomassen, Johns Hopkins University press, 2001
Further reading[link]
 |
Wikibooks has a book on the topic of |
- Norman L. Biggs, Discrete Mathematics 2nd ed. Oxford University Press. ISBN 0-19-850717-8, and companion web site including questions together with solutions.
- Ronald Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics
- Donald E. Knuth, The Art of Computer Programming ISBN 978-0-321-75104-1.
- Kenneth H. Rosen, Handbook of Discrete and Combinatorial Mathematics CRC Press. ISBN 0-8493-0149-1.
- Richard Johnsonbaugh, Discrete Mathematics 6th ed. Macmillan. ISBN 0-13-045803-1, and companion web site.
- John Dwyer & Suzy Jagger, Discrete Mathematics for Business & Computing, 1st ed. 2010 ISBN 978-1-907934-00-1.
- Kenneth H. Rosen, Discrete Mathematics and Its Applications 6th ed. McGraw Hill. ISBN 0-07-288008-2, and companion web site.
- Ralph P. Grimaldi, Discrete and Combinatorial Mathematics: An Applied Introduction 5th ed. Addison Wesley. ISBN 0-201-72634-3
- Susanna S. Epp, Discrete Mathematics with Applications Brooks Cole. ISBN 978-0-495-39132-6
- Jiří Matoušek & Jaroslav Nešetřil, Invitation to Discrete Mathematics, OUP, ISBN 978-0-19-850208-1.
- Mathematics Archives, Discrete Mathematics links to syllabi, tutorials, programs, etc.
- Andrew Simpson, Discrete Mathematics by Example McGraw Hill. ISBN 0-07-709840-4
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Wiki Summary
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (sets that have the same cardinality as subsets of the natural numbers, including rational numbers but not real numbers). However, there is no exact, universally agreed, definition of the term "discrete mathematics." Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.
The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business.
