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- ...t system is a '''sunflower''' if all its member sets intersect at the same set of elements. : A set family <math>\mathcal{F}\subseteq 2^X</math> is a '''sunflower''' of size < ...50 KB (8,991 words) - 12:23, 21 May 2023
- A set family <math>\mathcal{F}\subseteq 2^X</math> with the relation <math>\subse A set family <math>\mathcal{F}\subseteq 2^X</math> is an '''antichain''' (also ca ...25 KB (4,389 words) - 09:05, 12 January 2011
- A set family <math>\mathcal{F}\subseteq 2^X</math> with the relation <math>\subse A set family <math>\mathcal{F}\subseteq 2^X</math> is an '''antichain''' (also ca ...25 KB (4,408 words) - 01:20, 10 November 2011
- A set family <math>\mathcal{F}\subseteq 2^X</math> with the relation <math>\subse A set family <math>\mathcal{F}\subseteq 2^X</math> is an '''antichain''' (also ca ...24 KB (4,365 words) - 11:50, 5 June 2013
- A set family <math>\mathcal{F}\subseteq 2^X</math> with the relation <math>\subse A set family <math>\mathcal{F}\subseteq 2^X</math> is an '''antichain''' (also ca ...32 KB (5,800 words) - 07:57, 21 May 2014
- ...t system is a '''sunflower''' if all its member sets intersect at the same set of elements. : A set family <math>\mathcal{F}\subseteq 2^X</math> is a '''sunflower''' of size < ...25 KB (4,523 words) - 13:54, 27 December 2015
- ...t system is a '''sunflower''' if all its member sets intersect at the same set of elements. : A set family <math>\mathcal{F}\subseteq 2^X</math> is a '''sunflower''' of size < ...32 KB (5,780 words) - 02:49, 24 November 2016
- ...t system is a '''sunflower''' if all its member sets intersect at the same set of elements. : A set family <math>\mathcal{F}\subseteq 2^X</math> is a '''sunflower''' of size < ...32 KB (5,780 words) - 13:32, 2 December 2017
- ...t system is a '''sunflower''' if all its member sets intersect at the same set of elements. : A set family <math>\mathcal{F}\subseteq 2^X</math> is a '''sunflower''' of size < ...32 KB (5,780 words) - 07:54, 28 November 2019
- :Let <math>\mathcal{F}\subseteq 2^X</math> be set family and let <math>R\subseteq X</math> be a subset. The '''trace''' of <m ...nsion '''VC dimension'''] is defined by the power of a family to shatter a set. ...25 KB (4,480 words) - 04:58, 17 November 2010
- :Let <math>\mathcal{F}\subseteq 2^X</math> be set family and let <math>R\subseteq X</math> be a subset. The '''trace''' of <m ...nsion '''VC dimension'''] is defined by the power of a family to shatter a set. ...25 KB (4,480 words) - 08:23, 16 August 2011
Page text matches
- #REDIRECT [[Combinatorics (Fall 2010)/Extremal set theory II]] ...62 bytes (7 words) - 06:45, 3 November 2010
- This is a (temporary) homepage for the probability theory course at Zhiyuan college. *[[概率论 (Summer 2014)/Problem Set 1|Problem Set 1]], due on July 25, Friday, in class. ...203 bytes (26 words) - 11:41, 17 July 2014
- This is a (temporary) homepage for the probability theory course at Zhiyuan college. *[[概率论 (Summer 2013)/Problem Set 1|Problem Set 1]], due on July 11, Thursday, in class. ...572 bytes (68 words) - 09:02, 1 August 2013
- ...''B'' is ''greater than or equal to'' (or ''more than or equal'' members) set ''B'' if and only if there is an injective function from ''A'' to ''B''. ...set is only one way of giving a number to the ''size'' of a set. [[Measure theory|Measure]] is different. ...2 KB (375 words) - 06:00, 4 July 2015
- * 概率论（Probability Theory） *[[组合数学 (Fall 2016)/Problem Set 1|Problem Set 1]] due on Oct. 10 in class. ...6 KB (479 words) - 10:20, 12 September 2017
- ...'proper subset'''. We use the symbol ⊆ to say a set is a subset of another set. We can also use ⊂ if it is a proper subset. The symbols ⊃ ⊇ are opposite - ...[interval]] [0;1] is a proper subset of the set of [[real number]]s or the set of [[positive number]]s. ...980 bytes (159 words) - 10:02, 8 June 2017
- ...]], an '''alphabet''' is a [[finite set|finite]] [[empty set|non-empty]] [[set]]. The [[element]]s of an alphabet are called the '''letters''' or '''symbo The set of [[natural number]]s is '''not''' an alphabet because as it is not finite ...3 KB (400 words) - 06:51, 6 July 2016
- ...artesian product''' of ''A'' with ''B'' is written as ''A × B'' and is the set of all [[ordered pair|ordered pairs]] whose first element is a member of '' [[Category:Set theory]] ...497 bytes (82 words) - 22:03, 12 March 2013
- ...algebraic structure]]. It is a [[set]] with a [[binary operation]] on that set. ...that do not have to be different) and returning some other element of that set. ...1 KB (250 words) - 05:20, 8 November 2014
- ...of a [[set]] is the cardinal number that tells how many things are in the set. [[Category:Number theory]] ...1 KB (154 words) - 17:56, 8 October 2014
- * 概率论（Probability Theory） *[[组合数学 (Spring 2014)/Problem Set 1|Problem Set 1]], due on March 27, Thursday, in class. ...9 KB (998 words) - 05:12, 11 June 2014
- ...set. Both elements b and c of the first set map to element z of the second set.]] The relation is ''homogeneous'' when it is formed with one set. ...4 KB (674 words) - 06:35, 19 August 2017
- * 概率论（Probability Theory） *[[组合数学 (Spring 2013)/Problem Set 1|Problem Set 1]], due on April 4, Thursday, in class. ...11 KB (1,243 words) - 12:46, 15 September 2017
- * 概率论（Probability Theory） *[[组合数学 (Fall 2015)/Problem Set 1|Problem Set 1]] due on Oct. 13 in class. ...11 KB (1,070 words) - 12:46, 15 September 2017
- * 概率论（Probability Theory） *[[组合数学 (Fall 2017)/Problem Set 1|Problem Set 1]] due on Oct 9, in class. ...11 KB (1,223 words) - 07:38, 2 January 2018
- * 概率论（Probability Theory） * (2011/09/15) [[组合数学 (Fall 2011)/Problem set 1|Problem set 1]] due on Sept 22, in class. ...13 KB (1,447 words) - 12:47, 15 September 2017
- * 概率论（Probability Theory） *[[组合数学 (Fall 2019)/Problem Set 1|Problem Set 1]] due on Sept 25, in class. [[组合数学 (Fall 2019)/作业1已提交名单 | 当前作业1已提交名单]]. ...12 KB (1,290 words) - 06:43, 27 December 2019
- * 概率论（Probability Theory） * (2010/09/17) [[Combinatorics (Fall 2010)/Problem set 1|Problem set 1]] due on Sept 25, in class. ...12 KB (1,494 words) - 14:27, 3 September 2011
- * 概率论（Probability Theory） * [[组合数学 (Fall 2023)/Problem Set 1|Problem Set 1]]，请在 2023/03/24 上课之前提交到 [mailto:njucomb23@163.com njucomb23@163.com] (文件名 ...14 KB (1,438 words) - 18:58, 9 April 2024
- ...ey wouldn't miss any of the numbers. Sometimes when people say 'countable set' they mean countable and infinite. [[Georg Cantor]] coined the term. Countable sets include all sets with a [[finite set|finite]] number of members, no matter how many. ...2 KB (379 words) - 14:57, 30 April 2017
- * the set of all words over <math>{a, b}\,</math> * the set <math>\left \{ a^{n}\right\}</math>, where <math>n\,</math> is a [[natural ...5 KB (710 words) - 00:57, 25 February 2017
- ...set definition of Georg Cantor.png|thumb|Cantor's original definition of a set: ''By an aggregate (..) we are to understand any collection into a whole (. ...Anything'' can be a member of a set. A set can be a member of a set. (If a set is a member of itself, '''beware''' of [[Russell's paradox]].) ...10 KB (1,884 words) - 16:03, 30 June 2015
- ...''') is a concept from [[mathematics]]. In many ways, multisets are like [[set]]s. Certain items are either elements of that multiset, or they are not. Ho ...me]] factors of a number ''n''. Here the underlying set of elements is the set of prime [[divisor]]s of ''n''. For example the number ''120'' has the [[fa ...2 KB (326 words) - 23:54, 8 March 2013
- ...h theory]] a '''graph''' is used to represent the connection between two [[set]]s. One of the sets contains nodes (which are officially called '''vertices ...t, where an edge that connects to the vertex at both ends (a [[loop (graph theory)|loop]]) is counted twice. ...3 KB (488 words) - 18:43, 22 August 2017
- ..."traditional" or "economic" [[game theory]]. CGT arose in relation to the theory of impartial games, the two-player game of [[Nim]] in particular, with an e Combinatorial Game Theory is largely confined to the study of a subset of combinatorial games which a ...6 KB (1,097 words) - 23:35, 19 September 2015
- The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one *<math>\Omega</math> is a set, called the '''sample space'''. ...7 KB (1,250 words) - 06:01, 27 August 2011
- * 概率论（Probability Theory） * [[组合数学 (Fall 2024)/Problem Set 1|Problem Set 1]] [[组合数学 (Spring 2024)/第一次作业提交名单|第一次作业提交名单]] ...9 KB (950 words) - 18:42, 29 April 2024
- :''This article is about the basic notions. For advanced topics, see [[Group theory]].'' ...ake a third element, also in the set. A familiar example of a group is the set of [[integer]]s together with the [[addition]] operator. ...6 KB (1,063 words) - 03:41, 3 March 2017
- A '''time signature''' is a set of two [[numbers]], one on top of the other one, written right after the [[ {{music theory}} ...3 KB (480 words) - 10:12, 20 June 2016
- ...equations are called [[wikt:equivalent|equivalent]] if they have the same set of [[Algebraic solutions|solutions]]. This means that all solutions of the ...'''. When solving an equation, we need to say in which [[Set (mathematics)|set]] the solutions are allowed. For example, for an equation over the rational ...3 KB (410 words) - 06:25, 6 July 2016
- ...real number can simply be written as <math>a + 0 \cdot i</math> or as the set {{math|(''a'', 0)}}.<ref name="wa" /> The set of all complex numbers is usually written as <math>\mathbb{C}</math>. ...6 KB (917 words) - 00:53, 5 February 2017
- Usually the input "data stream" is from a massive data set (e.g. search engine data), so you cannot afford storing the entire input. M (Interviewing problem of Theory Group in Microsoft Research Asia for student interns.) ...3 KB (443 words) - 01:40, 19 September 2011
- In formal [[number theory]] a '''Gödel numbering''' is a [[function (mathematics)|function]] which as ...equivalence theorem]] states criteria for which those numberings of the set of computable functions are Gödel numberings. ...5 KB (850 words) - 23:57, 1 January 2015
- == Assumption throughout Problem Set 3== == Problem 3 (Probability meets graph theory) == ...7 KB (1,107 words) - 07:46, 25 April 2023
- | rowspan=2|[[set theory|set membership]] | a∈S means that a is an element of the set S ...8 KB (1,182 words) - 07:31, 3 November 2013
- ...osure of [[intersection]]. Open sets have infinite closure of [[union (set theory)|union]]. That is, in mathematical notation, if A<sub>0</sub>, and A<sub>1< ...2 KB (271 words) - 03:33, 15 January 2015
- ...a [[coordinate system]]. This means that every place can be specified by a set of three numbers, called coordinates. ...ne by the Babylonians; Ancient Greeks, like [[Ptolemy]] later extended the theory. ...2 KB (271 words) - 22:25, 4 November 2016
- The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one *<math>\Omega</math> is a set, called the '''sample space'''. ...16 KB (2,701 words) - 02:51, 18 August 2011
- <math>\mathbf{Z}</math> or <math>\mathbb{Z}</math> is the name of the [[set]] of integers. (<math>\mathbf{Z}</math>, +, 0) is an [[abelian group]]. [[Category:Number theory]] ...2 KB (304 words) - 08:49, 14 March 2017
- ...ht, "Chromatic Features of E♭-Major Works of the Classical Period" ''Music Theory Spectrum'' '''22''' 2 (2000): 178</ref> ...t to A and E a [[perfect fifth|fifth]] apart. In most other keys, they are set a [[perfect fourth|fourth]] apart. ...3 KB (414 words) - 12:26, 11 March 2013
- Normal [[hotel]]s have a set number of rooms. This number is [[finite]]. Once every room has been assign ...d-numbered rooms is the same as the cardinality of the [[Set (mathematics)|set]] of all rooms. Indeed, infinite sets are characterized as sets that have p ...5 KB (910 words) - 20:39, 8 September 2016
- *[[高级算法 (Fall 2020)/Problem Set 1|Problem Set 1]] due on 2020/10/14 课上交纸质版作业（可以打印电子稿）. [[高级算法 (Fall 2020)/第一次作业提交名单|第一次作 *[[高级算法 (Fall 2020)/Problem Set 2|Problem Set 2]] due on 2020/11/18 课上交纸质版作业（可以打印电子稿）. [[高级算法 (Fall 2020)/第二次作业提交名单|第二次作 ...10 KB (797 words) - 08:56, 13 January 2021
- *[[随机算法 (Spring 2014)/Problem Set 1|Problem Set 1]], due on March 18, Tuesday, in class. *[[随机算法 (Spring 2014)/Problem Set 2|Problem Set 2]], due on April 8, Tuesday, in class. ...9 KB (846 words) - 07:34, 2 June 2014
- ...[[Abstract algebra]] || [[Linear algebra]] || [[Order theory]] || [[Graph theory]] ...l equation]]s || [[Dynamical systems theory|Dynamical systems]] || [[Chaos theory]] ...9 KB (1,088 words) - 18:04, 22 August 2017
- ...'median''' is a [[number]]. This number has the property that it divides a set of observed values in two equal halves, so that half of the values are belo ...es while the other numbers of this set are kept the same, the mean of this set of numbers changes, but the median does not. ...4 KB (751 words) - 08:12, 21 August 2016
- '''Probability Theory''' <br> & '''Mathematical Statistics'''</font> This is the webpage for the ''Probability Theory and Mathematical Statistics'' (概率论与数理统计) class of Spring 2024. Students who ...15 KB (1,465 words) - 13:36, 30 April 2024
- *[[随机算法 (Spring 2013)/Problem Set 1|Problem Set 1]], due on March 26, Tuesday, in class. *[[随机算法 (Spring 2013)/Problem Set 2|Problem Set 2]], due on April 23, Tuesday, in class. ...10 KB (1,029 words) - 12:44, 15 September 2017
- *[[随机算法 (Fall 2015)/Problem Set 1|Problem Set 1]], due on Oct 9, Friday, in class. *[[随机算法 (Fall 2015)/Problem Set 2|Problem Set 2]], due on Nov 27, Friday, in class. ...9 KB (893 words) - 12:43, 15 September 2017
- * (06/29/2010) [[Randomized Algorithms (Spring 2010)/Problem Set 6 | Problem Set 6]] makeup assignment, hand in before the final exam. * (06/01/2010) [[Randomized Algorithms (Spring 2010)/Problem Set 5 | Problem Set 5]] due on June 15 , Tuesday, in class. 中英文不限。 ...12 KB (1,315 words) - 03:37, 18 July 2011
- ...nt of the second set''. This statement works well for sets with a [[Finite set|finite]] number of elements. It is less intuitive for sets with an infinite ...he natural numbers are called at most countable. With that definition, the set of rational numbers / fractions is countable. ...12 KB (1,234 words) - 02:16, 5 September 2016
- *[[高级算法 (Fall 2023)/Problem Set 1|Problem Set 1]] 请在 2023/11/14 上课之前(14:00 UTC+8)提交到 [mailto:njuadvalg23@163.com njuadva *[[高级算法 (Fall 2023)/Problem Set 2|Problem Set 2]] 请在 2023/12/19 上课之前(14:00 UTC+8)提交到 [mailto:njuadvalg23@163.com njuadva ...13 KB (1,427 words) - 15:57, 9 January 2024
- The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one *<math>\Omega</math> is a set, called the '''sample space'''. ...15 KB (2,672 words) - 09:10, 22 September 2016
- *[[set theory]] *[[Banach space]] theory ...16 KB (2,241 words) - 05:01, 18 January 2017
- == Assumption throughout Problem Set 2== ...(Hint: Try expressing the event [math]Y=aX+b\le y[/math] by countably many set operations on the events defined on [math]X[/math].)</li> ...14 KB (2,403 words) - 10:41, 7 April 2023
- ...llest number <math>R(k,\ell)</math> satisfying the condition in the Ramsey theory is called the '''Ramsey number'''. :Let <math>A_1,A_2,\ldots,A_n</math> be a set of events, and assume that the following hold: ...17 KB (3,025 words) - 11:53, 21 November 2011
- * (2011/09/19) [[随机算法 (Fall 2011)/Problem set 1|Problem set 1]] due on Sept 26, in class. * (2011/10/10) [[随机算法 (Fall 2011)/Problem set 2|Problem set 2]] due on Oct 24, in class. ...12 KB (1,037 words) - 12:45, 15 September 2017
- The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one *<math>\Omega</math> is a set, called the '''sample space'''. ...17 KB (3,030 words) - 10:34, 12 September 2023
- The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one *<math>\Omega</math> is a set, called the '''sample space'''. ...17 KB (3,030 words) - 11:25, 8 September 2020
- The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one *<math>\Omega</math> is a set, called the '''sample space'''. ...17 KB (3,030 words) - 11:58, 14 September 2018
- The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one *<math>\Omega</math> is a set, called the '''sample space'''. ...17 KB (3,030 words) - 09:07, 24 August 2021
- The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one *<math>\Omega</math> is a set, called the '''sample space'''. ...17 KB (3,030 words) - 00:13, 2 September 2019
- The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one *<math>\Omega</math> is a set, called the '''sample space'''. ...17 KB (3,030 words) - 07:31, 5 September 2022
- The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one *<math>\Omega</math> is a set, called the '''sample space'''. ...17 KB (3,030 words) - 07:28, 20 September 2017
- ...been used in proofs of many important results in computational complexity theory, such as [http://en.wikipedia.org/wiki/SL_(complexity) SL]=[http://en.wikip * The '''Edge Boundary''' of a set <math>S\subset V</math>, denoted <math>\partial S\,</math>, is <math>\parti ...8 KB (1,407 words) - 02:23, 25 July 2011
- ...ord.</ref><sup>p160</sup> It is a central idea in [[evolution|evolutionary theory]]. Fitness is usually equal to the proportion of the individual's [[gene]]s ...well an organism can reproduce, and spread its genes to its offspring. The theory of inclusive fitness says that the fitness of an organism is also increased ...5 KB (696 words) - 14:26, 23 June 2016
- A group <math>(G,\cdot)</math> is set <math>G</math> along with a binary operator <math>\cdot</math> which satisf :Let <math>S_n</math> denote the set of all permutations of <math>[n]</math>. It is easy to verify that <math>S_ ...19 KB (3,695 words) - 03:56, 27 October 2015
- A group <math>(G,\cdot)</math> is set <math>G</math> along with a binary operator <math>\cdot</math> which satisf :Let <math>S_n</math> denote the set of all permutations of <math>[n]</math>. It is easy to verify that <math>S_ ...19 KB (3,695 words) - 14:25, 29 March 2023
- A group <math>(G,\cdot)</math> is set <math>G</math> along with a binary operator <math>\cdot</math> which satisf :Let <math>S_n</math> denote the set of all permutations of <math>[n]</math>. It is easy to verify that <math>S_ ...19 KB (3,695 words) - 04:35, 17 October 2016
- A group <math>(G,\cdot)</math> is set <math>G</math> along with a binary operator <math>\cdot</math> which satisf :Let <math>S_n</math> denote the set of all permutations of <math>[n]</math>. It is easy to verify that <math>S_ ...19 KB (3,695 words) - 05:36, 19 March 2014
- A group <math>(G,\cdot)</math> is set <math>G</math> along with a binary operator <math>\cdot</math> which satisf :Let <math>S_n</math> denote the set of all permutations of <math>[n]</math>. It is easy to verify that <math>S_ ...19 KB (3,695 words) - 07:16, 8 October 2011
- A group <math>(G,\cdot)</math> is set <math>G</math> along with a binary operator <math>\cdot</math> which satisf :Let <math>S_n</math> denote the set of all permutations of <math>[n]</math>. It is easy to verify that <math>S_ ...19 KB (3,695 words) - 14:54, 28 September 2017
- A group <math>(G,\cdot)</math> is set <math>G</math> along with a binary operator <math>\cdot</math> which satisf :Let <math>S_n</math> denote the set of all permutations of <math>[n]</math>. It is easy to verify that <math>S_ ...19 KB (3,695 words) - 06:16, 8 October 2019
- A group <math>(G,\cdot)</math> is set <math>G</math> along with a binary operator <math>\cdot</math> which satisf :Let <math>S_n</math> denote the set of all permutations of <math>[n]</math>. It is easy to verify that <math>S_ ...19 KB (3,695 words) - 04:27, 27 March 2024
- A group <math>(G,\cdot)</math> is set <math>G</math> along with a binary operator <math>\cdot</math> which satisf :Let <math>S_n</math> denote the set of all permutations of <math>[n]</math>. It is easy to verify that <math>S_ ...19 KB (3,695 words) - 10:46, 17 April 2013
- == Assumption throughout Problem Set 2== ...(Hint: Try expressing the event [math]Y=aX+b\le y[/math] by countably many set operations on the events defined on [math]X[/math].)</li> ...14 KB (2,465 words) - 19:27, 13 April 2024
- The universal quantifier is used to claim that for elements in a set, the elements all match some criteria. Usually, this statement "for all ele The existential quantifier is used to claim that for elements in a set, there exists at least one element that matches some criteria. Usually, thi ...8 KB (1,219 words) - 06:11, 20 June 2016
- '''Probability Theory''' <br> & '''Mathematical Statistics'''</font> This is the webpage for the ''Probability Theory and Mathematical Statistics'' (概率论与数理统计) class of Spring 2023. Students who ...21 KB (2,167 words) - 07:44, 27 February 2024
- In the [[atomic theory]] [[solids]], [[liquids]] and [[gases]] are made of tiny particles called " ...gy to travel through the solid. (Rather like the way energy passes along a set of tumbling dominos). ...3 KB (414 words) - 07:43, 16 June 2017
- :Let <math>\mathcal{F}\subseteq 2^X</math> be set family and let <math>R\subseteq X</math> be a subset. The '''trace''' of <m ...nsion '''VC dimension'''] is defined by the power of a family to shatter a set. ...25 KB (4,480 words) - 04:58, 17 November 2010
- :Let <math>\mathcal{F}\subseteq 2^X</math> be set family and let <math>R\subseteq X</math> be a subset. The '''trace''' of <m ...nsion '''VC dimension'''] is defined by the power of a family to shatter a set. ...25 KB (4,480 words) - 08:23, 16 August 2011
- ...olecular orbital]]s. This is used in the [[Hückel method]], [[ligand field theory]], and the [[Woodward–Hoffmann rules]]. Another idea on a larger scale is t ...ic point groups was published in 1936 by Rosenthal and Murphy.<ref>''Group Theory and the Vibrations of Polyatomic Molecules'' Jenny E. Rosenthal and G. M. M ...19 KB (2,913 words) - 21:36, 4 November 2015
- ...(V,E)</math>, a '''matching''' <math>M\subseteq E</math> is an independent set for edges, that is, for any <math>e_1,e_2\in M</math> that <math>e_1\neq e_ ...override the notation as <math>N(S)=\bigcup_{v\in S}N(v)</math>, i.e. the set of vertices that are adjacent to one of the vertices in <math>S</math>. ...19 KB (3,610 words) - 08:59, 28 May 2014
- ...(V,E)</math>, a '''matching''' <math>M\subseteq E</math> is an independent set for edges, that is, for any <math>e_1,e_2\in M</math> that <math>e_1\neq e_ ...override the notation as <math>N(S)=\bigcup_{v\in S}N(v)</math>, i.e. the set of vertices that are adjacent to one of the vertices in <math>S</math>. ...19 KB (3,610 words) - 14:17, 19 June 2013
- ...t system is a '''sunflower''' if all its member sets intersect at the same set of elements. : A set family <math>\mathcal{F}\subseteq 2^X</math> is a '''sunflower''' of size < ...32 KB (5,780 words) - 13:32, 2 December 2017
- ...t system is a '''sunflower''' if all its member sets intersect at the same set of elements. : A set family <math>\mathcal{F}\subseteq 2^X</math> is a '''sunflower''' of size < ...32 KB (5,780 words) - 02:49, 24 November 2016
- ...t system is a '''sunflower''' if all its member sets intersect at the same set of elements. : A set family <math>\mathcal{F}\subseteq 2^X</math> is a '''sunflower''' of size < ...32 KB (5,780 words) - 07:54, 28 November 2019
- A set family <math>\mathcal{F}\subseteq 2^X</math> with the relation <math>\subse A set family <math>\mathcal{F}\subseteq 2^X</math> is an '''antichain''' (also ca ...32 KB (5,800 words) - 07:57, 21 May 2014
- Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can Let <math>A</math> be the largest independent set in <math>G</math> and let <math>\alpha=|A|</math>. ...18 KB (3,527 words) - 08:55, 4 May 2023
- Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can Let <math>A</math> be the largest independent set in <math>G</math> and let <math>\alpha=|A|</math>. ...18 KB (3,527 words) - 05:10, 9 November 2016
- Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can Let <math>A</math> be the largest independent set in <math>G</math> and let <math>\alpha=|A|</math>. ...18 KB (3,527 words) - 05:55, 12 November 2019
- Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can Let <math>A</math> be the largest independent set in <math>G</math> and let <math>\alpha=|A|</math>. ...18 KB (3,527 words) - 06:10, 22 November 2017
- Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can Let <math>A</math> be the largest independent set in <math>G</math> and let <math>\alpha=|A|</math>. ...19 KB (3,541 words) - 07:47, 25 December 2015
- A set family <math>\mathcal{F}\subseteq 2^X</math> with the relation <math>\subse A set family <math>\mathcal{F}\subseteq 2^X</math> is an '''antichain''' (also ca ...24 KB (4,365 words) - 11:50, 5 June 2013
- == Extremal Graph Theory == Extremal grap theory studies the problems like "how many edges that a graph <math>G</math> can ...21 KB (3,921 words) - 08:23, 13 November 2010
- associated with a set <math>S(v)</math> of at least <math>2\mathrm{e}r</math> colors, where <math Prove the following theorem on extremal graph theory: ...3 KB (522 words) - 11:43, 16 May 2023
- A set family <math>\mathcal{F}\subseteq 2^X</math> with the relation <math>\subse A set family <math>\mathcal{F}\subseteq 2^X</math> is an '''antichain''' (also ca ...25 KB (4,389 words) - 09:05, 12 January 2011
- ...t system is a '''sunflower''' if all its member sets intersect at the same set of elements. : A set family <math>\mathcal{F}\subseteq 2^X</math> is a '''sunflower''' of size < ...25 KB (4,523 words) - 13:54, 27 December 2015
- A set family <math>\mathcal{F}\subseteq 2^X</math> with the relation <math>\subse A set family <math>\mathcal{F}\subseteq 2^X</math> is an '''antichain''' (also ca ...25 KB (4,408 words) - 01:20, 10 November 2011
- Let <math>A_1,A_2,\ldots,A_n\subseteq U</math> be subsets of some finite set <math>U</math>. Here <math>U</math> is some universe of combinatorial objec ...ber of surjective mappings from an <math>n</math>-set to an <math>m</math>-set is given by ...19 KB (3,458 words) - 06:51, 12 October 2015
- Let <math>A_1,A_2,\ldots,A_n\subseteq U</math> be subsets of some finite set <math>U</math>. Here <math>U</math> is some universe of combinatorial objec ...ber of surjective mappings from an <math>n</math>-set to an <math>m</math>-set is given by ...19 KB (3,458 words) - 07:33, 12 March 2014
- Let <math>A_1,A_2,\ldots,A_n\subseteq U</math> be subsets of some finite set <math>U</math>. Here <math>U</math> is some universe of combinatorial objec ...ber of surjective mappings from an <math>n</math>-set to an <math>m</math>-set is given by ...19 KB (3,458 words) - 06:18, 20 March 2013
- Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can Let <math>A</math> be the largest independent set in <math>G</math> and let <math>\alpha=|A|</math>. ...21 KB (3,922 words) - 01:04, 3 November 2011
- Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can Let <math>A</math> be the largest independent set in <math>G</math> and let <math>\alpha=|A|</math>. ...21 KB (3,922 words) - 08:56, 20 May 2013
- Extremal graph theory studies the problems like "how many edges that a graph <math>G</math> can Let <math>A</math> be the largest independent set in <math>G</math> and let <math>\alpha=|A|</math>. ...21 KB (3,922 words) - 10:31, 16 April 2014
- The basic theory, widely agreed, is that highly conserved DNA sequences must have functional ...ved DNA sequences, consist of repeated sequence elements. If only one of a set of a repeated sequences was removed, and the repetitions were not needed, t ...3 KB (479 words) - 04:08, 5 October 2016
- Let <math>A_1,A_2,\ldots,A_n\subseteq U</math> be subsets of some finite set <math>U</math>. Here <math>U</math> is some universe of combinatorial objec ...ber of surjective mappings from an <math>n</math>-set to an <math>m</math>-set is given by ...33 KB (6,227 words) - 12:45, 16 March 2023
- Let <math>A_1,A_2,\ldots,A_n\subseteq U</math> be subsets of some finite set <math>U</math>. Here <math>U</math> is some universe of combinatorial objec ...ber of surjective mappings from an <math>n</math>-set to an <math>m</math>-set is given by ...33 KB (6,227 words) - 07:00, 29 September 2016
- Let <math>A_1,A_2,\ldots,A_n\subseteq U</math> be subsets of some finite set <math>U</math>. Here <math>U</math> is some universe of combinatorial objec ...ber of surjective mappings from an <math>n</math>-set to an <math>m</math>-set is given by ...33 KB (6,227 words) - 06:15, 30 September 2019
- Let <math>A_1,A_2,\ldots,A_n\subseteq U</math> be subsets of some finite set <math>U</math>. Here <math>U</math> is some universe of combinatorial objec ...ber of surjective mappings from an <math>n</math>-set to an <math>m</math>-set is given by ...33 KB (6,227 words) - 09:51, 19 March 2024
- Let <math>A_1,A_2,\ldots,A_n\subseteq U</math> be subsets of some finite set <math>U</math>. Here <math>U</math> is some universe of combinatorial objec ...ber of surjective mappings from an <math>n</math>-set to an <math>m</math>-set is given by ...33 KB (6,227 words) - 11:45, 15 October 2017
- ...he problem and an output is called a '''solution''' to that instance. The theory of complexity deals almost exclusively with [http://en.wikipedia.org/wiki/D ...h "yes" answers. A decision problem can be equivalently represented as the set of all positive instances, denoted <math>L</math>. We call <math>L</math> a ...11 KB (1,828 words) - 06:00, 27 August 2011
- ...egral]] taken over a [[surface]] (which may be a curve [[Set (mathematics)|set]] in [[Space (physics)|space]]). Just as a [[line integral]] handles one di ...egrals have applications in [[physics]], particularly with the [[classical theory]] of [[electromagnetism]]. ...10 KB (1,551 words) - 10:37, 24 June 2017
- ...(V,E)</math>, a '''matching''' <math>M\subseteq E</math> is an independent set for edges, that is, for any <math>e_1,e_2\in M</math> that <math>e_1\neq e_ ...override the notation as <math>N(S)=\bigcup_{v\in S}N(v)</math>, i.e. the set of vertices that are adjacent to one of the vertices in <math>S</math>. ...23 KB (4,382 words) - 05:07, 5 November 2010
- ...(V,E)</math>, a '''matching''' <math>M\subseteq E</math> is an independent set for edges, that is, for any <math>e_1,e_2\in M</math> that <math>e_1\neq e_ ...override the notation as <math>N(S)=\bigcup_{v\in S}N(v)</math>, i.e. the set of vertices that are adjacent to one of the vertices in <math>S</math>. ...23 KB (4,382 words) - 02:41, 17 August 2011
- ...n of compounds with more than two functional groups. He also developed the theory of polymer networks or [[gels]]. ...Development Company.<ref name=nb1 /> He developed a statistical mechanical theory for polymer mixtures there. ...9 KB (1,269 words) - 01:38, 22 March 2015
- == Principles in probability theory == * '''Basics of probability theory''': probability space, events, the union bound, independence, conditional p ...22 KB (3,591 words) - 10:45, 4 March 2013
- == Principles in probability theory == * '''Basics of probability theory''': probability space, events, the union bound, independence, conditional p ...22 KB (3,591 words) - 03:54, 17 February 2014
- ...t system is a '''sunflower''' if all its member sets intersect at the same set of elements. : A set family <math>\mathcal{F}\subseteq 2^X</math> is a '''sunflower''' of size < ...50 KB (8,991 words) - 12:23, 21 May 2023
- ...(V,E)</math>, a '''matching''' <math>M\subseteq E</math> is an independent set for edges, that is, for any <math>e_1,e_2\in M</math> that <math>e_1\neq e_ ...override the notation as <math>N(S)=\bigcup_{v\in S}N(v)</math>, i.e. the set of vertices that are adjacent to one of the vertices in <math>S</math>. ...33 KB (6,636 words) - 05:50, 13 June 2023
- ...(V,E)</math>, a '''matching''' <math>M\subseteq E</math> is an independent set for edges, that is, for any <math>e_1,e_2\in M</math> that <math>e_1\neq e_ ...override the notation as <math>N(S)=\bigcup_{v\in S}N(v)</math>, i.e. the set of vertices that are adjacent to one of the vertices in <math>S</math>. ...33 KB (6,643 words) - 11:00, 20 December 2019
- ...(V,E)</math>, a '''matching''' <math>M\subseteq E</math> is an independent set for edges, that is, for any <math>e_1,e_2\in M</math> that <math>e_1\neq e_ ...override the notation as <math>N(S)=\bigcup_{v\in S}N(v)</math>, i.e. the set of vertices that are adjacent to one of the vertices in <math>S</math>. ...33 KB (6,643 words) - 07:40, 18 December 2017
- ...(V,E)</math>, a '''matching''' <math>M\subseteq E</math> is an independent set for edges, that is, for any <math>e_1,e_2\in M</math> that <math>e_1\neq e_ ...override the notation as <math>N(S)=\bigcup_{v\in S}N(v)</math>, i.e. the set of vertices that are adjacent to one of the vertices in <math>S</math>. ...33 KB (6,643 words) - 03:26, 22 December 2015
- ...(V,E)</math>, a '''matching''' <math>M\subseteq E</math> is an independent set for edges, that is, for any <math>e_1,e_2\in M</math> that <math>e_1\neq e_ ...override the notation as <math>N(S)=\bigcup_{v\in S}N(v)</math>, i.e. the set of vertices that are adjacent to one of the vertices in <math>S</math>. ...33 KB (6,643 words) - 10:48, 4 December 2016
- ...交通大学）|| Introduction to approximation algorithms.<br> Approximation ratio; set cover; rounding an LP; LP duality; greedy algorithms.||Chap. 1 of<br>Willia |align="center"|陈嘉|| Algorithmic Game Theory.<br>Strategic games; mechanism design; combinatorial auction; Vickrey aucti ...4 KB (455 words) - 07:53, 28 December 2011
- ...the chromatic number can be related to the size of the maximum independent set. ...h>, denoted <math>\alpha(G)</math>, is the size of the largest independent set in <math>G</math>. Formally, ...23 KB (4,153 words) - 08:30, 12 October 2010
- ...the chromatic number can be related to the size of the maximum independent set. ...h>, denoted <math>\alpha(G)</math>, is the size of the largest independent set in <math>G</math>. Formally, ...23 KB (4,153 words) - 08:18, 16 August 2011
- ...he problem and an output is called a '''solution''' to that instance. The theory of complexity deals almost exclusively with [http://en.wikipedia.org/wiki/D ...h "yes" answers. A decision problem can be equivalently represented as the set of all positive instances, denoted <math>L</math>. We call <math>L</math> a ...25 KB (4,263 words) - 08:43, 7 June 2010
- ...4mhz.de/cook.html|title = Proceedings of the Third Annual ACM Symposium on Theory of Computing|pages = 151–158}}</ref> ...if you have a problem, and someone says "The answer to your problem is the set of numbers 1, 2, 3, 4, 5", a computer may be able, quickly, to figure out i ...15 KB (2,427 words) - 22:24, 16 August 2017
- ...llest number <math>R(k,\ell)</math> satisfying the condition in the Ramsey theory is called the '''Ramsey number'''. :Let <math>A_1,A_2,\ldots,A_n</math> be a set of events, and assume that the following hold: ...16 KB (2,818 words) - 05:42, 11 December 2019
- ...llest number <math>R(k,\ell)</math> satisfying the condition in the Ramsey theory is called the '''Ramsey number'''. :Let <math>A_1,A_2,\ldots,A_n</math> be a set of events, and assume that the following hold: ...16 KB (2,818 words) - 11:41, 10 December 2017
- ...llest number <math>R(k,\ell)</math> satisfying the condition in the Ramsey theory is called the '''Ramsey number'''. :Let <math>A_1,A_2,\ldots,A_n</math> be a set of events, and assume that the following hold: ...16 KB (2,818 words) - 11:51, 5 June 2013
- ...llest number <math>R(k,\ell)</math> satisfying the condition in the Ramsey theory is called the '''Ramsey number'''. :Let <math>A_1,A_2,\ldots,A_n</math> be a set of events, and assume that the following hold: ...16 KB (2,818 words) - 12:03, 15 December 2015
- ...llest number <math>R(k,\ell)</math> satisfying the condition in the Ramsey theory is called the '''Ramsey number'''. :Let <math>A_1,A_2,\ldots,A_n</math> be a set of events, and assume that the following hold: ...16 KB (2,818 words) - 01:52, 4 December 2016
- ...llest number <math>R(k,\ell)</math> satisfying the condition in the Ramsey theory is called the '''Ramsey number'''. :Let <math>A_1,A_2,\ldots,A_n</math> be a set of events, and assume that the following hold: ...16 KB (2,818 words) - 07:52, 21 May 2014
- Note that in probability theory, the "mutual independence" is <font color="red">not</font> equivalent with ...require the random variables to be independent, thus can be applied to any set of random variables. For example: ...5 KB (860 words) - 03:28, 23 July 2011
- The real numbers are [[countable set|uncountable]]. That means that there is no way to put all the real numbers [[Category:Number theory]] ...6 KB (971 words) - 01:36, 21 August 2017
- Let <math>A_1,A_2,\ldots,A_n\subseteq U</math> be subsets of some finite set <math>U</math>. Here <math>U</math> is some universe of combinatorial objec ...ber of surjective mappings from an <math>n</math>-set to an <math>m</math>-set is given by ...33 KB (6,205 words) - 01:11, 22 September 2011
- ...4D]]) [[polytope]] which is both [[regular polytope|regular]] and [[convex set|convex]]. These are the four-dimensional analogs of the [[Platonic solid]]s Each convex regular 4-polytope is bounded by a set of 3-dimensional ''[[cell (mathematics)|cells]]'' which are all Platonic so ...8 KB (1,007 words) - 05:56, 15 September 2016
- ...that the [[range (mathematics)|range]] and codomain of ''f'' are the same set.<ref>{{cite web|url=http://mathworld.wolfram.com/Surjection.html| last1=Wei ...end}}. The range of ''x''² is <nowiki>[0,+∞) </nowiki>, that is, the set of non-negative numbers. (Also, this function is not an injection.) ...10 KB (1,438 words) - 06:38, 8 October 2016
- so the conjugation mapping is a permutation on the set of partitions of <math>n</math>. This fact is very useful in proving theore Let <math>A_1,A_2,\ldots,A_n\subseteq U</math> be subsets of some finite set <math>U</math>. Here <math>U</math> is some universe of combinatorial objec ...29 KB (5,077 words) - 04:54, 7 October 2010
- ...math>\mathcal{G}_n=2^{V\choose 2}</math>, where <math>|V|=n</math>, be the set of all possible graphs on <math>n</math> vertices. A '''graph property''' i ...ath>\{0,1\}</math>, a monotone property is just a monotonically increasing set function. ...11 KB (2,031 words) - 01:33, 24 July 2011
- The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one *<math>\Omega</math> is a set, called the '''sample space'''. ...30 KB (5,405 words) - 09:12, 17 September 2015
- A fundamental fact in flow theory is that cuts always upper bound flows. ...aph <math>G(V,E)</math> and two distinct vertices <math>s,t\in V</math>, a set of edges <math>C\subseteq E</math> is called an '''<math>s</math>-<math>t</ ...15 KB (3,049 words) - 03:53, 17 August 2011
- ...been used in proofs of many important results in computational complexity theory, such as [http://en.wikipedia.org/wiki/SL_(complexity) SL]=[http://en.wikip * The '''Edge Boundary''' of a set <math>S\subset V</math>, denoted <math>\partial S\,</math>, is <math>\parti ...15 KB (2,745 words) - 10:19, 4 January 2011
- ...near system''') is a collection of [[linear equation]]s involving the same set of [[variable]]s. In mathematics, the theory of linear systems is a branch of [[linear algebra]], a subject which is fun ...10 KB (1,386 words) - 15:51, 19 June 2016
- .../math>. The time complexity of fastest matrix multiplication algorithm (in theory) is <math>O(n^{2.376})</math>, and so is the time complexity of this method ...math>r_1, \ldots , r_n</math> independently and uniformly at random from a set <math>S</math>; ...16 KB (2,886 words) - 07:49, 13 November 2011
- ...counting principle states the following obvious fact: if the elements of a set are counted in two different ways, the answers are the same. ...K%C3%B6nigsberg Seven Bridges of Königsberg] that began the study of graph theory. ...26 KB (4,583 words) - 04:53, 7 October 2010
- ...llest number <math>R(k,\ell)</math> satisfying the condition in the Ramsey theory is called the '''Ramsey number'''. Consider a set of "bad" events <math>A_1,A_2,\ldots,A_n</math>. Suppose that <math>\Pr[A_i ...23 KB (4,275 words) - 12:28, 1 December 2010
- As with classical potentials, the potential energy is usually set to zero at infinity, leading to a negative potential energy for [[bound sta * [[Perturbation theory (quantum mechanics)]] ...5 KB (736 words) - 02:59, 18 October 2014
- .../math>. The time complexity of fastest matrix multiplication algorithm (in theory) is <math>O(n^{2.376})</math>, and so is the time complexity of this method ...math>r_1, \ldots , r_n</math> independently and uniformly at random from a set <math>S</math>; ...20 KB (3,617 words) - 01:24, 8 June 2010
- * [http://theory.stanford.edu/~yuhch123/ Huacheng Yu] (Harvard) |align="center"|[http://theory.stanford.edu/~yuhch123/ Huacheng Yu] (俞华程)<br> ...14 KB (1,850 words) - 01:51, 7 May 2018
- An '''eigenspace''' of ''A'' is the set of all eigenvectors with the same eigenvalue together with the [[zero vecto | title = Schaum's outline of theory and problems of linear algebra ...13 KB (1,723 words) - 07:13, 11 November 2016
- ...been used in proofs of many important results in computational complexity theory, such as [http://en.wikipedia.org/wiki/SL_(complexity) SL]=[http://en.wikip * The '''Edge Boundary''' of a set <math>S\subset V</math>, denoted <math>\partial S\,</math>, is <math>\parti ...34 KB (6,244 words) - 15:28, 8 June 2013
- ...been used in proofs of many important results in computational complexity theory, such as [http://en.wikipedia.org/wiki/SL_(complexity) SL]=[http://en.wikip * The '''Edge Boundary''' of a set <math>S\subset V</math>, denoted <math>\partial S\,</math>, is <math>\parti ...37 KB (6,824 words) - 02:20, 29 December 2015
- ...counting principle states the following obvious fact: if the elements of a set are counted in two different ways, the answers are the same. ...K%C3%B6nigsberg Seven Bridges of Königsberg] that began the study of graph theory. ...14 KB (2,455 words) - 09:24, 19 April 2013
- ...counting principle states the following obvious fact: if the elements of a set are counted in two different ways, the answers are the same. ...K%C3%B6nigsberg Seven Bridges of Königsberg] that began the study of graph theory. ...14 KB (2,455 words) - 09:37, 9 November 2015
- ...counting principle states the following obvious fact: if the elements of a set are counted in two different ways, the answers are the same. ...K%C3%B6nigsberg Seven Bridges of Königsberg] that began the study of graph theory. ...14 KB (2,455 words) - 12:56, 18 April 2023
- ...counting principle states the following obvious fact: if the elements of a set are counted in two different ways, the answers are the same. ...K%C3%B6nigsberg Seven Bridges of Königsberg] that began the study of graph theory. ...14 KB (2,455 words) - 03:49, 24 October 2016
- ...counting principle states the following obvious fact: if the elements of a set are counted in two different ways, the answers are the same. ...K%C3%B6nigsberg Seven Bridges of Königsberg] that began the study of graph theory. ...14 KB (2,455 words) - 09:36, 2 April 2014
- ...counting principle states the following obvious fact: if the elements of a set are counted in two different ways, the answers are the same. ...K%C3%B6nigsberg Seven Bridges of Königsberg] that began the study of graph theory. ...14 KB (2,455 words) - 08:14, 16 October 2019
- ...counting principle states the following obvious fact: if the elements of a set are counted in two different ways, the answers are the same. ...K%C3%B6nigsberg Seven Bridges of Königsberg] that began the study of graph theory. ...14 KB (2,455 words) - 13:27, 9 April 2024
- ...counting principle states the following obvious fact: if the elements of a set are counted in two different ways, the answers are the same. ...K%C3%B6nigsberg Seven Bridges of Königsberg] that began the study of graph theory. ...14 KB (2,455 words) - 02:36, 31 October 2017
- In probability theory class we have learned the basic concepts of '''events''' and '''random vari ...f events and random variables are related to the foundation of probability theory, which is not the topic of this class. Here we only give an informal descri ...34 KB (5,979 words) - 13:52, 20 September 2010
- ...llest number <math>R(k,\ell)</math> satisfying the condition in the Ramsey theory is called the '''Ramsey number'''. :Let <math>A_1,A_2,\ldots,A_n</math> be a set of events, and assume that the following hold: ...25 KB (4,530 words) - 12:14, 26 May 2023
- ...blem] is the problem of finding the <math>k</math>th smallest element in a set <math>S</math>. A typical case of selection problem is finding the '''media :The median of a set <math>S</math> is the <math>(\lceil n/2\rceil)</math>th element in the sort ...22 KB (3,809 words) - 05:34, 19 March 2014
- ...been used in proofs of many important results in computational complexity theory, such as [http://en.wikipedia.org/wiki/SL_(complexity) SL]=[http://en.wikip * The '''Edge Boundary''' of a set <math>S\subset V</math>, denoted <math>\partial S\,</math>, is <math>\parti ...41 KB (7,547 words) - 09:24, 22 May 2023
- [[Cardinality]] is the ''number'' of elements in a set. The cardinality of ''A''={X,Y,Z,W} is 4. We write #''A''=4.<ref name=T ...an injection. However, we can restrict both its domain and codomain to the set of non-negative numbers (0,+∞) to get an (invertible) bijection (see examp ...11 KB (1,621 words) - 07:51, 17 July 2016
- * A set of valid '''instances''' (inputs). * Each instance <math>I</math> defines a set of '''feasible solutions''' <math>\mathcal{F}_I</math>. ...20 KB (3,484 words) - 05:14, 11 June 2010
- ...r) that forms the basis of [[quantum mechanics]], one of the most accurate theory of how [[subatomic particle]]s behave. It is a mathematical equation that w ...(for example: an [[electron]] in an [[atom]]), the second equation of the set of separated differential equations presented above is often used. This equ ...7 KB (1,105 words) - 21:06, 26 June 2017
- == Assumption throughout Problem Set 4== <li>[<strong>Chernoff bound meets graph theory</strong>] ...13 KB (2,150 words) - 08:49, 7 June 2023
- ...been used in proofs of many important results in computational complexity theory, such as [http://en.wikipedia.org/wiki/SL_(complexity) SL]=[http://en.wikip * The '''Edge Boundary''' of a set <math>S\subset V</math>, denoted <math>\partial S\,</math>, is <math>\parti ...35 KB (6,195 words) - 08:39, 7 June 2010
- For any fixed set <math>S</math> of <math>k</math> vertices, let <math>\mathcal{E}_S</math> b ...math> be a set. Let <math>2^{S}=\{A\mid A\subseteq S\}</math> be the power set of <math>S</math>, and let <math>{S\choose k}=\{A\mid A\subseteq S\mbox{ an ...33 KB (6,039 words) - 08:41, 7 June 2010
- In probability theory, the '''total variation distance''' measures the difference between two pro Coupling is a powerful proof technique in probability theory. It allows us to compare two unrelated variables (or processes) by forcing ...23 KB (4,166 words) - 05:41, 22 December 2015
- A fundamental fact in flow theory is that cuts always upper bound flows. ...aph <math>G(V,E)</math> and two distinct vertices <math>s,t\in V</math>, a set of edges <math>C\subseteq E</math> is called an '''<math>s</math>-<math>t</ ...21 KB (4,167 words) - 09:57, 4 January 2011
- Therefore, the Prüfer code establishes a bijection between the set of trees on <math>n</math> distinct vertices and the tuples from <math>\{1, ...counting principle states the following obvious fact: if the elements of a set are counted in two different ways, the answers are the same. ...21 KB (3,832 words) - 15:23, 7 October 2011
- ...s]] (like [[light]]) work. It is also called "quantum physics" or "quantum theory". ...e to explain the light that comes from glowing [[hydrogen]]. The quantum [[theory]] of the [[atom]] also had to explain why the [[electron]] stays in its [[o ...36 KB (5,991 words) - 08:00, 24 August 2017
- In probability theory, the word "condition" is a verb. "Conditioning on the event ..." means that * fix an arbitrary set <math>S\subseteq \mathbb{F}</math> whose size to be fixed; ...22 KB (4,084 words) - 10:22, 6 March 2013
- <math>\mathbf{N}</math> or <math>\mathbb{N}</math> is the way to write the [[set]] of all natural numbers. Because some people say 0 is a natural number, an [[Category:Number theory]] ...7 KB (924 words) - 01:12, 17 January 2017
- Consider a problem as follows: We have a set of <math>n</math> points in a high-dimensional Euclidean space <math>\mathb This problem has various important applications in both theory and practice. In many tasks, the data points are drawn from a high dimensio ...14 KB (2,413 words) - 15:11, 12 May 2013
- Consider a problem as follows: We have a set of <math>n</math> points in a high-dimensional Euclidean space <math>\mathb This problem has various important applications in both theory and practice. In many tasks, the data points are drawn from a high dimensio ...14 KB (2,413 words) - 02:32, 25 November 2011
- [[Image:Standard deviation illustration.gif|frame|right|A data set with a mean of 50 (shown in blue) and a standard deviation (σ) of 20.]] ...icted number (with the distance measured in standard deviations), then the theory being tested may not be right. See [[prediction interval]]. ...12 KB (1,881 words) - 09:42, 29 May 2017
- Let Bi be the set of items hashed to the it h bucket. ...erfect hashing. In Proceedings of the thirty-third annual ACM symposium on Theory of computing， pages 425{432. ACM NewYork， NY， USA， 2001.] ...17 KB (967 words) - 06:08, 9 November 2010
- In probability theory, the '''total variation distance''' measures the difference between two pro Coupling is a powerful proof technique in probability theory. It allows us to compare two unrelated variables (or processes) by forcing ...27 KB (4,881 words) - 07:04, 2 June 2014
- In probability theory, the '''total variation distance''' measures the difference between two pro Coupling is a powerful proof technique in probability theory. It allows us to compare two unrelated variables (or processes) by forcing ...27 KB (4,881 words) - 13:52, 31 July 2013
- One of the most exciting results in spectral graph theory is the following theorem which relate the graph expansion to the spectral g Let <math>\chi_S</math> be the '''characteristic vector''' of the set <math>S</math> such that ...14 KB (2,683 words) - 15:16, 13 December 2011
- ...blem] is the problem of finding the <math>k</math>th smallest element in a set <math>S</math>. A typical case of selection problem is finding the '''media :The median of a set <math>S</math> is the <math>(\lceil n/2\rceil)</math>th element in the sort ...29 KB (5,238 words) - 05:34, 13 November 2015
- A fundamental fact in flow theory is that cuts always upper bound flows. ...aph <math>G(V,E)</math> and two distinct vertices <math>s,t\in V</math>, a set of edges <math>C\subseteq E</math> is called an '''<math>s</math>-<math>t</ ...30 KB (5,740 words) - 05:12, 11 June 2014
- A fundamental fact in flow theory is that cuts always upper bound flows. ...aph <math>G(V,E)</math> and two distinct vertices <math>s,t\in V</math>, a set of edges <math>C\subseteq E</math> is called an '''<math>s</math>-<math>t</ ...30 KB (5,740 words) - 14:29, 19 June 2013
- Suppose that we are give a set of "bad" events <math>A_1,A_2,\ldots,A_n</math>. We want to know that it is ...nt under ''limited dependencies''. The structure of dependencies between a set of events is described by a '''dependency graph''', which is a graph with e ...31 KB (5,614 words) - 12:29, 8 December 2015
- ...th>r_i</math> chosen independently and uniformly at random from some fixed set <math>S</math>. ...m)</math> time. The more sophisticated KMP algorithm inspired by automaton theory runs in <math>O(n+m)</math> time. ...9 KB (1,662 words) - 07:05, 15 November 2011
- This is found if we set ''z'' = 1/2 in the reflection or duplication formulas. If we ===Analytic number theory=== ...8 KB (1,133 words) - 09:45, 26 June 2017
- ...math>\mathcal{G}_n=2^{V\choose 2}</math>, where <math>|V|=n</math>, be the set of all possible graphs on <math>n</math> vertices. A '''graph property''' i ...ath>\{0,1\}</math>, a monotone property is just a monotonically increasing set function. ...21 KB (3,837 words) - 09:44, 1 April 2013
- In probability theory, the word "condition" is a verb. "Conditioning on the event ..." means that * fix an arbitrary set <math>S\subseteq \mathbb{F}</math> whose size to be fixed; ...26 KB (4,800 words) - 06:38, 3 March 2014
- ...a random variable changing over time. Let <math>\mathcal{S}</math> be the set of values assumed by the random variables <math>X_t</math>. We call each el :The index set <math>T</math> is countable. Specifically, we assume that <math>T=\{0,1,2,\ ...37 KB (6,516 words) - 08:40, 7 June 2010
- Note that in probability theory, the "mutual independence" is <font color="red">not</font> equivalent with ...require the random variables to be independent, thus can be applied to any set of random variables. For example: ...26 KB (4,811 words) - 10:33, 11 March 2013
- ...th>r_i</math> chosen independently and uniformly at random from some fixed set <math>S</math>. ...m)</math> time. The more sophisticated KMP algorithm inspired by automaton theory runs in <math>O(n+m)</math> time. ...21 KB (3,794 words) - 09:47, 10 September 2015
- ...ue of a random variable changing over time. Let <math>\Omega</math> be the set of values assumed by the random variables <math>X_t</math>. We call each el :The index set <math>T</math> is countable. Specifically, we assume that <math>T=\{0,1,2,\ ...40 KB (7,046 words) - 08:04, 2 June 2014
- ...ue of a random variable changing over time. Let <math>\Omega</math> be the set of values assumed by the random variables <math>X_t</math>. We call each el :The index set <math>T</math> is countable. Specifically, we assume that <math>T=\{0,1,2,\ ...40 KB (7,046 words) - 10:00, 13 December 2015
- ...ue of a random variable changing over time. Let <math>\Omega</math> be the set of values assumed by the random variables <math>X_t</math>. We call each el :The index set <math>T</math> is countable. Specifically, we assume that <math>T=\{0,1,2,\ ...40 KB (7,049 words) - 15:11, 8 June 2013
- Coupling is a powerful proof technique in probability theory. It allows us to compare two unrelated variables (or processes) by forcing ...tp://en.wikipedia.org/wiki/Hypercube '''hypercube'''] is a graph on vertex set <math>\{0,1\}^n</math>. For any <math>x,y\in\{0,1\}^n</math>, there is an e ...13 KB (2,540 words) - 13:57, 10 August 2011
- === Set balancing === This problem is called set balancing for a reason. ...31 KB (5,481 words) - 03:52, 9 November 2010
- Note that in probability theory, the "mutual independence" is <font color="red">not</font> equivalent with ...require the random variables to be independent, thus can be applied to any set of random variables. For example: ...26 KB (4,614 words) - 07:53, 10 March 2014
- ...ath>\mathbb{F}</math>. Let <math>S\subset\mathbb{F}</math> be an arbitrary set and <math>x\in S</math> is chosen uniformly at random from <math>S</math>. * fix an arbitrary set <math>S\subseteq \mathbb{F}</math> whose size to be fixed; ...28 KB (5,169 words) - 11:26, 13 September 2015
- However, right now this beautiful theory holds only when the transition matrix <math>P</math> is symmetric. In some ...at it can be lower bounded by computing the maximum congestion caused by a set of paths. ...27 KB (4,860 words) - 03:17, 22 March 2011
- ...ie, Leipzig: Hirzel English translation The Physical Principles of Quantum Theory. Chicago: University of Chicago Press, 1930.</ref> ...o idea that he was going to stumble over a hidden secret of nature when he set off to discover the explanation for the intensities of the colored lines ch ...42 KB (7,065 words) - 02:42, 24 August 2017
- Consider a problem as follows: We have a set of <math>n</math> points in a high-dimensional Euclidean space <math>\mathb This problem has various important applications in both theory and practice. In many tasks, the data points are drawn from a high dimensio ...26 KB (4,623 words) - 08:28, 31 March 2014
- ...th> be the set of vertices, and let <math>[m]={[n]\choose 2}</math> be the set of all possible edges. For convenience, we enumerate these potential edges ...number, number of triangles, the size of the largest clique or independent set, etc. Denote that <math>m={n\choose 2}</math>. Fix an arbitrary numbering o ...50 KB (9,096 words) - 06:09, 8 December 2015
- ...math>\mathbb{F}</math>. If <math>f\not\equiv 0</math>, then for any finite set <math>S\subset\mathbb{F}</math>, and <math>r_1,r_2\ldots,r_n\in S</math> ch ...th>r_i</math> chosen independently and uniformly at random from some fixed set <math>S\subseteq\mathbb{F}</math>. ...37 KB (6,665 words) - 12:21, 19 September 2023
- ...x|index]] set up by the Vancouver Stock Exchange in 1982. It was initially set at 1000.000, and after 22 months had fallen to about 520 — whereas [[stock ..., 0.732) that gets rounded down, by the same amount. When rounding a large set of numbers with [[random]] fractional parts, these rounding errors would st ...46 KB (7,060 words) - 01:36, 21 August 2017