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== 大挑战 (Lovász's simplified version of the Kruskal-Katona theorem)== ...

...[mass]]. However, this is a simplified version of his equation. The longer version of his equation is E²=m²c⁴+p²c< ...

...pdf slides] | [http://lamda.nju.edu.cn/yinyt/notes/random1-print.pdf print version] ...pdf slides] | [http://lamda.nju.edu.cn/yinyt/notes/random2-print.pdf print version] ...

...that the opposite number of a number, is the [[negative number|negative]] version of the [[positive]] number. This is what the opposite of 7 is or the opposi ...

An '''approximation''' is a version of a piece of [[information]] that does not describe it exactly, but is clo ...

* ''Introduction to Fluid Mechanics'' Fourth Edition, Wiley, SI Version, R.W. Fox & A.T. McDonald ...

...ansversely Vibrating Beams using four Engineering Theories|version = final version|publisher = Academic Press|date = March 22, 1999|url = http://csxe.rutgers. ...

==Time independent Version== ...

...ect, and ''v'' is the [[velocity]] of the object. An alternate but correct version of this formula is ...

...sources/the-world-factbook/graphics/ref_maps/political/pdf/world.pdf large version] (pdf, 1.8MB)]] ...

...was given. In this way, it can solve a problem by first solving a simpler version of the problem. ...

|Definition (martingale, general version)| ==Azuma's inequality -- general version== ...

...eir real name, so he used the name "Student" to hide his identity. Another version is that the brewery did not want their competitors to know that they were u ...

This is another version, which is also true. ...

Prove the following independent set version of the Turan theorem: ...

Prove the following independent set version of the Turan theorem: ...

Prove the following independent set version of the Turan theorem: ...

....php/Complexity_Zoo:N#nl NL]. In fact USTCON is complete for the symmetric version of nondeterministic log-space. That is, every problem in the class of [http ...

...Qualification authority specifically mentions this skill in document 13004 version 2, dated 17 October 2003 titled [http://www.nzqa.govt.nz/nqfdocs/units/pdf/ ...

...tz conjecture seems to be hard to prove, consider the following randomized version: ...

Usually, [[Dirichlet]] is credited with the version used in schools until the second half of the 20th century: ...[[Bourbaki]] generalized the Dirichlet definition and gave a set theoretic version of the definition as a correspondence between inputs and outputs; this was ...

...ijn graphs are defined as directed graphs, here we consider the undirected version.) ...

#:This version of the Chebyshev inequality is sometimes referred to as the <b>Chebyshev-Ca ...

|Definition (martingale, general version)| |Azuma's Inequality (general version)| ...

...valent definitions of the set cover problem. We take the '''hitting set''' version.) ...

Let's first consider the decision version of the problem. That is, given as input a graph <math>G(V,E)</math>, decide ...

...al|generalizes]] readily into higher [[dimension]]s. The three-dimensional version is called [[Pascal's pyramid]] or ''Pascal's tetrahedron''. A higher-dimens ...

We consider the non-lazy version of random walk. We observe that the random walk has the following stationar ...

A stricter version of the big O is the little-o. The difference between big O and little-o is ...

We will prove a general version of the local lemma, where the events <math>A_i</math> are not symmetric. Th This gives the symmetric version of local lemma. ...

McLaughlin also created directions for an approximate version which can be done with just [[mental calculation]]. In 1995, Dale and Chall published a new version of their formula with an upgraded word list.<ref name=Dale-Chall2>Chall J.S ...

|Definition (martingale, general version)| |Azuma's Inequality (general version)| ...

Consider the following independent set version of the Turan theorem: ...

...For any <math>\lambda<0</math>, by the same analysis as in the upper tail version, |Definition (martingale, general version)| ...

...For any <math>\lambda<0</math>, by the same analysis as in the upper tail version, |Definition (martingale, general version)| ...

...For any <math>\lambda<0</math>, by the same analysis as in the upper tail version, |Definition (martingale, general version)| ...

...For any <math>\lambda<0</math>, by the same analysis as in the upper tail version, |Definition (martingale, general version)| ...

...oordinates|cylindrical polar coordinates]] (''r'', θ, ''z''), an unbounded version of the Möbius strip can be represented by the equation: *[http://www.toroidalsnark.net/mkmb.html Knitted version] ...

|Definition (martingale, general version)| |Azuma's Inequality (general version)| ...

|Definition (martingale, general version)| |Azuma's Inequality (general version)| ...

|Definition (martingale, general version)| |Azuma's Inequality (general version)| ...

|Definition (martingale, general version)| ====Azuma's inequality -- general version==== ...

...ue to Sauer, and independently due to Shelah and Perles. A slightly weaker version is found by Vapnik and Chervonenkis, who use the framework to develop a the And the general version of the Kruskal-Katona theorem can be deduced. ...

...ue to Sauer, and independently due to Shelah and Perles. A slightly weaker version is found by Vapnik and Chervonenkis, who use the framework to develop a the And the general version of the Kruskal-Katona theorem can be deduced. ...

We consider the the optimization version of SAT, which ask for an assignment that the number of satisfied clauses is We will prove a general version of the local lemma, where the events <math>A_i</math> are not symmetric. Th ...

We consider the the optimization version of SAT, which ask for an assignment that the number of satisfied clauses is We will prove a general version of the local lemma, where the events <math>A_i</math> are not symmetric. Th ...

{{Theorem|FPRAS (equivalent simple version)| ...dent to more than one edges in <math>M</math>, i.e. a matching is an "edge-version" of independent set. ...

...valent definitions of the set cover problem. We take the '''hitting set''' version.) ...

The following is the weighted version of set cover problem: ...

In the hitting set version, the frequency should be defined for each set: for a set <math>S_i</math> i The vertex cover problem is '''NP-hard'''. Its decision version is among [https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problems Ka ...

...entical parallel machines'''. It can be described as the following simpler version as a load balancing problem: {{Theorem|Minimum Makespan on Identical Parallel Machines (load balancing version)| ...

In the hitting set version, the frequency should be defined for each set: for a set <math>S_i</math> i The vertex cover problem is '''NP-hard'''. Its decision version is among [https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problems Ka ...

...entical parallel machines'''. It can be described as the following simpler version as a load balancing problem: {{Theorem|Minimum Makespan on Identical Parallel Machines (load balancing version)| ...

In the hitting set version, the frequency should be defined for each set: for a set <math>S_i</math> i The vertex cover problem is '''NP-hard'''. Its decision version is among [https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problems Ka ...

In the hitting set version, the frequency should be defined for each set: for a set <math>S_i</math> i The vertex cover problem is '''NP-hard'''. Its decision version is among [https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problems Ka ...

In the hitting set version, the frequency should be defined for each set: for a set <math>S_i</math> i The vertex cover problem is '''NP-hard'''. Its decision version is among [https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problems Ka ...

The following is a general asymmetric version of the local lemma. This generalization is due to Spencer. ...he dependency graph has <math>d\ge 2</math>, we can have another symmetric version of the local lemma. ...

...valent definitions of the set cover problem. We take the '''hitting set''' version.) ...

The following is the asymmetric version of the Lovász Local Lemma. This generalization is due to Spencer. This generalized version of the local lemma immediately implies the symmetric version of the lemma. Namely, <math>\Pr\left[\bigwedge_{i}\overline{A_i}\right]>0</ ...

The following is the weighted version of set cover problem: ...

We will prove a general version of the local lemma, where the events <math>A_i</math> are not symmetric. Th ...

...se is also true: we can find the optimum by making queries to the decision version. Through binary search, this can be done efficiently. ...imization problem is an '''NP-hard''' optimization problem if its decision version is '''NP-hard'''. Many interesting optimization problems are '''NP-hard'''. ...

We consider the following randomized version of the quicksort. ...

The following is the weighted version of set cover problem: ...

...unt is configured so your solutions are not publicly visible. Many popular version control systems provide free repositories to students. ...

The following is the weighted version of set cover problem: ...

...unt is configured so your solutions are not publicly visible. Many popular version control systems provide free repositories to students. ...

...valent definitions of the set cover problem. We take the '''hitting set''' version.) ...

...valent definitions of the set cover problem. We take the '''hitting set''' version.) ...

The following is the weighted version of set cover problem: ...

...T) problem], which is NP-hard. The decision version of one of the weighted version of the problem is one of the [http://en.wikipedia.org/wiki/Karp%27s_21_NP-c ...

...s hard to compute, because it is '''NP-complete'''. Actually, the weighted version of max-cut is among the [http://en.wikipedia.org/wiki/Karp's_21_NP-complete ...clauses are true. The maximum satisfiability (MAXSAT) is the optimization version of SAT, which ask for an assignment that the number of satisfied clauses is ...

...a diagram of dots (or squares), called a '''Ferrers diagram''' (the square version of Ferrers diagram is also called a '''Young diagram''', named after a stru |align=center|Ferrers diagram (''dot version'') of (5,4,2,1)|| ...

...ath>S</math>. The expansion property can be interpreted as a combinatorial version of [http://en.wikipedia.org/wiki/Isoperimetric_inequality isoperimetric ine ...

|Definition (martingale, general version)| |Azuma's Inequality (general version)| ...

...a diagram of dots (or squares), called a '''Ferrers diagram''' (the square version of Ferrers diagram is also called a '''Young diagram''', named after a stru |align=center|Ferrers diagram (''dot version'') of (5,4,2,1)|| ...

[https://arxiv.org/pdf/1709.05143.pdf Variable Version Lovász Local Lemma: Beyond Shearer's Bound]<br>[https://arxiv.org/pdf/1804. [https://arxiv.org/abs/1709.05143 Variable Version Lovász Local Lemma: Beyond Shearer's Bound] ...

...d his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653. Cavalieri first used polar coordinates to solve a proble ...

We consider the the optimization version of SAT, which ask for an assignment that the number of satisfied clauses is ...

...For any <math>\lambda<0</math>, by the same analysis as in the upper tail version, ...

We consider the the optimization version of SAT, which ask for an assignment that the number of satisfied clauses is ...

The following is the weighted version of set cover problem: ...

...a diagram of dots (or squares), called a '''Ferrers diagram''' (the square version of Ferrers diagram is also called a '''Young diagram''', named after a stru |align=center|Ferrers diagram (''dot version'') of (5,4,2,1)|| ...

...a diagram of dots (or squares), called a '''Ferrers diagram''' (the square version of Ferrers diagram is also called a '''Young diagram''', named after a stru |align=center|Ferrers diagram (''dot version'') of (5,4,2,1)|| ...

...a diagram of dots (or squares), called a '''Ferrers diagram''' (the square version of Ferrers diagram is also called a '''Young diagram''', named after a stru |align=center|Ferrers diagram (''dot version'') of (5,4,2,1)|| ...

...a diagram of dots (or squares), called a '''Ferrers diagram''' (the square version of Ferrers diagram is also called a '''Young diagram''', named after a stru |align=center|Ferrers diagram (''dot version'') of (5,4,2,1)|| ...

...a diagram of dots (or squares), called a '''Ferrers diagram''' (the square version of Ferrers diagram is also called a '''Young diagram''', named after a stru |align=center|Ferrers diagram (''dot version'') of (5,4,2,1)|| ...

...a diagram of dots (or squares), called a '''Ferrers diagram''' (the square version of Ferrers diagram is also called a '''Young diagram''', named after a stru |align=center|Ferrers diagram (''dot version'') of (5,4,2,1)|| ...

...a diagram of dots (or squares), called a '''Ferrers diagram''' (the square version of Ferrers diagram is also called a '''Young diagram''', named after a stru |align=center|Ferrers diagram (''dot version'') of (5,4,2,1)|| ...

...a diagram of dots (or squares), called a '''Ferrers diagram''' (the square version of Ferrers diagram is also called a '''Young diagram''', named after a stru |align=center|Ferrers diagram (''dot version'') of (5,4,2,1)|| ...

...a that describes the spring is exactly the same as the Newtonian mechanics version except only certain energies or energy levels are allowed. Think of the ene ...

|Definition (martingale, general version)| ...

|Definition (martingale, general version)| ...

|Definition (martingale, general version)| ...

|Definition (martingale, general version)| ...

|Definition (martingale, general version)| ...

We will prove a general version of the local lemma, where the events <math>A_i</math> are not symmetric. Th This gives the symmetric version of local lemma. ...

...s hard to compute, because it is '''NP-complete'''. Actually, the weighted version of max-cut is among the [http://en.wikipedia.org/wiki/Karp's_21_NP-complete ...

...nce of 1 can be replaced by arbitrary constants, which gives a generalized version of Lipschitz condition. |Definition (Lipschitz condition, general version)| ...

:Define the '''decision version''' of the minimum cut problem as follows. For a graph <math>G</math>, <math ...

We will prove a general version of the local lemma, where the events <math>A_i</math> are not symmetric. Th This gives the symmetric version of local lemma. ...

We will prove a general version of the local lemma, where the events <math>A_i</math> are not symmetric. Th This gives the symmetric version of local lemma. ...

We will prove a general version of the local lemma, where the events <math>A_i</math> are not symmetric. Th This gives the symmetric version of local lemma. ...

We will prove a general version of the local lemma, where the events <math>A_i</math> are not symmetric. Th This gives the symmetric version of local lemma. ...

We will prove a general version of the local lemma, where the events <math>A_i</math> are not symmetric. Th This gives the symmetric version of local lemma. ...

We will prove a general version of the local lemma, where the events <math>A_i</math> are not symmetric. Th This gives the symmetric version of local lemma. ...

We will prove a general version of the local lemma, where the events <math>A_i</math> are not symmetric. Th This gives the symmetric version of local lemma. ...

...ue to Sauer, and independently due to Shelah and Perles. A slightly weaker version is found by Vapnik and Chervonenkis, who use the framework to develop a the And the general version of the Kruskal-Katona theorem can be deduced. ...

We will prove a general version of the local lemma, where the events <math>A_i</math> are not symmetric. Th This gives the symmetric version of local lemma. ...

We will prove a general version of the local lemma, where the events <math>A_i</math> are not symmetric. Th This gives the symmetric version of local lemma. ...

Here we prefer to use our version, because it is convenient to use the tools for norms to analyze it. Very informally, the conductance can be seen as the weighted normalized version of expansion ratio. ...

| title = Elementary Linear Algebra (Applications Version) ...

{{Theorem|Polynomial Identity Testing (equivalent version)| ...

...Holant problems and quantum theory. He is also interested in the decision version of Holant problems, known as edge-CSP, and approximation algorithms for cou ...

...ed this in a paradox called ''Russell's paradox''. An easier to understand version, closer to real life, is called the ''Barber paradox'': ...

...s at √s=7 and 8 TeV with the ATLAS and CMS Experiments|date=26 March 2015 |version= }}</ref> ...

...T) problem], which is NP-hard. The decision version of one of the weighted version of the problem is one of the [http://en.wikipedia.org/wiki/Karp%27s_21_NP-c ...

...T) problem], which is NP-hard. The decision version of one of the weighted version of the problem is one of the [http://en.wikipedia.org/wiki/Karp%27s_21_NP-c ...

...For any <math>\lambda<0</math>, by the same analysis as in the upper tail version, ...

...ath>S</math>. The expansion property can be interpreted as a combinatorial version of [http://en.wikipedia.org/wiki/Isoperimetric_inequality isoperimetric ine Here we prefer to use our version, because it is convenient to use the tools for norms to analyze it. ...

<li><p>[<strong>Conditional version of the total probability</strong>] Let <math>C_1,\cdots, C_n</math> be disj ...

We consider the non-lazy version of random walk. We observe that the random walk has the following stationar ...wiki/Complexity_Zoo:N#nl NL]. In fact USTCON is complete for the symmetric version of nondeterministic log-space. That is, every problem in the class of [http ...

...ath>S</math>. The expansion property can be interpreted as a combinatorial version of [http://en.wikipedia.org/wiki/Isoperimetric_inequality isoperimetric ine ...

...h color occurs precisely <math>n_i</math> times. The following generalized version of multinomial theorem generates <math>|X^{\mathbf{v}}_\pi|</math> for all ...

...h color occurs precisely <math>n_i</math> times. The following generalized version of multinomial theorem generates <math>|X^{\mathbf{v}}_\pi|</math> for all ...

...h color occurs precisely <math>n_i</math> times. The following generalized version of multinomial theorem generates <math>|X^{\mathbf{v}}_\pi|</math> for all ...

...h color occurs precisely <math>n_i</math> times. The following generalized version of multinomial theorem generates <math>|X^{\mathbf{v}}_\pi|</math> for all ...

...h color occurs precisely <math>n_i</math> times. The following generalized version of multinomial theorem generates <math>|X^{\mathbf{v}}_\pi|</math> for all ...

...h color occurs precisely <math>n_i</math> times. The following generalized version of multinomial theorem generates <math>|X^{\mathbf{v}}_\pi|</math> for all ...

...h color occurs precisely <math>n_i</math> times. The following generalized version of multinomial theorem generates <math>|X^{\mathbf{v}}_\pi|</math> for all ...

...h color occurs precisely <math>n_i</math> times. The following generalized version of multinomial theorem generates <math>|X^{\mathbf{v}}_\pi|</math> for all ...

...h color occurs precisely <math>n_i</math> times. The following generalized version of multinomial theorem generates <math>|X^{\mathbf{v}}_\pi|</math> for all ...

:'''Abstract''':I will talk about a version of the quantum de Finetti theorem: permutation-invariant quantum states are ...

..._2\in M</math> shares a vertex. In other words, a matching is just an edge version of independent set. ...

..._2\in M</math> shares a vertex. In other words, a matching is just an edge version of independent set. ...

We consider the non-lazy version of random walk. We observe that the random walk has the following stationar ....php/Complexity_Zoo:N#nl NL]. In fact USTCON is complete for the symmetric version of nondeterministic log-space. That is, every problem in the class of [http ...

We consider the non-lazy version of random walk. We observe that the random walk has the following stationar ....php/Complexity_Zoo:N#nl NL]. In fact USTCON is complete for the symmetric version of nondeterministic log-space. That is, every problem in the class of [http ...

We consider the non-lazy version of random walk. We observe that the random walk has the following stationar ....php/Complexity_Zoo:N#nl NL]. In fact USTCON is complete for the symmetric version of nondeterministic log-space. That is, every problem in the class of [http ...

...For any <math>\lambda<0</math>, by the same analysis as in the upper tail version, ...

...For any <math>\lambda<0</math>, by the same analysis as in the upper tail version, ...

...For any <math>\lambda<0</math>, by the same analysis as in the upper tail version, ...

...For any <math>\lambda<0</math>, by the same analysis as in the upper tail version, ...

...For any <math>\lambda<0</math>, by the same analysis as in the upper tail version, ...

We consider the following randomized version of the quicksort. ...

...T) problem], which is NP-hard. The decision version of one of the weighted version of the problem is one of the [http://en.wikipedia.org/wiki/Karp%27s_21_NP-c ...

The problem is a typical MAX-CSP, an optimization version of the [https://en.wikipedia.org/wiki/Constraint_satisfaction_problem const ...be [https://en.wikipedia.org/wiki/NP-hardness '''NP-hard''']. Its decision version is among the [https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problem ...

The problem is a typical MAX-CSP, an optimization version of the [https://en.wikipedia.org/wiki/Constraint_satisfaction_problem const ...be [https://en.wikipedia.org/wiki/NP-hardness '''NP-hard''']. Its decision version is among the [https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problem ...

The problem is a typical MAX-CSP, an optimization version of the [https://en.wikipedia.org/wiki/Constraint_satisfaction_problem const ...be [https://en.wikipedia.org/wiki/NP-hardness '''NP-hard''']. Its decision version is among the [https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problem ...

The problem is a typical MAX-CSP, an optimization version of the [https://en.wikipedia.org/wiki/Constraint_satisfaction_problem const ...be [https://en.wikipedia.org/wiki/NP-hardness '''NP-hard''']. Its decision version is among the [https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problem ...

The problem is a typical MAX-CSP, an optimization version of the [https://en.wikipedia.org/wiki/Constraint_satisfaction_problem const ...be [https://en.wikipedia.org/wiki/NP-hardness '''NP-hard''']. Its decision version is among the [https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problem ...

The problem is a typical MAX-CSP, an optimization version of the [https://en.wikipedia.org/wiki/Constraint_satisfaction_problem const ...be [https://en.wikipedia.org/wiki/NP-hardness '''NP-hard''']. Its decision version is among the [https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problem ...

The problem is a typical MAX-CSP, an optimization version of the [https://en.wikipedia.org/wiki/Constraint_satisfaction_problem const ...be [https://en.wikipedia.org/wiki/NP-hardness '''NP-hard''']. Its decision version is among the [https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problem ...

The problem is a typical MAX-CSP, an optimization version of the [https://en.wikipedia.org/wiki/Constraint_satisfaction_problem const ...be [https://en.wikipedia.org/wiki/NP-hardness '''NP-hard''']. Its decision version is among the [https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problem ...

It is more helpful to consider its integer version: ...

...For any <math>\lambda<0</math>, by the same analysis as in the upper tail version, ...

...ial mechanism and its existing generalizations. We also develop an on-line version of our algorithm, that can be seen as a generalization of the sparse vector ...

The problem is a typical MAX-CSP, an optimization version of the [https://en.wikipedia.org/wiki/Constraint_satisfaction_problem const ...be [https://en.wikipedia.org/wiki/NP-hardness '''NP-hard''']. Its decision version is among the [https://en.wikipedia.org/wiki/Karp%27s_21_NP-complete_problem ...

By Taylor series, we get a generalized version of the binomial theorem known as [http://en.wikipedia.org/wiki/Binomial_coe ...

We consider the following randomized version of the quicksort. ...

...ue to Sauer, and independently due to Shelah and Perles. A slightly weaker version is found by Vapnik and Chervonenkis, who use the framework to develop a the ...

...ue to Sauer, and independently due to Shelah and Perles. A slightly weaker version is found by Vapnik and Chervonenkis, who use the framework to develop a the ...

...ue to Sauer, and independently due to Shelah and Perles. A slightly weaker version is found by Vapnik and Chervonenkis, who use the framework to develop a the ...

...ue to Sauer, and independently due to Shelah and Perles. A slightly weaker version is found by Vapnik and Chervonenkis, who use the framework to develop a the ...

:Define the '''decision version''' of the minimum cut problem as follows. For a graph <math>G</math>, <math ...

It is more helpful to consider its integer version: ...

...For any <math>\lambda<0</math>, by the same analysis as in the upper tail version, ...

It is more helpful to consider its integer version: ...

It is more helpful to consider its integer version: ...

In the fascicle version of Volume 4 of Don Knuth's [http://www-cs-faculty.stanford.edu/~uno/taocp.h ...

By Taylor series, we get a generalized version of the binomial theorem known as [http://en.wikipedia.org/wiki/Binomial_coe ...

By Taylor series, we get a generalized version of the binomial theorem known as [http://en.wikipedia.org/wiki/Binomial_coe ...

By Taylor series, we get a generalized version of the binomial theorem known as [http://en.wikipedia.org/wiki/Binomial_coe ...

By Taylor series, we get a generalized version of the binomial theorem known as [http://en.wikipedia.org/wiki/Binomial_coe ...

By Taylor series, we get a generalized version of the binomial theorem known as [http://en.wikipedia.org/wiki/Binomial_coe ...

By Taylor series, we get a generalized version of the binomial theorem known as [http://en.wikipedia.org/wiki/Binomial_coe ...

Very informally, the conductance can be seen as the weighted normalized version of expansion ratio. ...

By Taylor series, we get a generalized version of the binomial theorem known as [http://en.wikipedia.org/wiki/Binomial_coe ...

By Taylor series, we get a generalized version of the binomial theorem known as [http://en.wikipedia.org/wiki/Binomial_coe ...

By Taylor series, we get a generalized version of the binomial theorem known as [http://en.wikipedia.org/wiki/Binomial_coe ...

Let's first consider the decision version of the problem. That is, given as input a graph <math>G(V,E)</math>, decide ...

Let's first consider the decision version of the problem. That is, given as input a graph <math>G(V,E)</math>, decide ...

..._2\in M</math> shares a vertex. In other words, a matching is just an edge version of independent set. ...

..._2\in M</math> shares a vertex. In other words, a matching is just an edge version of independent set. ...

..._2\in M</math> shares a vertex. In other words, a matching is just an edge version of independent set. ...

..._2\in M</math> shares a vertex. In other words, a matching is just an edge version of independent set. ...

..._2\in M</math> shares a vertex. In other words, a matching is just an edge version of independent set. ...

...For any <math>\lambda<0</math>, by the same analysis as in the upper tail version, ...

...For any <math>\lambda<0</math>, by the same analysis as in the upper tail version, ...

First let's consider the following simplified version of '''Polynomial Identity Testing (PIT)''' which takes only the single-vari ...

We consider the following randomized version of the quicksort. ...

...ath>S</math>. The expansion property can be interpreted as a combinatorial version of [http://en.wikipedia.org/wiki/Isoperimetric_inequality isoperimetric ine ...

...ath>S</math>. The expansion property can be interpreted as a combinatorial version of [http://en.wikipedia.org/wiki/Isoperimetric_inequality isoperimetric ine ...

...ath>S</math>. The expansion property can be interpreted as a combinatorial version of [http://en.wikipedia.org/wiki/Isoperimetric_inequality isoperimetric ine ...

...xistence of satisfying assignments) is '''NP-complete''', and its counting version <nowiki>#</nowiki>SAT now ask for the exact number of satisfying assignment ...

...towards zero). The tie-breaking method may vary depending the language and version, and/or may be selectable by the programmer. Several languages follow the l ...

...form, equation (10) in Aitchison et al., p. 5.</ref><ref>Heisenberg's own version of this equation appears very near the end of the first numbered section of ...