Search results

A '''mathematical proof''' is a way to show that a [[mathematics|math]] [[theorem]] is true. One mu == Proof by Induction == ...

...[http://en.wikipedia.org/wiki/VC_dimension '''VC dimension'''] is defined by the power of a family to shatter a set. ...F})</math>, is the size of the largest <math>R\subseteq X</math> shattered by <math>\mathcal{F}</math>. ...

...[http://en.wikipedia.org/wiki/VC_dimension '''VC dimension'''] is defined by the power of a family to shatter a set. ...F})</math>, is the size of the largest <math>R\subseteq X</math> shattered by <math>\mathcal{F}</math>. ...

...antor|Cantor]] published articles on it in 1877, 1891 and 1899. His first proof of the diagonal argument was published in 1890 in the [[journal]] of the [[ By omitting fractions that can still be simplified, there is a [[Bijective fun ...

* <math>R(4,3)\leq 9</math>. (Hint: Proof by contradiction. Color the edges of <math>K_9</math> in red and blue, and assume that there ...</math> and that all the vertices of <math>T\subseteq B</math> are matched by <math>M'</math>. Prove that <math>G</math> contains a matching that matches ...

=== First proof (shadows)=== We first introduce the original proof by Sperner, which uses concepts called '''shadows''' and '''shades''' of set s ...

=== First proof (shadows)=== We first introduce the original proof by Sperner, which uses concepts called '''shadows''' and '''shades''' of set s ...

{{Proof| We proceed by induction on <math>k</math>. For <math>k=1</math>, <math>\mathcal{F}\subset ...

=== First proof (shadows)=== We first introduce the original proof by Sperner, which uses concepts called '''shadows''' and '''shades''' of set s ...

{{Proof| We proceed by induction on <math>k</math>. For <math>k=1</math>, <math>\mathcal{F}\subset ...

{{Proof| We proceed by induction on <math>k</math>. For <math>k=1</math>, <math>\mathcal{F}\subset ...

{{Proof| We proceed by induction on <math>k</math>. For <math>k=1</math>, <math>\mathcal{F}\subset ...

{{Proof| We proceed by induction on <math>k</math>. For <math>k=1</math>, <math>\mathcal{F}\subset ...

=== First proof (shadows)=== We first introduce the original proof by Sperner, which uses concepts called '''shadows''' and '''shades''' of set s ...

...proof uses the famous Cauchy-Schwarz inequality in analysis. And the third proof uses another famous inequality: the inequality of the arithmetic and geomet {{Prooftitle|First proof. (pigeonhole principle)| ...

...proof uses the famous Cauchy-Schwarz inequality in analysis. And the third proof uses another famous inequality: the inequality of the arithmetic and geomet {{Prooftitle|First proof. (pigeonhole principle)| ...

...proof uses the famous Cauchy-Schwarz inequality in analysis. And the third proof uses another famous inequality: the inequality of the arithmetic and geomet {{Prooftitle|First proof. (pigeonhole principle)| ...

...proof uses the famous Cauchy-Schwarz inequality in analysis. And the third proof uses another famous inequality: the inequality of the arithmetic and geomet {{Prooftitle|First proof. (pigeonhole principle)| ...

...proof uses the famous Cauchy-Schwarz inequality in analysis. And the third proof uses another famous inequality: the inequality of the arithmetic and geomet {{Prooftitle|First proof. (pigeonhole principle)| ...

...and how to efficiently find a stable matching. Both questions are answered by the following proposal algorithm due to Gale and Shapley. ...hus <math>m'</math> becomes single again and considers himself as rejected by <math>w</math>) and is married to <math>m</math>; ...

...proof uses the famous Cauchy-Schwarz inequality in analysis. And the third proof uses another famous inequality: the inequality of the arithmetic and geomet {{Prooftitle|First proof. (pigeonhole principle)| ...

...proof uses the famous Cauchy-Schwarz inequality in analysis. And the third proof uses another famous inequality: the inequality of the arithmetic and geomet {{Prooftitle|First proof. (pigeonhole principle)| ...

...proof uses the famous Cauchy-Schwarz inequality in analysis. And the third proof uses another famous inequality: the inequality of the arithmetic and geomet {{Prooftitle|First proof. (pigeonhole principle)| ...

...proof uses the famous Cauchy-Schwarz inequality in analysis. And the third proof uses another famous inequality: the inequality of the arithmetic and geomet {{Prooftitle|First proof. (pigeonhole principle)| ...

...ary condition for the existence of SDR is also sufficient, which is stated by the Hall's theorem, also called the '''mariage theorem'''. {{Proof| ...

...ary condition for the existence of SDR is also sufficient, which is stated by the Hall's theorem, also called the '''mariage theorem'''. {{Proof| ...

...>\{a\in\Omega\mid X(a)=x\}</math>, and denote the probability of the event by ...m variable <math>X</math>, denoted by <math>\mathbf{E}[X]</math>, is given by ...

...ary condition for the existence of SDR is also sufficient, which is stated by the Hall's theorem, also called the '''mariage theorem'''. {{Proof| ...

...ary condition for the existence of SDR is also sufficient, which is stated by the Hall's theorem, also called the '''mariage theorem'''. {{Proof| ...

The axiom foundation of probability theory is laid by [http://en.wikipedia.org/wiki/Andrey_Kolmogorov Kolmogorov], one of the gre ...f <math>\Omega</math>. But in general, a probability space is well-defined by any <math>\Sigma</math> satisfying (K1) and (K2). Such <math>\Sigma</math> ...

...ary condition for the existence of SDR is also sufficient, which is stated by the Hall's theorem, also called the '''mariage theorem'''. {{Proof| ...

...ary condition for the existence of SDR is also sufficient, which is stated by the Hall's theorem, also called the '''mariage theorem'''. {{Proof| ...

...ary condition for the existence of SDR is also sufficient, which is stated by the Hall's theorem, also called the '''mariage theorem'''. {{Proof| ...

...ary condition for the existence of SDR is also sufficient, which is stated by the Hall's theorem, also called the '''mariage theorem'''. {{Proof| ...

...ary condition for the existence of SDR is also sufficient, which is stated by the Hall's theorem, also called the '''mariage theorem'''. {{Proof| ...

...non-degenerate eingenstates of a parity-conserving Hamiltonian. This is in contradiction to the existence of chiral molecules—a fact known as as the Hund paradox. T ...pairs. In a two-dimensional (2D) plane, we cool the Mott-insulator samples by immersing them into removable superfluid reservoirs and a record-low entrop ...

We first analyze this by counting. There are totally <math>n^m</math> ways of assigning <math>m</mat Thus the probability is given by: ...

Alternatively, we can consider the following equivalent problem by comparing the polynomial <math>f-g</math> (whose degree is at most <math>d< ...ath>x</math> from the field <math>\mathbb{F}</math>, <math>x</math> chosen by the algorithm. ...

Alternatively, we can consider the following equivalent problem by comparing the polynomial <math>f-g</math> (whose degree is at most <math>d< ...ath>x</math> from the field <math>\mathbb{F}</math>, <math>x</math> chosen by the algorithm. ...

Alternatively, we can consider the following equivalent problem by comparing the polynomial <math>f-g</math> (whose degree is at most <math>d< ...ath>x</math> from the field <math>\mathbb{F}</math>, <math>x</math> chosen by the algorithm. ...

Alternatively, we can consider the following equivalent problem by comparing the polynomial <math>f-g</math> (whose degree is at most <math>d< ...ath>x</math> from the field <math>\mathbb{F}</math>, <math>x</math> chosen by the algorithm. ...

Alternatively, we can consider the following equivalent problem by comparing the polynomial <math>f-g</math> (whose degree is at most <math>d< ...ath>x</math> from the field <math>\mathbb{F}</math>, <math>x</math> chosen by the algorithm. ...

Alternatively, we can consider the following equivalent problem by comparing the polynomial <math>f-g</math> (whose degree is at most <math>d< ...ath>x</math> from the field <math>\mathbb{F}</math>, <math>x</math> chosen by the algorithm. ...

Alternatively, we can consider the following equivalent problem by comparing the polynomial <math>f-g</math> (whose degree is at most <math>d< ...ath>x</math> from the field <math>\mathbb{F}</math>, <math>x</math> chosen by the algorithm. ...

...cies''. The structure of dependencies between a set of events is described by a '''dependency graph''', which is a graph with events as vertices and each ...ing theorem was proved by Erdős and Lovász in 1975 and then later improved by Lovász in 1977. Now it is commonly referred as the '''Lovász local lemma''' ...

...yset</math> and <math>S\cup T=V</math>. A cut <math>C</math> is specified by this bipartition as ...d in the same way as before, and the cost of a cut <math>C</math> is given by the total number of edges (including parallel edges) in <math>C</math>. Equ ...

...<math>x</math> with <math>S_i</math> if and only if <math>x\in S_i</math>. By this translation the set cover problem is precisely the problem of given as By alternating the roles of sets and elements in the above interpretation of s ...

...<math>x</math> with <math>S_i</math> if and only if <math>x\in S_i</math>. By this translation the set cover problem is precisely the problem of given as By alternating the roles of sets and elements in the above interpretation of s ...

The <math>\alpha|\beta|\gamma</math> notation was introduced by [http://www.math.ucsd.edu/~ronspubs/79_03_scheduling_survey.pdf Ron Graham {{Proof| ...

...<math>x</math> with <math>S_i</math> if and only if <math>x\in S_i</math>. By this translation the set cover problem is precisely the problem of given as By alternating the roles of sets and elements in the above interpretation of s ...

...<math>x</math> with <math>S_i</math> if and only if <math>x\in S_i</math>. By this translation the set cover problem is precisely the problem of given as By alternating the roles of sets and elements in the above interpretation of s ...

...<math>x</math> with <math>S_i</math> if and only if <math>x\in S_i</math>. By this translation the set cover problem is precisely the problem of given as By alternating the roles of sets and elements in the above interpretation of s ...

The <math>\alpha|\beta|\gamma</math> notation was introduced by [http://www.math.ucsd.edu/~ronspubs/79_03_scheduling_survey.pdf Ron Graham {{Proof| ...

...trassen_algorithm Strassen's algorithm] discovered in 1969 now implemented by many numerical libraries runs in time <math>O(n^{\log_2 7})\approx O(n^{2.8 The product <math>A(Br)</math> is computed by first multiplying <math>Br</math> and then <math>A(Br)</math>. ...