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...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. ...