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Proof of Stein's example

Stein's example is an important result in decision theory which can be stated as The ordinary decision rule for estimating the mean of a multivariate Gaussian distribution is inadmissible under mean s

Proof that 22/7 exceeds π

Proofs of the mathematical result that the rational number 22/7 is greater than π (pi) date back to antiquity. One of these proofs, more recently developed but requiring only elementary techniques fro

Proofs involving the addition of natural numbers

This article contains mathematical proofs for some properties of addition of the natural numbers: the additive identity, commutativity, and associativity. These proofs are used in the article Addition

Proofs of convergence of random variables

This article is supplemental for “Convergence of random variables” and provides proofs for selected results. Several results will be established using the portmanteau lemma: A sequence {Xn} converges

Proofs of quadratic reciprocity

In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred proofs of the law of quadratic reciprocity hav

Derivation of the Schwarzschild solution

The Schwarzschild solution describes spacetime under the influence of a massive, non-rotating, spherically symmetric object. It is considered by some to be one of the simplest and most useful solution

Proofs of Fermat's little theorem

This article collects together a variety of proofs of Fermat's little theorem, which states that for every prime number p and every integer a (see modular arithmetic).

Furstenberg's proof of the infinitude of primes

In mathematics, particularly in number theory, Hillel Furstenberg's proof of the infinitude of primes is a topological proof that the integers contain infinitely many prime numbers. When examined clos

Proofs related to chi-squared distribution

The following are proofs of several characteristics related to the chi-squared distribution.

Proof that π is irrational

In the 1760s, Johann Heinrich Lambert was the first to prove that the number π is irrational, meaning it cannot be expressed as a fraction , where and are both integers. In the 19th century, Charles H

Proofs involving ordinary least squares

The purpose of this page is to provide supplementary materials for the ordinary least squares article, reducing the load of the main article with mathematics and improving its accessibility, while at

♯P-completeness of 01-permanent

The #P-completeness of 01-permanent, sometimes known as Valiant's theorem, is a mathematical proof about the permanent of matrices, considered a seminal result in computational complexity theory. In a

Dual of BCH is an independent source

A certain family of BCH codes have a particularly useful property, which is thattreated as linear operators, their dual operators turns their input into an -wise independent source. That is, the set o

Proof of the Euler product formula for the Riemann zeta function

Leonhard Euler proved the Euler product formula for the Riemann zeta function in his thesis Variae observationes circa series infinitas (Various Observations about Infinite Series), published by St Pe

Proof of Bertrand's postulate

In mathematics, Bertrand's postulate (actually a theorem) states that for each there is a prime such that . It was first proven by Chebyshev, and a short but advanced proof was given by Ramanujan. The

Proof that e is irrational

The number e was introduced by Jacob Bernoulli in 1683. More than half a century later, Euler, who had been a student of Jacob's younger brother Johann, proved that e is irrational; that is, that it c

Analyticity of holomorphic functions

In complex analysis, a complex-valued function of a complex variable :
* is said to be holomorphic at a point if it is differentiable at every point within some open disk centered at , and
* is said

Proof of Fermat's Last Theorem for specific exponents

Fermat's Last Theorem is a theorem in number theory, originally stated by Pierre de Fermat in 1637 and proved by Andrew Wiles in 1995. The statement of the theorem involves an integer exponent n large

Original proof of Gödel's completeness theorem

The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 (and a shorter version of the proof, published as an article in 1930, titled "The completeness of the

Derivation of the Routh array

The Routh array is a tabular method permitting one to establish the stability of a system using only the coefficients of the characteristic polynomial. Central to the field of control systems design,

Proofs of trigonometric identities

There are several equivalent ways for defining trigonometric functions, and the proof of the trigonometric identities between them depend on the chosen definition. The oldest and somehow the most elem

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