#mathematical theorems

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[mymathematics]

An introduction to mathematical theorems

Euclid of Alexandria revolutionized the way that mathematics is written, presented or thought about, and introduced the concept of mathematical proofs.  Discover what it takes to move from a loose theory or idea to a universally convincing proof.

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Eli Stein was the first to appreciate the interplay among representation theory, classical Fourier analysis, and partial differential equations, and to perceive the fundamental insights in each field arising from that interplay. How did Stein’s scope and originality contribute to numerical analysis?

Image: ‘E8 Petrie projection’ by Jgmoxness. CC BY-SA 3.0 via Wikimedia Commons.

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Mathematical theorems

Mathematical Theorems are a part of Maths.

Maths contain Euclidean Geometry.

Euclidian Geometry was created by Euclid.

Euclid is a Foreigner Class Servant in Fate/Requiem.

Fate/Requiem is owned by Type Moon.

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Amitsur–Levitzki theorem

In algebra, the Amitsur–Levitzki theorem states that the algebra of n by n matrices satisfies a certain identity of degree 2n. It was proved by Amitsur and Levitsky (1950). In particular matrix rings are polynomial identity rings such that the smallest identity they satisfy has degree exactly 2n. More details Android, Windows

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Hadamard's inequality

Another inequality is called the Hermite–Hadamard inequality. In mathematics, Hadamard's inequality, first published by Jacques Hadamard in 1893, is a bound on the determinant of a matrix whose entries are complex numbers in terms of the lengths of its column vectors. In geometrical terms, when restricted to real numbers, it bounds the volume in Euclidean space of n dimensions marked out by n vectors vi for 1 ≤ i ≤ n in terms of the lengths of these vectors ||vi||. Specifically, Hadamard's inequality states that if N is the matrix having columns vi, then | det ( N ) | ≤ ∏ i = 1 n ∥ v i ∥ . {\displaystyle |\det(N)|\leq \prod _{i=1}^{n}\|v_{i}\|.} If the n vectors are linearly independent, equality in Hadamard's inequality is achieved if and only if the vectors are orthogonal. More details Android, Windows

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Eisenstein's criterion

In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers—that is, for it to be unfactorable into the product of non-constant polynomials with rational coefficients. This criterion is not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but it does allow in certain important cases to prove irreducibility with very little effort. It may apply either directly or after transformation of the original polynomial. This criterion is named after Gotthold Eisenstein. In the early 20th century, it was also known as the Schönemann–Eisenstein theorem because Theodor Schönemann was the first to publish it. More details Android, Windows

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Schur's lemma

For other uses, see Schur's lemma (disambiguation). In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations of a group G and φ is a linear map from M to N that commutes with the action of the group, then either φ is invertible, or φ = 0. An important special case occurs when M = N and φ is a self-map. The lemma is named after Issai Schur who used it to prove Schur orthogonality relations and develop the basics of the representation theory of finite groups. Schur's lemma admits generalisations to Lie groups and Lie algebras, the most common of which is due to Jacques Dixmier. More details Android, Windows