Schur's theorem

In discrete mathematics, Schur's theorem is either of two different theorems of the mathematician Issai Schur. In differential geometry, Schur's theorem is a theorem of A. Schur.

Ramsey theory

In Ramsey theory, Schur's theorem states that for any partition of the positive integers into a finite number of parts, one of the parts contains three integers "x", "y", "z" with

:"x" + "y" = "z".

Moreover, for every positive integer "c", there exists a number "S"("c"), called "Schur's number", such that for every partition of the integers

:{1, ..., "S"("c")}

into "c" parts, one of the parts contains integers "x", "y", and "z" with

:"x" + "y" = "z".

Combinatorics

In combinatorics, Schur's theorem tells the number of ways for expressing a given number as a linear combination of a fixed set of relatively prime numbers. In particular, if $\{a\_1,ldots,a\_n\}$ is a set of integers such that $gcd(a\_1,ldots,a\_n)=1$, the number of different tuples of non-negative integer numbers $(c\_1,ldots,c\_n)$ such that $x=c\_1a\_1\; +\; cdots\; +\; c\_na\_n$ when $x$ goes to infinity is:

:$frac\{x^\{n-1\{(n-1)!a\_1ldots\; a\_n\}.$

As a result, for every set of relatively prime numbers $\{a\_1,ldots,a\_n\}$ there exists a value of $x$ such that every larger number is representable as a linear combination of $\{a\_1,ldots,a\_n\}$ in at least one way. This consequence of the theorem can be recast in a familiar context considering the problem of changing an amount using a set of coins. If the denominations of the coins are relatively prime numbers (such as 2 and 5) then any sufficiently large amount can be changed using only these coins.

Differential geometry

In differential geometry, Schur's theorem compares the distance between the endpoints of a space curve $C^*$ to the distance between the endpoints of a corresponding plane curve $C$ of less curvature.

Suppose $C(s)$ is a plane curve with curvature $kappa(s)$ which makes a convex curve when closed by the chord connecting its endpoints, and $C^*(s)$ is a curve of the same length with curvature $kappa^*(s)$. Let $d$ denote the distance between the endpoints of $C$ and $d^*$ denote the distance between the endpoints of $C^*$. If $kappa^*(s)\; leq\; kappa(s)$ then $d^*\; geq\; d$.

Schur's theorem is usually stated for $C^2$ curves, but John M. Sullivan has observed that Schur's theorem applies to curves of finite total curvature (the statement is slightly different).

Linear Algebra

In linear algebra Schur’s theorem is referred to as either the triangularization of a square matrix with complex entries, or of a square matrix with real entries and real eigenvalues.

References

* Herbert S. Wilf (1994). [http://www.cs.utsa.edu/~wagner/CS3343/resources/gfology.pdf generatingfunctionology] . Academic Press.
* Daniel Panario (2005). [http://www.math.carleton.ca/~zgao/MATH5819/Notes/NoteT4.pdf Integer Partition and The Money Changing Problem] .
* Dany Breslauer and Devdatt P. Dubhashi (1995). [http://www.brics.dk/LS/95/4/BRICS-LS-95-4/BRICS-LS-95-4.html Combinatorics for Computer Scientists]
* Shiing-Shen Chern (1967). Curves and Surfaces in Euclidean Space. In "Studies in Global Geometry and Analysis." Prentice-Hall.
* John M. Sullivan (2006). [http://arxiv.org/pdf/math.GT/0606007 Curves of Finite Total Curvature] . arXiv.