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 Axel Schur. In functional analysis, Schur's theorem is often called Schur's property, also due to Issai Schur.
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Moreover, for every positive integer c, there exists a number S(c), called Schur's number, such that for every partition of the integers
into c parts, one of the parts contains integers x, y, and z with
Folkman's theorem generalizes Schur's theorem by stating that there exist arbitrarily large sets of integers all of whose nonempty sums belong to the same part.
In combinatorics, Schur's theorem tells the number of ways for expressing a given number as a (non-negative, integer) linear combination of a fixed set of relatively prime numbers. In particular, if is a set of integers such that , the number of different tuples of non-negative integer numbers such that when goes to infinity is:
As a result, for every set of relatively prime numbers there exists a value of such that every larger number is representable as a linear combination of 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. (See Coin problem.)
In differential geometry, Schur's theorem compares the distance between the endpoints of a space curve to the distance between the endpoints of a corresponding plane curve of less curvature.
Suppose is a plane curve with curvature which makes a convex curve when closed by the chord connecting its endpoints, and is a curve of the same length with curvature . Let denote the distance between the endpoints of and denote the distance between the endpoints of . If then .
Schur's theorem is usually stated for curves, but John M. Sullivan has observed that Schur's theorem applies to curves of finite total curvature (the statement is slightly different).
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.
- Herbert S. Wilf (1994). generatingfunctionology. Academic Press.
- Shiing-Shen Chern (1967). Curves and Surfaces in Euclidean Space. In Studies in Global Geometry and Analysis. Prentice-Hall.