Schur's theorem

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In discrete mathematics, Schur's theorem is any of several 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.

Ramsey theory[edit]

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

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.

Combinatorics[edit]

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.)

Differential geometry[edit]

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).

Linear algebra[edit]

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.

Functional analysis[edit]

In functional analysis and the study of Banach spaces, Schur's theorem, due to J. Schur, often refers to Schur's property, that for certain spaces, weak convergence[disambiguation needed] implies convergence in the norm.

Number theory[edit]

In number theory, Issai Schur showed in 1912 that for every nonconstant polynomial p(x) with integer coefficients, if S is the set of all nonzero values , then the set of primes that divide some member of S is infinite.

See also[edit]

References[edit]

  • 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.
  • Issai Schur (1912). Über die Existenz unendlich vieler Primzahlen in einigen speziellen arithmetischen Progressionen, Sitzungsberichte der Berliner Math.

Further reading[edit]