Kronecker's theorem

Kronecker's theorem

In mathematics, Kronecker's theorem, named after Leopold Kronecker, is a result in diophantine approximations applying to several real numbers "xi", for 1 ≤ "i" ≤ "N", that generalises dubious the equidistribution theorem, which implies that an infinite cyclic subgroup of the unit circle group is a dense subset. In terms of physical systems, it has the consequence that planets in circular orbits moving uniformly around a star will, over time, assume all alignments, unless there is an exact dependency between their orbital periods.

In the case of "N" numbers, taken as a single "N"-tuple and point "P" of the torus

:"T" = "RN/ZN",

the closure of the subgroup <"P"> generated by "P" will be finite, or some torus "T&prime;" contained in "T". The original Kronecker's theorem (Leopold Kronecker, 1884) stated that the necessary condition for

:"T&prime;" = "T",

which is that the numbers "xi" together with 1 should be linearly independent over the rational numbers, is also sufficient. Here it is easy to see that if some linear combination of the "xi" and 1 with non-zero rational number coefficients is zero, then the coefficients may be taken as integers, and a character χ of the group "T" other than the trivial character takes the value 1 on "P". By Pontryagin duality we have "T&prime;" contained in the kernel of χ, and therefore not equal to "T".

In fact a thorough use of Pontryagin duality here shows that the whole Kronecker theorem describes the closure of <"P"> as the intersection of the kernels of the χ with

:χ("P") = 1.

This gives an (antitone) Galois connection between "monogenic" closed subgroups of "T" (those with a single generator, in the topological sense), and sets of characters with kernel containing a given point. Not all closed subgroups occur as monogenic; for example a subgroup that has a torus of dimension ≥ 1 as connected component of the identity element, and that is not connected, cannot be such a subgroup.

The theorem leaves open the question of how well (uniformly) the multiples "mP" of "P" fill up the closure. In the one-dimensional case, the distribution is uniform by the equidistribution theorem.

ee also

* Kronecker set
* Weyl's criterion


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Kronecker–Weber theorem — In algebraic number theory, the Kronecker–Weber theorem states that every finite abelian extension of the field of rational numbers Q, or in other words every algebraic number field whose Galois group over Q is abelian, is a subfield of a… …   Wikipedia

  • Kronecker — Cette page d’homonymie répertorie les différents sujets et articles partageant un même nom. Kronecker peut désigner Leopold Kronecker (1823 1891), un mathématicien et logicien allemand, et dont les articles suivants portent le nom :… …   Wikipédia en Français

  • Leopold Kronecker — Infobox Scientist name = Leopold Kronecker caption = Leopold Kronecker birth date = birth date|1823|12|7|mf=y birth place = Liegnitz, Prussian province of Silesia residence = Prussian nationality = Prussian death date = death date and… …   Wikipedia

  • Leopold Kronecker — Pour les articles homonymes, voir Kronecker. Leopold Kronecker Leopold Kronecker Naissance 7 décembre 1823 Legn …   Wikipédia en Français

  • Théorème de Kronecker-Weber — Le théorème de Kronecker Weber établit en théorie algébrique des nombres le résultat suivant : toute extension abélienne finie du corps des nombres rationnels, c est à dire tout corps de nombres algébriques dont le groupe de Galois sur est… …   Wikipédia en Français

  • Hilbert–Speiser theorem — In mathematics, the Hilbert–Speiser theorem is a result on cyclotomic fields, characterising those with a normal integral basis. More generally, it applies to any abelian extension K of the rational field Q . The Kronecker–Weber theorem… …   Wikipedia

  • Wick's theorem — is a method of reducing high order derivatives to a combinatorics problem.(Philips, 2001) It is named after Gian Carlo Wick. It is used extensively in quantum field theory to reduce arbitrary products of creation and annihilation operators to… …   Wikipedia

  • Noether's theorem — This article discusses Emmy Noether s first theorem, which derives conserved quantities from symmetries. For her related theorem on infinite dimensional Lie algebras and differential equations, see Noether s second theorem. For her unrelated… …   Wikipedia

  • Equipartition theorem — [ Thermal motion of an α helical peptide. The jittery motion is random and complex, and the energy of any particular atom can fluctuate wildly. Nevertheless, the equipartition theorem allows the average kinetic energy of each atom to be computed …   Wikipedia

  • Parallel axis theorem — In physics, the parallel axis theorem can be used to determine the moment of inertia of a rigid body about any axis, given the moment of inertia of the object about the parallel axis through the object s center of mass and the perpendicular… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”