Coherence (signal processing): Difference between revisions

Revision as of 16:53, 26 September 2013

In mathematics, the supernatural numbers (sometimes called generalized natural numbers or Steinitz numbers) are a generalization of the natural numbers. They were used by Ernst Steinitz^[1] in 1910 as a part of his work on field theory.

A supernatural number $ω$ is a formal product:

ω = \prod_{p} p^{n_{p}},

where $p$ runs over all prime numbers, and each $n_{p}$ is zero, a natural number or infinity. Sometimes we write $v_{p} (ω)$ for $n_{p}$ . If no $n_{p} = \infty$ and there are only a finite number of non-zero $n_{p}$ then we recover the positive integers. Slightly less intuitively, if all $n_{p}$ are $\infty$ , we get zero. Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide $ω$ "infinitely often," by taking that prime's corresponding exponent to be the symbol $\infty$ .

There is no natural way to add supernatural numbers, but they can be multiplied, with $\prod_{p} p^{n_{p}} \cdot \prod_{p} p^{m_{p}} = \prod_{p} p^{n_{p} + m_{p}}$ . Similarly, the notion of divisibility extends to the supernaturals with $ω_{1} ∣ ω_{2}$ if $v_{p} (ω_{1}) \leq v_{p} (ω_{2})$ for all $p$ . The notion of the least common multiple and greatest common divisor can also be generalized for supernatural numbers, by defining

lcm ({ω_{i}}) = \prod_{p} p^{\sup (v_{p} (ω_{i}))}

\gcd ({ω_{i}}) = \prod_{p} p^{\inf (v_{p} (ω_{i}))}

With these definitions, the gcd or lcm of infinitely many natural numbers (or supernatural numbers) is supernatural number. We can also extend the usual $p$ -adic order functions to supernatural numbers by defining $v_{p} (ω) = n_{p}$ for each $p$

Supernatural numbers are used to define orders and indices of profinite groups and subgroups, in which case many of the theorems from finite group theory carry over exactly. They are also used implicitly in many number-theoretical proofs, such as the density of the square-free integers and bounds for odd perfect numbers.

References

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External links

Planet Math: Supernatural number

Template:Number Systems

Template:Mathlogic-stub

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+In [[mathematics]], the '''supernatural numbers''' (sometimes called '''generalized natural numbers''' or '''Steinitz numbers''') are a generalization of the [[natural number]]s. They were used by [[Ernst Steinitz]]<ref>{{cite journal | first=Ernst | last=Steinitz | authorlink=Ernst Steinitz | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002167042 | title=Algebraische Theorie der Körper | journal=Journal für die reine und angewandte Mathematik | year=1910 | pages=167–309 | jfm=41.0445.03 | issn=0075-4102 }}</ref> in 1910 as a part of his work on [[Field theory (mathematics)|field theory]].
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+A supernatural number <math>\omega</math> is a [[Formal calculation|formal]] [[Product (mathematics)|product]]:
+: <math>\omega = \prod_p p^{n_p},</math>
+where <math>p</math> runs over all [[prime number]]s, and each <math>n_p</math> is zero, a natural number or [[infinity]]. Sometimes we write <math>v_p(\omega)</math> for <math>n_p</math>. If no <math>n_p = \infty</math> and there are only a finite number of non-zero <math>n_p</math> then we recover the positive integers. Slightly less intuitively, if all <math>n_p</math> are <math>\infty</math>, we get zero. Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide <math>\omega</math> "infinitely often," by taking that prime's corresponding exponent to be the symbol <math>\infty</math>.
+There is no natural way to add supernatural numbers, but they can be multiplied, with <math>\prod_p p^{n_p}\cdot\prod_p p^{m_p}=\prod_p p^{n_p+m_p}</math>. Similarly, the notion of divisibility extends to the supernaturals with <math>\omega_1\mid\omega_2</math> if <math>v_p(\omega_1)\leq v_p(\omega_2)</math> for all <math>p</math>. The notion of the [[least common multiple]] and [[greatest common divisor]] can also be generalized for supernatural numbers, by defining
+: <math>\displaystyle \operatorname{lcm}(\{\omega_i\}) \displaystyle =\prod_p p^{\sup(v_p(\omega_i))}</math>
+: <math>\displaystyle \operatorname{gcd}(\{\omega_i\}) \displaystyle =\prod_p p^{\inf(v_p(\omega_i))}</math>
+With these definitions, the gcd or lcm of infinitely many natural numbers (or supernatural numbers) is supernatural number.
+We can also extend the usual [[p-adic number|<math>p</math>-adic]] order functions to supernatural numbers by defining <math>v_p(\omega)=n_p</math> for each <math>p</math>
+Supernatural numbers are used to define orders and indices of [[profinite group]]s and subgroups, in which case many of the theorems from [[finite group theory]] carry over exactly. They are also used implicitly in many [[Number theory|number-theoretical]] proofs, such as the density of the [[square-free integer]]s and bounds for odd [[perfect number]]s.
+==References==
+{{reflist}}
+* {{cite book | last=Efrat | first=Ido | title=Valuations, orderings, and Milnor ''K''-theory | series=Mathematical Surveys and Monographs | volume=124 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2006 | isbn=0-8218-4041-X | zbl=1103.12002 | page=125 }}
+==External links==
+* [http://planetmath.org/supernaturalnumber Planet Math: Supernatural number]
+{{Number Systems}}
+[[Category:Number theory]]
+{{mathlogic-stub}}

Coherence (signal processing): Difference between revisions

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