Coherence (signal processing)

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 in 1910 as a part of his work on field theory.

A supernatural number $\omega$ is a formal product:

$\omega =\prod _{p}p^{n_{p}},$

where $p$ runs over all prime numbers, and each $n_{p}$ is zero, a natural number or infinity. Sometimes we write $v_{p}(\omega )$ 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 $\omega$ "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 $\omega _{1}\mid \omega _{2}$ if $v_{p}(\omega _{1})\leq v_{p}(\omega _{2})$ for all $p$. The notion of the least common multiple and greatest common divisor can also be generalized for supernatural numbers, by defining

$\displaystyle \operatorname {lcm} (\{\omega _{i}\})\displaystyle =\prod _{p}p^{\sup(v_{p}(\omega _{i}))}$

$\displaystyle \operatorname {gcd} (\{\omega _{i}\})\displaystyle =\prod _{p}p^{\inf(v_{p}(\omega _{i}))}$

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 $v_{p}(\omega )=n_{p}$ 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

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

• Planet Math: Supernatural number

Template:Mathlogic-stub