User:Richard Pinch/sandbox-5

Gray map

A map from $\mathbf{Z}_4$ to $\mathbf{F}_2^2$, extended in the obvious way to $\mathbf{Z}_4^n$ and $\mathbf{F}_2^n$ which maps Lee distance to Hamming distance. Explicitly, $$ 0 \mapsto 00 \ ,\ \ 1 \mapsto 01 \ ,\ \ 2 \mapsto 11 \ ,\ \ 3 \mapsto 10 \ . $$

The map instantiates a Gray code in dimension 2.

Möbius inversion for arithmetic functions

The original form of Möbius inversion developed by A. Möbius for arithmetic functions.

Möbius considered also inversion formulas for finite sums running over the divisors of a natural number $n$: $$ F(n) = \sum_{d | n} f(d) \ ,\ \ \ f(n) = \sum_{d | n} \mu(d) F(n/d) \ . $$ The correspondence $f \mapsto F$ is the Möbius transform, and $F \mapsto f$ the inverse Möbius transform.

Another inversion formula: If $P(n)$ is a totally multiplicative function for which $P(1) = 1$, and $f(x)$ is a function defined for all real $x > 0$, then $$ g(x) = \sum_{n \le x} P(n) f(x/n) $$ implies $$ f(x) = \sum_{n \le x} \mu(n) P(n) g(x/n) \ . $$

All these (and many other) inversion formulas follow from the basic property of the Möbius function that it is the inverse of the unit arithmetic function $E(n) \equiv 1$ under Dirichlet convolution, cf. (the editorial comments to) Möbius function and Multiplicative arithmetic function.

References

[1]	A. Möbius, "Ueber eine besondere Art der Umkehrung der Reihen" J. Reine Angew. Math. , 9 (1832) pp. 105–123 DOI 10.1515/crll.1832.9.105 Zbl 009.0333cj
[2]	I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian) Zbl 0057.28201
[3]	K. Prachar, "Primzahlverteilung" , Springer (1957) Zbl 0080.25901
[4]	Hua Loo Keng, "Introduction to number theory" Springer-Verlag (1982) ISBN 3-540-10818-1 Zbl 0483.10001
[5]	Alan Baker, "A concise introduction to the theory of numbers" Cambridge University Press (1984) ISBN 0-521-28654-9 Zbl 0554.10001
[6]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" (6th edition, edited and revised by D. R. Heath-Brown and J. H. Silverman with a foreword by Andrew Wiles) Oxford University Press (2008) ISBN 978-0-19-921986-5 Zbl 1159.11001

Gilbreath conjecture

A conjecture on the distribution of prime numbers.

For any sequence $(x_n)$, define the absolute difference sequence $\delta^1_n = |x_{n+1} - x_n|$, and the iterated differences $\delta^{k+1} = \delta^1 \delta^k$. In 1958 N. L. Gilbreath conjectured that when applied to the sequence of prime numbers, the first term in each iterated sequence $\delta^k$ is always $1$. Odlyzko has verified the conjecture for the primes $\le 10^{13}$.

References

[1]	Andrew M. Odlyzko, "Iterated absolute values of differences of consecutive primes", Math. Comput. 61, no.203 (1993) pp.373-380 DOI 10.2307/2152962 Zbl 0781.11037
[2]	Richard K. Guy, "Unsolved problems in number theory" (3rd ed.) Springer-Verlag (2004) ISBN 0-387-20860-7 Zbl 1058.11001
[2]	Norman Gilbreath, "Processing process: the Gilbreath conjecture", J. Number Theory 131 (2011) pp.2436-2441 DOI 10.1016/j.jnt.2011.06.008 Zbl 1254.11006

Fréchet filter

The filter on an infinite set $A$ consisting of all cofinite subsets of $A$: that is, all subsets of $A$ such that the relative complement is finite. More generally, the filter on a set $A$ of cardinality $\mathfrak{a}$ consisting of all subsets of $A$ with complement of cardinality strictly less than $\mathfrak{a}$. The Fréchet filter is not principal.

The Fréchet ideal is the ideal dual to the Fréchet filter: it is the collection of all finite subsets of $A$, or all subsets of cardinality strictly less than $\mathfrak{a}$, respectively.

References

[1]	Thomas Jech, Set Theory (3rd edition), Springer (2003) ISBN 3-540-44085-2 Zbl 1007.03002

Principal filter

A filter on a set $A$ consisting of all subsets of $A$ containing a given subset $X$. If $X$ is a singleton $\{x\}$ then the principal filter on $\{x\}$ is a principal ultrafilter. The Fréchet filter is an example of a non-principal filter.

References

[1]	Thomas Jech, Set Theory (3rd edition), Springer (2003) ISBN 3-540-44085-2 Zbl 1007.03002

Fréchet filter

The filter on an infinite set $A$ consisting of all cofinite subsets of $A$: that is, all subsets of $A$ such that the relative complement is finite. More generally, the filter on a set $A$ of cardinality $\mathfrak{a}$ consisting of all subsets of $A$ with complement of cardinality strictly less than $\mathfrak{a}$. The Fréchet filter is not principal.

The Fréchet ideal is the ideal dual to the Fréchet filter: it is the collection of all finite subsets of $A$, or all subsets of cardinality strictly less than $\mathfrak{a}$, respectively.

References

[1]	Thomas Jech, Set Theory (3rd edition), Springer (2003) ISBN 3-540-44085-2 Zbl 1007.03002

Puiseux series

over a field $K$

A formal power series or Laurent series with coefficients in $K$ and formal variable $Z^{1/k}$ for some natural number $k$. The Puiseaux series form a field $K\{\{Z\}\} = \bigcup_{k\ge1} K(( Z ^{1/k} ))$.

For complex analytic functions, the functions with branch points are represented by Puiseux series.

The Newton–Puiseux theorem states that if $K$ is an algebraically closed field of characteristic zero then the algebraic closure of the field of formal power series $K((Z))$ is the field of Puiseux series $K\{\{Z\}\}$. This is not the case for positive characteristic.

References

[1]

Kedlaya, Kiran S. "The algebraic closure of the power series field in positive characteristic" Proc. Am. Math. Soc. 129 (2001) DOI 10.1090/S0002-9939-01-06001-4 Zbl 1012.12007