User:Richard Pinch/sandbox-5

Unital ring

unitary ring, ring with identity, ring-with-a-one

A ring with a multiplicative identity: an element $1$ such that $1x = x = x1$ for all elements $x$ of the ring. In many developments of the theory of rings, the existence of such an identity is taken as part of the definition of a ring. The term rng has been coined to denote rings in which the existence of an identity is not assumed. A unital ring homomorphism is a ring homomorphism between unital rings which respects the multiplicative identities. Any ring $R$ can be embedded in a ring $R^1$ with an identity by taking $R^1 = \mathbf{Z} \oplus R$ with multiplication $(m,r) \cdot (n,s) = (mn, ms + nr + rs)$ which has $(1,0)$ as a multiplicative identity.

A unital algebra $A$ over a field $K$ is an algebra over $K$ which is unital as a ring. As with rings, any $K$-algebra $A$ can be embedded in a unital $K$-algebra $A^1 = K \oplus A$.

A unital or unitary module is a module over a unital ring in which the identity element of the ring acts as the identity on the module.

References

• Nathan Jacobson, "Basic Algebra I" (2 ed) Dover (2012) ISBN 0486135225

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

Metabelian

A term with several meanings.

A metabelian group is a solvable group of derived length two, i.e. a group whose commutator subgroup is Abelian, see Meta-Abelian group.

In the Russian mathematical literature, by a metabelian group one sometimes means a nilpotent group of nilpotency class 2.

A metabelian algebra satisfies the identity $[[x,y],z] = 0$ where $[x,y] = x \cdot y - y \cdot x$ is the commutator.

A metabelian Lie algebra is an extension of an abelian algebra by an abelian algebra.

References

[1]

Knuth equivalence

Plactic equivalence

An equivalence relation on words over a totally ordered alphabet $(A,{<})$ generated by the equivalences $$ a c b \leftrightarrow c a b $$ when $a \le b < c$ and $$ b a c \leftrightarrow b c a $$ when $a < b \le c$.

The set of equivalence classes under concatenation forms the plactic monoid over $A$.

References

[1]

D.E. Knuth, "Permutations, matrices and generalized Young tableaux" Pacific J. Math. , 34 (1970) pp. 709–727

Mould calculus

The mould algebra over a set $X$ is the total monoid algebra $M^*(X,A)$ of the free monoid on $X$ over a complex unital commutative algebra $A$. A mould is a function $w \mapsto v^w$ from words $w$ over the alphabet $X$ to $A$, formally written as $\sum_{w \in X^*} v^w$. Addition is defined pointwise, $(u+v)^w = u^w + v^w$ and multiplication by convolution $$ u*v : w \mapsto \sum_{w = xy} u^x v^y $$ where the sum is over all finitely many factorisations of the word $w$ into non-trivial words $x,y$.

References

[1]	Ebrahimi-Fard, Kurusch, Fauvet, Frédéric (edd.) Faà di Bruno Hopf algebras, Dyson-Schwinger equations, and Lie-Butcher series. IRMA Lectures in Mathematics and Theoretical Physics 21 European Mathematical Society (2015) ISBN 978-3-03719-143-9 Zbl 1318.16001
[2]	Sauzin, David; "Mould expansions for the saddle-node and resurgence monomials"; Renormalization and Galois theories Connes, Alain et al (edd.) IRMA Lectures in Mathematics and Theoretical Physics 15 European Mathematical Society (2009) ISBN 978-3-03719-073-9 pp.83-163 Zbl 1191.34104