User:Richard Pinch/sandbox

Dilworth number

A numerical invariant of a graph. The Dilworth number of the graph $G$ is the maximum of vertices in a set $D$ such that for any pair of such vertices $x,y \in D$, the neighbourhoods $N(x)$, N(y)$ each has at least one element not contained in the other. It is the [[Width of a partially ordered set|width]] of the set of neighbourhoods partially ordered by inclusion. The [[diameter]] of a graph exceeds its Dilworth number by at most 1. There is a polynomial time algorithm for computing the Dilworth number of a finite graph. =='"`UNIQ--h-1--QINU`"'References== * Andreas Brandstädt, Van Bang Le; Jeremy P. Spinrad, "Graph classes: a survey". SIAM Monographs on Discrete Mathematics. and Applications '''3'''. Society for Industrial and Applied Mathematics (1999) ISBN 978-0-898714-32-6 {ZBL|0919.05001}} ='"`UNIQ--h-2--QINU`"'Quaternion algebra= An associative algebra over a field that generalises the construction of the [[quaternion]]s over the field of real numbers. The quaternion algebra $(a,b)_F$ over a field $F$ is the four-dimensional vector space with basis $\mathbf{1}, \mathbf{i}, \mathbf{j}, \mathbf{k}$ and multiplication defined by '"`UNIQ-MathJax2-QINU`"' It follows that $\mathbf{k}^2 = -ab\mathbf{1}$ and that any two of $\mathbf{i}, \mathbf{j}, \mathbf{k}$ anti-commute. The construction is symmetric: $(a,b)_F = (b,a)_F$. The algebra so constructed is a [[central simple algebra]] over $F$. The algebra $(1,1)_F$ is isomorphic to the $2 \times 2$ matrix ring $M_2(F)$. Such a quaternion algebra is termed ''split''. =='"`UNIQ--h-3--QINU`"'References== * Tsit-Yuen Lam, ''Introduction to Quadratic Forms over Fields'', Graduate Studies in Mathematics '''67''' , American Mathematical Society (2005) ISBN 0-8218-1095-2 [https://zbmath.org/?q=an%3A1068.11023 Zbl 1068.11023] [https://mathscinet.ams.org/mathscinet/article?mr=2104929 MR2104929] ='"`UNIQ--h-4--QINU`"'Square-free= Containing only a trivial square factor. A natural number $n$ is square-free if the only natural number $d$ such that $d^2$ divides $n$ is $d=1$. The prime factorisation of such a number $n$ has all exponents equal to 1. Similarly a polynomial $f$ is square-free if the only factors $g$ such that $g^2$ divides $f$ are constants. For polynomials over the real or complex numbers, this is equivalent to having no repeated roots. An element $x$ of a [[monoid]] $M$ is square-free if the only $y \in M$ such that $y^2$ divides $x$ are units. A word $x$ over an alphabet $A$, that is, an element of the free monoid $A^*$, is square-free if $x=uwwv$ implies that $w$ is the empty string. ='"`UNIQ--h-5--QINU`"'Root of unity= An element $\zeta$ of a ring $R$ with unity $1$ such that $\zeta^m = 1$ for some $m \ge 1$. The least such $m$ is the order of $\zeta$. A primitive root of unity of order $m$ is an element $\zeta$ such that $\zeta^m = 1$ and $\zeta^r \neq 1$ for any positive integer $r < m$. The element $\zeta$ generates the [[cyclic group]] $\mu_m$ of roots of unity of order $m$. In the field of [[complex number]]s, there are roots of unity of every order: those of order $m$ take the form '"`UNIQ-MathJax3-QINU`"' where $0 < k < m$ and $k$ is relatively prime to $m$. ===='"`UNIQ--h-6--QINU`"'References==== <table> <tr><td valign="top">[1]</td> <td valign="top"> S. Lang, "Algebra" , Addison-Wesley (1984)</td></tr> </table> ='"`UNIQ--h-7--QINU`"'Law of quadratic reciprocity= =='"`UNIQ--h-8--QINU`"'Gauss reciprocity law== A relation connecting the values of the Legendre symbols (cf. [[Legendre symbol|Legendre symbol]]) $(p/q)$ and $(q/p)$ for different odd prime numbers $p$ and $q$ (cf. [[Quadratic reciprocity law|Quadratic reciprocity law]]). In addition to the principal reciprocity law of Gauss for quadratic residues, which may be expressed as the relation '"`UNIQ-MathJax4-QINU`"' there are two more additions to this law, viz.: '"`UNIQ-MathJax5-QINU`"' The reciprocity law for quadratic residues was first stated in 1772 by L. Euler. A. Legendre in 1785 formulated the law in modern form and proved a part of it. C.F. Gauss in 1801 was the first to give a complete proof of the law [[#References|[1]]]; he also gave no less than eight different proofs of the reciprocity law, based on various principles, during his lifetime. Attempts to establish the reciprocity law for [[Cubic residue|cubic]] and [[biquadratic residue]]s led Gauss to introduce the ring of [[Gaussian integer]]s. ===='"`UNIQ--h-9--QINU`"'References==== <table> <tr><td valign="top">[1]</td> <td valign="top"> C.F. Gauss, "Disquisitiones Arithmeticae" , Yale Univ. Press (1966) (Translated from Latin)</td></tr> <tr><td valign="top">[2]</td> <td valign="top"> I.M. [I.M. Vinogradov] Winogradow, "Elemente der Zahlentheorie" , R. Oldenbourg (1956) (Translated from Russian)</td></tr> <tr><td valign="top">[3]</td> <td valign="top"> H. Hasse, "Vorlesungen über Zahlentheorie" , Springer (1950)</td></tr> </table> ===='"`UNIQ--h-10--QINU`"'Comments==== Attempts to generalize the quadratic reciprocity law (as Gauss' reciprocity law is usually called) have been an important driving force for the development of [[algebraic number theory]] and [[class field theory]]. A far-reaching generalization of the quadratic reciprocity law is known as Artin's reciprocity law. =='"`UNIQ--h-11--QINU`"'Quadratic reciprocity law== The relation '"`UNIQ-MathJax6-QINU`"' connecting the Legendre symbols (cf. [[Legendre symbol|Legendre symbol]]) '"`UNIQ-MathJax7-QINU`"' for different odd prime numbers $p$ and $q$. There are two additions to this quadratic reciprocity law, namely: '"`UNIQ-MathJax8-QINU`"' and '"`UNIQ-MathJax9-QINU`"' C.F. Gauss gave the first complete proof of the quadratic reciprocity law, which for this reason is also called the [[Gauss reciprocity law|Gauss reciprocity law]]. It immediately follows from this law that for a given square-free number $d$, the primes $p$ for which $d$ is a quadratic residue modulo $p$ ly in certain arithmetic progressions with common difference $2|d|$ or $4|d|$. The number of these progressions is $\phi(2|d|)/2$ or $\phi(4|d|)/2$, where $\phi(n)$ is the [[Euler function|Euler function]]. The quadratic reciprocity law makes it possible to establish factorization laws in quadratic extensions $\mathbf Q(\sqrt d)$ of the field of rational numbers, since the factorization into prime factors in $\mathbf Q(\sqrt d)$ of a prime number that does not divide $d$ depends on whether or not $x^2-d$ is reducible modulo $p$. ===='"`UNIQ--h-12--QINU`"'References==== <table><tr><td valign="top">[1]</td> <td valign="top"> I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)</td></tr><tr><td valign="top">[2]</td> <td valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966)</td></tr></table> ===='"`UNIQ--h-13--QINU`"'Comments==== See also [[Quadratic residue|Quadratic residue]]; [[Dirichlet character|Dirichlet character]]. ===='"`UNIQ--h-14--QINU`"'References==== <table><tr><td valign="top">[a1]</td> <td valign="top"> G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XIII</td></tr></table> ='"`UNIQ--h-15--QINU`"'Euler function= =='"`UNIQ--h-16--QINU`"'Euler function== The arithmetic function $\phi$ whose value at $n$ is equal to the number of positive integers not exceeding $n$ and relatively prime to $n$ (the "totatives" of $n$). The Euler function is a [[multiplicative arithmetic function]], that is $\phi(1)=1$ and $\phi(mn)=\phi(m)\phi(n)$ for $(m,n)=1$. The function $\phi(n)$ satisfies the relations '"`UNIQ-MathJax10-QINU`"' '"`UNIQ-MathJax11-QINU`"' '"`UNIQ-MathJax12-QINU`"' It was introduced by L. Euler (1763). The function $\phi(n)$ can be evaluated by $\phi(n)=n\prod_{p|n}(1-p^{-1})$, where the product is taken over all primes dividing $n$, cf. [[#References|[a1]]]. For a derivation of the asymptotic formula in the article above, as well as of the formula '"`UNIQ-MathJax13-QINU`"' where $\gamma$ is the [[Euler constant]], see also [[#References|[a1]]], Chapts. 18.4 and 18.5. The Carmichael conjecture on the Euler totient function states that if $\phi(x) = m$ for some $m$, then $\phi(y) = m$ for some $y \neq x$; i.e. no value of the Euler function is assumed once. This has been verified for $x < 10^{1000000}$, [[#References|[c1]]]. D. H. Lehmer asked whether whether there is any composite number $n$ such that $\phi(n)$ divides $n-1$. This is true of every [[prime number]], and Lehmer conjectured in 1932 that there are no composite numbers with this property: he showed that if any such $n$ exists, it must be odd, square-free, and divisible by at least seven primes [[#References|[c2]]]. For some results on this still (1996) largely open problem, see [[#References|[c3]]] and the references therein. ===='"`UNIQ--h-17--QINU`"'References==== <table> <tr><td valign="top">[1]</td> <td valign="top"> K. Chandrasekharan, "Introduction to analytic number theory" , Springer (1968) [https://mathscinet.ams.org/mathscinet/article?mr=0249348 MR0249348] [https://zbmath.org/?q=an%3A0169.37502 Zbl 0169.37502]</td></tr> <tr><td valign="top">[a1]</td> <td valign="top"> G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 [https://mathscinet.ams.org/mathscinet/article?mr=0568909 MR0568909] [https://zbmath.org/?q=an%3A0423.10001 Zbl 0423.10001]</td></tr> <tr><td valign="top">[c1]</td> <td valign="top"> A. Schlafly, S. Wagon, "Carmichael's conjecture on the Euler function is valid below $10^{1000000}$" ''Math. Comp.'' , '''63''' (1994) pp. 415–419</td></tr> <tr><td valign="top">[c2]</td> <td valign="top"> D.H. Lehmer, "On Euler's totient function", ''Bulletin of the American Mathematical Society'' '''38''' (1932) 745–751 [https://doi.org/10.1090/s0002-9904-1932-05521-5 DOI 10.1090/s0002-9904-1932-05521-5] [https://zbmath.org/?q=an%3A0005.34302 Zbl 0005.34302]</td></tr> <tr><td valign="top">[c3]</td> <td valign="top"> V. Siva Rama Prasad, M. Rangamma, "On composite $n$ for which $\phi(n)|n-1$" ''Nieuw Archief voor Wiskunde (4)'' , '''5''' (1987) pp. 77–83</td></tr> <tr><td valign="top">[c4]</td> <td valign="top"> M.V. Subbarao, V. Siva Rama Prasad, "Some analogues of a Lehmer problem on the totient function" ''Rocky Mount. J. Math.'' , '''15''' (1985) pp. 609–620</td></tr> <tr><td valign="top">[c5]</td> <td valign="top"> R. Sivamarakrishnan, "The many facets of Euler's totient II: generalizations and analogues" ''Nieuw Archief Wiskunde (4)'' , '''8''' (1990) pp. 169–188</td></tr> <tr><td valign="top">[c6]</td> <td valign="top"> R. Sivamarakrishnan, "The many facets of Euler's totient I" ''Nieuw Archief Wiskunde (4)'' , '''4''' (1986) pp. 175–190</td></tr> <tr><td valign="top">[c7]</td> <td valign="top"> L.E. Dickson, "History of the theory of numbers I: Divisibility and primality" , Chelsea, reprint (1971) pp. Chapt. V; 113–155</td></tr> </table> =='"`UNIQ--h-18--QINU`"'Totient function== ''Euler totient function, Euler totient'' Another frequently used named for the [[Euler function]] $\phi(n)$, which counts a [[reduced system of residues]] modulo $n$: the natural numbers $k \in \{1,\ldots,n\}$ that are relatively prime to $n$. A natural generalization of the Euler totient function is the [[Jordan totient function]] $J_k(n)$, which counts the number of $k$-tuples $(a_1,\ldots,a_k)$, $a_i \in \{1,\ldots,n\}$, such that $\mathrm{hcf}\{n,a_1,\ldots,a_k\} = 1$. Clearly, $J_1 = \phi$. The $J_k$ are [[multiplicative arithmetic function]]s. One has '"`UNIQ-MathJax14-QINU`"' where $p$ runs over the prime numbers dividing $n$, and '"`UNIQ-MathJax15-QINU`"' where $\mu$ is the [[Möbius function]] and $d$ runs over all divisors of $n$. For $k=1$ these formulae reduce to the well-known formulae for the Euler function. ===='"`UNIQ--h-19--QINU`"'References==== <table> <tr><td valign="top">[a1]</td> <td valign="top"> A. Schlafly, S. Wagon, "Carmichael's conjecture on the Euler function is valid below $10^{1000000}$" ''Math. Comp.'' , '''63''' (1994) pp. 415–419</td></tr> <tr><td valign="top">[a2]</td> <td valign="top"> D.H. Lehmer, "On Euler's totient function" ''Bull. Amer. Math. Soc.'' , '''38''' (1932) pp. 745–751</td></tr> <tr><td valign="top">[a3]</td> <td valign="top"> V. Siva Rama Prasad, M. Rangamma, "On composite $n$ for which $\phi(n)|n-1$" ''Nieuw Archief voor Wiskunde (4)'' , '''5''' (1987) pp. 77–83</td></tr> <tr><td valign="top">[a4]</td> <td valign="top"> M.V. Subbarao, V. Siva Rama Prasad, "Some analogues of a Lehmer problem on the totient function" ''Rocky Mount. J. Math.'' , '''15''' (1985) pp. 609–620</td></tr> <tr><td valign="top">[a5]</td> <td valign="top"> R. Sivamarakrishnan, "The many facets of Euler's totient II: generalizations and analogues" ''Nieuw Archief Wiskunde (4)'' , '''8''' (1990) pp. 169–188</td></tr> <tr><td valign="top">[a6]</td> <td valign="top"> R. Sivamarakrishnan, "The many facets of Euler's totient I" ''Nieuw Archief Wiskunde (4)'' , '''4''' (1986) pp. 175–190</td></tr> <tr><td valign="top">[a7]</td> <td valign="top"> L.E. Dickson, "History of the theory of numbers I: Divisibility and primality" , Chelsea, reprint (1971) pp. Chapt. V; 113–155</td></tr> </table> =='"`UNIQ--h-20--QINU`"'Jordan totient function== An arithmetic function $J_k(n)$ of a [[natural number]] $n$, named after Camille Jordan, counting the $k$-tuples of positive integers all less than or equal to $n$ that form a [[Coprime numbers|coprime]] $(k + 1)$-tuple together with $n$. This is a generalisation of Euler's [[totient function]], which is $J_1$. Jordan's totient function is [[Multiplicative arithmetic function|multiplicative]] and may be evaluated as '"`UNIQ-MathJax16-QINU`"' By [[Möbius inversion]] we have $\sum_{d | n } J_k(d) = n^k $. The [[Average order of an arithmetic function|average order]] of $J_k(n)$ is $c n^k$ for some $c$. The analogue of Lehmer's problem for the Jordan totient function (and $k>1$) is easy, [[#References|[c4]]]: For $k>1$, $J_k(n) | n^k-1$ if and only if $n$ is a prime number. Moreover, if $n$ is a prime number, then $J_k(n) = n^k-1$. For much more information on the Euler totient function, the Jordan totient function and various other generalizations, see [[#References|[c5]]], [[#References|[c6]]]. ===='"`UNIQ--h-21--QINU`"'References==== * Dickson, L.E. ''History of the Theory of Numbers I'', Chelsea (1971) p. 147, ISBN 0-8284-0086-5 * Ram Murty, M. ''Problems in Analytic Number Theory'', Graduate Texts in Mathematics '''206''' Springer-Verlag (2001) p. 11 ISBN 0387951431 [https://zbmath.org/?q=an%3A0971.11001 Zbl 0971.11001] * Sándor, Jozsef; Crstici, Borislav, edd. Handbook of number theory II. Dordrecht: Kluwer Academic (2004). pp.32–36. ISBN 1-4020-2546-7. [https://zbmath.org/?q=an%3A1079.11001 Zbl 1079.11001] ='"`UNIQ--h-22--QINU`"'Matrix multiplication= A [[binary operation]] on compatible [[matrix|matrices]] over a [[ring]] $R$. There are several such operations. ==='"`UNIQ--h-23--QINU`"'Cayley multiplication=== Most usually what is referred to as "matrix multiplication". The product of an $m \times n$ matrix $A$ and an $n \times p$ matrix $B$ is the $m \times p$ matrix $AB$ with entries '"`UNIQ-MathJax17-QINU`"' The multiplication corresponds to composition of linear maps. If $A$ is the matrix of a linear map $\alpha : R^m \rightarrow R^n$ and $B$ is the matrix of a linear map $\beta : R^n \rightarrow R^p$, then $AB$ is the matrix of the linear map $\alpha\beta : R^m \rightarrow R^p$. ==='"`UNIQ--h-24--QINU`"'Hadamard multiplication=== The Hadamard product, or Schur product, of two $m \times n$ matrices $A$ and $B$ is the $m \times n$ matrix $AB$ with '"`UNIQ-MathJax18-QINU`"' ==='"`UNIQ--h-25--QINU`"'Kronecker multiplication=== The Kronecker product, also tensor product or direct product, of an $m \times n$ matrix $A$ and an $p \times q$ matrix $B$ is the $mp \times nq$ matrix $AB$ with entries '"`UNIQ-MathJax19-QINU`"' ===='"`UNIQ--h-26--QINU`"'References==== * Gene H. Golub, Charles F. Van Loan, ''Matrix Computations'', Johns Hopkins Studies in the Mathematical Sciences '''3''', JHU Press (2013) ISBN 1421407949 * James E. Gentle, ''Matrix Algebra: Theory, Computations, and Applications in Statistics'', Springer Texts in Statistics, Springer (2007) ISBN 0-387-70872-3 * Manfred Schroeder, ''Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity'', Springer (2008) ISBN 3-540-85297-2 ='"`UNIQ--h-27--QINU`"'Multiplicative sequence= Also ''m''-sequence, a sequence of [[polynomial]]s associated with a formal group structure. They have application in the [[cobordism|cobordism ring]] in [[algebraic topology]]. =='"`UNIQ--h-28--QINU`"'Definition== Let $K_n$ be polynomials over a ring $A$ in indeterminates $p_1,\ldots$ weighted so that $p_i$ has weight $i$ (with $p_0=1$) and all the terms in $K_n$ have weight $n$ (so that $K_n$ is a polynomial in $p_1,\ldots,p_n$). The sequence $K_n$ is ''multiplicative'' if an identity '"`UNIQ-MathJax20-QINU`"' implies '"`UNIQ-MathJax21-QINU`"' The power series '"`UNIQ-MathJax22-QINU`"' is the ''characteristic power series'' of the $K_n$. A multiplicative sequence is determined by its characteristic power series $Q(z)$, and every power series with constant term 1 gives rise to a multiplicative sequence. To recover a multiplicative sequence from a characteristic power series $Q(z)$ we consider the coefficient of ''z''^''j'' in the product '"`UNIQ-MathJax23-QINU`"' for any $m>j$. This is symmetric in the $\beta_i$ and homogeneous of weight ''j'': so can be expressed as a polynomial $K_j(p_1,\ldots,p_j)$ in the [[elementary symmetric function]]s $p$ of the $\beta$. Then $K_j$ defines a multiplicative sequence. =='"`UNIQ--h-29--QINU`"'Examples== As an example, the sequence $K_n = p_n$ is multiplicative and has characteristic power series $1+z$. Consider the power series '"`UNIQ-MathJax24-QINU`"' where $B_k$ is the $k$-th [[Bernoulli number]]. The multiplicative sequence with $Q$ as characteristic power series is denoted $L_j(p_1,\ldots,p_j)$. The multiplicative sequence with characteristic power series '"`UNIQ-MathJax25-QINU`"' is denoted $A_j(p_1,\ldots,p_j)$. The multiplicative sequence with characteristic power series '"`UNIQ-MathJax26-QINU`"' is denoted $T_j(p_1,\ldots,p_j)$: the ''[[Todd polynomial]]s''. =='"`UNIQ--h-30--QINU`"'Genus== The '''genus''' of a multiplicative sequence is a [[ring homomorphism]], from the [[cobordism|cobordism ring]] of smooth oriented [[compact manifold]]s to another [[ring]], usually the ring of [[rational number]]s. For example, the [[Todd genus]] is associated to the Todd polynomials $T_j$ with characteristic power series '"`UNIQ-MathJax27-QINU`"' and the [[L-genus]] is associated to the polynomials $L_j$ with charac\teristic polynomial '"`UNIQ-MathJax28-QINU`"' =='"`UNIQ--h-31--QINU`"'References== * Hirzebruch, Friedrich. ''Topological methods in algebraic geometry'', Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel. Reprint of the 2nd, corr. print. of the 3rd ed. [1978] (Berlin: Springer-Verlag, 1995). ISBN 3-540-58663-6. [https://zbmath.org/?q=an%3A0843.14009 Zbl 0843.14009]. ='"`UNIQ--h-32--QINU`"'Nagao's theorem= A result, named after Hirosi Nagao, about the structure of the [[group]] of 2-by-2 [[Invertible matrix|invertible matrices]] over the [[ring of polynomials]] over a [[field]]. It has been extended by [[Jean-Pierre Serre|Serre]] to give a description of the structure of the corresponding matrix group over the [[coordinate ring]] of a [[projective curve]]. =='"`UNIQ--h-33--QINU`"'Nagao's theorem== For a [[Ring (mathematics)|general ring]] $R$ we let $GL_2(R)$ denote the group of invertible 2-by-2 matrices with entries in $R$, and let $R^*$ denote the [[group of units]] of $R$, and let '"`UNIQ-MathJax29-QINU`"' Then $B(R)$ is a subgroup of $GL_2(R)$. Nagao's theorem states that in the case that $R$ is the ring $K[t]$ of polynomials in one variable over a field $K$, the group $GL_2(R)$ is the [[amalgamated product]] of $GL_2(K)$ and $B(K[t])$ over their intersection $B(K)$. =='"`UNIQ--h-34--QINU`"'Serre's extension== In this setting, $C$ is a [[Singular point of an algebraic variety|smooth]] projective curve over a field $K$. For a [[closed point]] $P$ of $C$ let $R$ be the corresponding coordinate ring of $C$ with $P$ removed. There exists a [[graph of groups]] $(G,T)$ where $T$ is a [[tree]] with at most one non-terminal vertex, such that $GL_2(R)$ is isomorphic to the [[fundamental group]] $\pi_1(G,T)$. =='"`UNIQ--h-35--QINU`"'References== * Mason, A.. "Serre's generalization of Nagao's theorem: an elementary approach". ''Transactions of the American Mathematical Society'' '''353''' (2001) 749–767. [https://doi.org/10.1090/S0002-9947-00-02707-0 DOI 10.1090/S0002-9947-00-02707-0] [https://zbmath.org/?q=an%3A0964.20027 Zbl 0964.20027]. * Nagao, Hirosi. "On $GL(2, K[x])$". J. Inst. Polytechn., Osaka City Univ., Ser. A '''10''' (1959) 117–121. [https://mathscinet.ams.org/mathscinet/article?mr=0114866 MR0114866]. [https://zbmath.org/?q=an%3A0092.02504 Zbl 0092.02504]. * Serre, Jean-Pierre. ''Trees''. (Springer, 2003) ISBN 3-540-44237-5. ='"`UNIQ--h-36--QINU`"'Erdős–Wintner theorem= A result in [[probabilistic number theory]] characterising those [[additive function]]s that possess a limiting distribution. =='"`UNIQ--h-37--QINU`"'Limiting distribution== A distribution function $F$ is a non-decreasing function from the real numbers to the unit interval [0,1] which is right-continuous and has limits 0 at $-\infty$ and 1 at $+\infty$. Let $f$ be a complex-valued function on natural numbers. We say that $F$ is a limiting distribution for $f$ if $F$ is a distribution function and the sequence $F_N$ defined by '"`UNIQ-MathJax30-QINU`"' [[Weak convergence of probability measures|converges weakly]] to $F$. =='"`UNIQ--h-38--QINU`"'Statement of the theorem== Let $f$ be an additive function. There is a limiting distribution for $f$ if and only if the following three series converge: '"`UNIQ-MathJax31-QINU`"' When these conditions are satisfied, the distribution is given by '"`UNIQ-MathJax32-QINU`"' =='"`UNIQ--h-39--QINU`"'References== * Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. ''Handbook of number theory I''. Dordrecht: Springer-Verlag (2006). pp. 564–566. ISBN 1-4020-4215-9. [https://zbmath.org/?q=an%3A1151.11300 Zbl 1151.11300] * Tenenbaum, Gérald ''Introduction to Analytic and Probabilistic Number Theory''. Cambridge studies in advanced mathematics '''46'''. Cambridge University Press (1995). ISBN 0-521-41261-7. [https://zbmath.org/?q=an%3A0831.11001 Zbl 0831.11001] ='"`UNIQ--h-40--QINU`"'Brauer–Wall group= A [[group]] classifying graded [[central simple algebra]]s over a field. It was first defined by Wall (1964) as a generalisation of the [[Brauer group]]. The Brauer group $\mathrm{B}(F)$ of a field $F$ is defined on the isomorphism classes of central simple algebras over ''F''. The analogous construction for $\mathbf{Z}/2$-[[graded algebra]]s defines the Brauer–Wall group $\mathrm{BW}(F)$.[[#Lam (2005) pp.98–99|[Lam (2005) pp.98–99]]] =='"`UNIQ--h-41--QINU`"'Properties== * The Brauer group $\mathrm{B}(F)$ injects into $\mathrm{BW}(F)$ by mapping a CSA $A$ to the graded algebra which is $A$ in grade zero. There is an exact sequence '"`UNIQ-MathJax33-QINU`"' where $Q(F)$ is the group of graded quadratic extensions of $F$, defined as $\mathbf{Z}/2 \times F^*/(F^*)^2$ with multiplication $(e,x)(f,y) = (e+f,(-1)^{ef}xy$. The map from W to BW is the '''[[Clifford invariant]]''' defined by mapping an algebra to the pair consisting of its grade and [[Determinant of a quadratic form|determinant]]. There is a map from the additive group of the [[Witt–Grothendieck ring]] to the Brauer–Wall group obtained by sending a quadratic space to its [[Clifford algebra]]. The map factors through the [[Witt group]][[#Lam (2005) p.113|[Lam (2005) p.113]]] which has kernel $I^3$, where $I$ is the fundamental ideal of $W(F)$.[[#Lam (2005) p.115|[Lam (2005) p.115]]] =='"`UNIQ--h-42--QINU`"'Examples== * $\mathrm{BW}(\mathbf{R})$ is isomorphic to $\mathbf{Z}/8$. This is an algebraic aspect of [[Bott periodicity]]. =='"`UNIQ--h-43--QINU`"'References== * Lam, Tsit-Yuen, ''Introduction to Quadratic Forms over Fields'', Graduate Studies in Mathematics '''67''', (American Mathematical Society, 2005) ISBN 0-8218-1095-2 [https://mathscinet.ams.org/mathscinet/article?mr=2104929 MR2104929], [https://zbmath.org/?q=an%3A1068.11023 Zbl 1068.11023] * Wall, C. T. C., "Graded Brauer groups", ''Journal für die reine und angewandte Mathematik'' '''213''' (1964) 187–199, ISSN 0075-4102, [https://zbmath.org/?q=an%3A0125.01904 Zbl 0125.01904], [https://mathscinet.ams.org/mathscinet/article?mr=0167498 MR0167498] ='"`UNIQ--h-44--QINU`"'Factor system= A function on a [[group]] giving the data required to construct an [[algebra]]. A factor system constitutes a realisation of the cocycles in the second [[cohomology group]] in [[group cohomology]]. Let $G$ be a group and $L$ a field on which $G$ acts as automorphisms. A ''cocycle'' or ''factor system'' is a map $c : G \times G \rightarrow L^*$ satisfying '"`UNIQ-MathJax34-QINU`"' Cocycles are ''equivalent'' if there exists some system of elements $a : G \rightarrow L^*$ with '"`UNIQ-MathJax35-QINU`"' Cocycles of the form '"`UNIQ-MathJax36-QINU`"' are called ''split''. Cocycles under multiplication modulo split cocycles form a group, the second cohomology group $H^2(G,L^*)$. =='"`UNIQ--h-45--QINU`"'Crossed product algebras== Let us take the case that $G$ is the [[Galois group]] of a [[field extension]] $L/K$. A factor system $c$ in $H^2(G,L^*)$ gives rise to a ''crossed product algebra'' $A$, which is a $K$-algebra containing $L$ as a subfield, generated by the elements $\lambda \in L$ and $u_g$ with multiplication '"`UNIQ-MathJax37-QINU`"' '"`UNIQ-MathJax38-QINU`"' Equivalent factor systems correspond to a change of basis in $A$ over $K$. We may write '"`UNIQ-MathJax39-QINU`"' Every [[central simple algebra]] over$K$ that splits over $L$ arises in this way. The tensor product of algebras corresponds to multiplication of the corresponding elements in$H^2$. We thus obtain an identification of the [[Brauer group]], where the elements are classes of CSAs over $K$, with $H^2$.[[#Saltman (1999) p.44|[Saltman (1999) p.44]]] =='"`UNIQ--h-46--QINU`"'Cyclic algebra== Let us further restrict to the case that $L/K$ is [[Cyclic extension|cyclic]] with Galois group $G$ of order $n$ generated by $t$. Let $A$ be a crossed product $(L,G,c)$ with factor set $c$. Let $u=u_t$ be the generator in $A$ corresponding to $t$. We can define the other generators '"`UNIQ-MathJax40-QINU`"' and then we have $u^n = a$ in $K$. This element $a$ specifies a cocycle $c$ by '"`UNIQ-MathJax41-QINU`"' It thus makes sense to denote $A$ simply by $(L,t,a)$. However $a$ is not uniquely specified by $A$ since we can multiply $u$ by any element $\lambda$ of $L^*$ and then $a$ is multiplied by the product of the conjugates of λ. Hence $A$ corresponds to an element of the norm residue group $(K^*/N_{L/K}L^*$. We obtain the isomorphisms $$ \mathop{Br}(L/K) \equiv K^*/\mathrm{N}_{L/K} L^* \equiv \mathrm{H}^2(G,L^*) \ . $$

References

• Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Universitext. Translated from the German by Silvio Levy. With the collaboration of the translator. Springer-Verlag. ISBN 978-0-387-72487-4. Zbl 1130.12001.
• Saltman, David J. (1999). Lectures on division algebras. Regional Conference Series in Mathematics 94. Providence, RI: American Mathematical Society. ISBN 0-8218-0979-2. Zbl 0934.16013.