Latest revision as of 17:39, 6 November 2016

Analytic subgroup theorem

A significant result in modern transcendence theory, a generalisation of Baker's theorem on linear forms in logarithms.

Let $G$ be a commutative algebraic group defined over a number field $K$ and let $B$ be a subgroup of the complex points $G(\mathbb{C})$ defined over $K$. There are points of $B$ defined over the field of algebraic numbers if and only if there is a non-trivial analytic subgroup $H$ of $G$' defined over a number field such that $H(\mathbb{C})$ is contained in $B$.

References

Alan Baker and Gisbert Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs 9, Cambridge University Press (2007) ISBN 978-0-521-88268-2. Chapter 6, pp.109-146.

Conductor of an abelian variety

In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is. It is connected to the ramification in the field generated by the division points.

For an abelian variety A defined over a field F with ring of integers R, consider the Néron model of A, which is a 'best possible' model of A defined over R. This model may be represented as a scheme over

Spec(R)

(cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism

Spec(F) → Spec(R)

gives back A. Let A⁰ denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a residue field k, A⁰_k is a group variety over k, hence an extension of an abelian variety by a linear group. This linear group is an extension of a torus by a unipotent group. Let u be the dimension of the unipotent group and t the dimension of the torus. The order of the conductor is

\[ f = 2u + t + \delta , \, \]

where δ is a measure of wild ramification.

Properties

If A has good reduction then f = u = t = δ = 0.
If A has semistable reduction or, more generally, acquires semistable reduction over a Galois extension of F of degree prime to p, the residue characteristic, then δ = 0.
If p > 2d + 1, where d is the dimension of A, then δ = 0.

References

Cubic reciprocity

Various results connecting the solvability of two related cubic equations in modular arithmetic.

The law of cubic reciprocity is most naturally expressed in terms of the Eisenstein integers, that is, the ring $E$ of complex numbers of the form $$ \alpha = a + b\,\omega $$ where and $a$ and $b$ are integers and $\omega = \frac{1}{2}(-1 + i\sqrt 3) = e^{2\pi i/3}$ is a complex cube root of unity. The field norm $\mathbf{N}(\alpha) = \alpha \bar\alpha$.

If $\pi$ is a prime element of $E$ of norm $P$ and $\alpha$ is an element coprime to $\pi$, we define the cubic residue symbol $\left(\frac{\alpha}{\pi}\right)_3$ to be the cube root of unity (power of $\omega$) satisfying $$ \alpha^{(P-1)/3} \equiv \left(\frac{\alpha}{\pi}\right)_3 \pmod \pi $$

We further define a primary prime to be one which is congruent to $-1 \pmod 3$. Then for distinct primary primes $\pi$ and $\theta$ the law of cubic reciprocity is simply $$ \left(\frac{\pi}{\theta}\right)_3 = \left(\frac{\theta}{\pi}\right)_3 $$ with the supplementary laws for the units and for the prime $1-\omega$ of norm 3 that if $\pi = -1 + 3(m+n\omega)$ then $$ \left(\frac{\omega}{\pi}\right)_3 = \omega^{m+n}\,, $$ $$ \left(\frac{1-\omega}{\pi}\right)_3 = \omega^{2m} \ . $$

References

David A. Cox, Primes of the form $x^2+ny^2$, Wiley, 1989, ISBN 0-471-50654-0.
K. Ireland and M. Rosen, A classical introduction to modern number theory, 2nd ed, Graduate Texts in Mathematics 84, Springer-Verlag, 1990.
Franz Lemmermeyer, Reciprocity laws: From Euler to Eisenstein, Springer Verlag, 2000, ISBN 3-540-66957-4.

Genus field

In algebraic number theory, the genus field $G$ of a number field $K$ is the maximal abelian extension of $K$ which is obtained by composing an absolutely abelian field with $K$ and which is unramified at all finite primes of $K$. The genus number of $K$ is the degree $[G:K]$ and the genus group is the Galois group of $G$ over $K$.

If $K$ is itself absolutely abelian, the genus field may be described as the maximal absolutely abelian extension of $K$ unramified at all finite primes: this definition was used by Leopoldt and Hasse.

If $K = \mathbb{Q}(\sqrt m)$ ($m$ squarefree) is a quadratic field of discriminant $D$, the genus field of $K$ is a composite of quadratic fields. Let $p_i$ run over the prime factors of $D$. For each such prime $p$, define $p^*$ as follows:

$$p^* = \pm p \equiv 1 \pmod 4 \text{ if } p \text{ is odd} ; $$ $$2^* = -4, 8, -8 \text{ according as } m \equiv 3 \pmod 4, 2 \pmod 8, -2 \pmod 8 . $$

Then the genus field is the composite $K(\sqrt{p^*_i})$.

References

Template:Cite book

Group isomorphism problem

In abstract algebra, the group isomorphism problem is the decision problem of determining whether two group presentations present isomorphic groups.

The isomorphism problem was identified by Max Dehn in 1911 as one of three fundamental decision problems in group theory; the other two being the word problem and the conjugacy problem.

References

Hall algebra

The Hall polynomials in mathematics were developed by Philip Hall in the 1950s in the study of group representations.

A finite abelian p-group M is a direct sum of cyclic p-power components $C_{p^\lambda_i}$ where $\lambda=(\lambda_1,\lambda_2,\ldots)$ is a partition of $n$ called the type of M. Let $g^\lambda_{\mu,\nu}(p)$ be the number of subgroups N of M such that N has type $\nu$ and the quotient M/N has type $\mu$. Hall showed that the functions g are polynomial functions of p with integer coefficients: these are the Hall polynomials.

Hall next constructs an algebra $H(p)$ with symbols $u_\lambda$ a generators and multiplication given by the $g^\lambda_{\mu,\nu}$ as structure constants

\[ u_\mu u_\nu = \sum_\lambda g^\lambda_{\mu,\nu} u_\lambda \]

which is freely generated by the $u_{\mathbf1_n}$ corresponding to the elementary p-groups. The map from $H(p)$ to the algebra of symmetric functions $e_n$ given by $u_{\mathbf 1_n} \mapsto p^{-n(n-1)}e_n$ is a homomorphism and its image may be interpreted as the Hall-Littlewood polynomial functions. The theory of Schur functions is thus closely connected with the theory of Hall polynomials.

References

I.G. Macdonald, Symmetric functions and Hall polynomials, (Oxford University Press, 1979) ISBN 0-19-853530-9

Hall–Littlewood polynomials

In mathematics, the Hall–Littlewood polynomials encode combinatorial data relating to the representations of the general linear group. They are named for Philip Hall and Dudley E. Littlewood.

References

I.G. Macdonald; Symmetric Functions and Hall Polynomials, (1979), pp. 101-104, Oxford University Press ISBN: 0-19-853530-9
D.E. Littlewood; On certain symmetric functions, Proc. London Math. Soc., 43 (1961), pp. 485–498, DOI: 10.1112/plms/s3-11.1.485

Hutchinson operator

In mathematics, in the study of fractals, a Hutchinson operator is a collection of functions on an underlying space E. The iteration on these functions gives rise to an iterated function system, for which the fixed set is self-similar.

Definition

Formally, let f_i be a finite set of N functions from a set X to itself. We may regard this as defining an operator H on the power set P X as

\[H : A \mapsto \bigcup_{i=1}^N f_i[A],\,\]

where A is any subset of X.

A key question in the theory is to describe the fixed sets of the operator H. One way of constructing such a fixed set is to start with an initial point or set S₀ and iterate the actions of the f_i, taking S_n+1 to be the union of the images of S_n under the operator H; then taking S to be the union of the S_n, that is,

\[S_{n+1} = \bigcup_{i=1}^N f_i[S_n] \]

and

\[S = \bigcup_{n=0}^\infty S_n . \]

Properties

Hutchinson (1981) considered the case when the f_i are contraction mappings on a Euclidean space X = R^d. He showed that such a system of functions has a unique compact (closed and bounded) fixed set S.

The collection of functions $f_i$ together with composition form a monoid. With N functions, then one may visualize the monoid as a full N-ary tree or a Cayley tree.

References

Kneser theorem

A statement about set addition in finite groups. It may be regarded as an extension of the Cauchy-Davenport theorem on sumsets in groups of prime order.

Let $G$ be a non-trivial abelian group and $A,B$ finite non-empty subsets. If $|A| + |B| \le |G|$ then there is a finite subgroup $H$ of $G$ such that $$ |A+B| \ge |A+H| + |B+H| - |H| \ge |A| + |B| - |H| \ . $$

The subgroup $H$ can be taken to be the stabiliser of $A+B$ $$ H = \lbrace g \in G : g + (A+B) = (A+B) \rbrace \ . $$

References

Combinatorial number theory and additive group theory, ser. Advanced Courses in Mathematics CRM Barcelona (2009), Birkhäuser, Zbl 1177.11005 ISBN: 978-3-7643-8961-1
Kneser, Martin; Abschätzungen der asymptotischen Dichte von Summenmengen, Math. Zeitschr., 58 (1953), pp. 459–484, Zbl 0051.28104
Nathanson, Melvyn B.; Additive Number Theory: Inverse Problems and the Geometry of Sumsets, ser. Graduate Texts in Mathematics 165 (1996), pp. 109–132, Springer-Verlag, Zbl 0859.11003 ISBN: 0-387-94655-1
Tao, Terence; Vu, Van H.; Additive Combinatorics, (2010), Cambridge University Press, Zbl 1179.11002 ISBN: 978-0-521-13656-3

Linked field

A field for which the quadratic forms attached to quaternion algebras have a common property.

Let $F$ be a field of characteristic not equal to 2. Let $A = (a_1,a_2)_F$, $B = (b_1,b_2)_F$ be quaternion algebras over F. The algebras $A$ and $B$ are linked quaternion algebras over $F$ if there is $x \in F$ such that $A$ is equivalent to $(x,y)_F$ and $B$ is equivalent to $(x,z)_F$.

The Albert form for $A,B$ is $$ q = \left\langle{ -a_1, -a_2, a_1a_2, b_1, b_2, -b_1b_2 }\right\rangle \ . $$ It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of $A$ and $B$. The quaternion algebras are linked if and only if the Albert form is isotropic.

The field $F$ is linked if any two quaternion algebras over $F$ are linked. Every global and local field is linked since all quadratic forms of degree 6 over such fields are isotropic.

The following properties of F are equivalent:

$F$ is linked.
Any two quaternion algebras over F are linked.
Every Albert form (dimension six form of discriminant −1) is isotropic.
The quaternion algebras form a subgroup of the Brauer group of F.
Every dimension five form over F is a Pfister neighbour.
No biquaternion algebra over F is a division algebra.

A nonreal linked field has u-invariant equal to 1,2,4 or 8.

References

Pinch point

A pinch point or cuspidal point is a type of singular point on an algebraic surface. It is one of the three types of ordinary singularity of a surface.

The equation for the surface near a pinch point may be put in the form

\[ f(u,v,w) = u^2 - vw^2 + [4] \, \]

where [4] denotes terms of degree 4 or more.

References

P. Griffiths; J. Harris; Principles of Algebraic Geometry, ser. Wiley Classics Library (1994), p. 617, Wiley Interscience ISBN: 0-471-05059-8

Quaternionic structure

Q-structure

An axiomatic system that abstracts the concept of a quaternion algebra over a field.

A quaternionic structure is a triple $(G,Q,q)$ where $G$ is an elementary abelian group of exponent 2 with a distinguished element $−1$, $Q$ is a pointed set with distinguished element $1$, and $q$ is a symmetric surjection from $G \times G \rightarrow Q$ satisfying axioms

$q(a,(-1)a) = 1$;
$q(a,b) = q(a,c) \Leftrightarrow q(q,bc) = 1$;
$q(a,b) = q(c,d) \Rightarrow \exists x\in Q\,:\, q(a,b) = q(a,x),\, q(c,d) = q(c,x)$.

Every field $F$ supports a quaternionic structure by taking $G$ to be $F^*/(F^*)^2$, $Q$ the set of Brauer classes of quaternion algebras in the Brauer group of $F$ with the split quaternion algebra as distinguished element and $q(a,b)$ the quaternion algebra $(a,b)_F$.

References

Tsit-Yuen Lam, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67 , American Mathematical Society (2005) ISBN 0-8218-1095-2 Zbl 1068.11023 MR2104929

Siegel identity

One of two formulae that are used in the resolution of Diophantine equations.

The first formula is $$ \frac{x_3 - x_1}{x_2 - x_1} + \frac{x_2 - x_3}{x_2 - x_1} = 1 \ . $$ The second is $$ \frac{x_3 - x_1}{x_2 - x_1} \cdot\frac{t - x_2}{t - x_3} + \frac{x_2 - x_3}{x_2 - x_1} \cdot \frac{t - x_1}{t - x_3} = 1 \ . $$

The identities are used in translating Diophantine problems connected with integral points on hyperelliptic curves into S-unit equations.

References

Splicing rule

A transformation on formal languages which formalises the action of gene splicing in molecular biology. A splicing language is a language generated by iterated application of a splicing rule: the splicing languages form a proper subset of the regular languages.

Let $A$ be an alphabet and $L$ a language over $A$, that is, a subset of the free monoid $A^*$. A splicing rule is a quadruple $r = (a,b,c,d)$ of elements of $A^*$, and the action of the rule $r$ on $L$ is to produce the language $$ r(L) = \{ xady : xabq, pcdy \in L \} \ . $$ If $R$ is a set of rules then $R(L)$ is the union of the languages produced by the rules of $R$. We say that $R$ respects $L$ if $R(L)$ is a subset of $L$. The $R$-closure of $L$ is the union of $L$ and all iterates of $R$ on$L$: clearly it is respected by $R$. A splicing language is an $R$-closure of a finite language.

A rule set $R$ is reflexive if $(a,b,c,d)$ in $R$ implies that $(a,b,a,b)$ and $(c,d,c,d)$ are in $R$. A splicing language is reflexive if it is defined by a reflexive rule set.

Let $A = \{a,b,c\}$. The rule $(caba,a,cab,a)$ applied to the finite set $\{cabb,cabab,cabaab\}$ generates the regular language $caba^*b$.

Properties

All splicing languages are regular.
Not all regular languages are splicing. An example is $(aa)^*$ over $\{a,b\}$.
If $L$ is a regular language on the alphabet $A$, and $z$ is a letter not in $A$, then the language $\{ zw : w \in L \}$ is a splicing language.
There is an algorithm to determine whether a given regular language is a reflexive splicing language.
The set of splicing rules that respect a regular language can be determined from the syntactic monoid of the language.

References

James A. Anderson, "Automata theory with modern applications" (with contributions by Tom Head) Cambridge University Press (2006) ISBN 0-521-61324-8 Zbl 1127.68049

How to Cite This Entry:
Richard Pinch/sandbox-WP. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-WP&oldid=30234

@@ Line 1: / Line 1: @@
-=Essential subgroup=
+=Analytic subgroup theorem=
+A significant result in modern [[transcendence theory]], a generalisation of [[Baker's theorem]] on linear forms in logarithms.
-A [[subgroup]] that determines much of the structure of its containing group.  The concept may be generalized to [[essential submodule]]s.
+Let $G$ be a [[Abelian group|commutative]] [[algebraic group]] defined over a [[number field]] $K$ and let $B$ be a [[subgroup]] of the complex points $G(\mathbb{C})$ defined over $K$.  There are points of $B$ defined over the field of algebraic numbers if and only if there is a non-trivial [[analytic subgroup]] $H$ of $G$' defined over a number field such that $H(\mathbb{C})$ is contained in $B$.
+==References==
+* Alan Baker and Gisbert Wüstholz, ''Logarithmic Forms and Diophantine Geometry'', New Mathematical Monographs '''9''', Cambridge University Press (2007) ISBN 978-0-521-88268-2.  Chapter 6, pp.109-146.
+=Conductor of an abelian variety=
+In [[mathematics]], in [[Diophantine geometry]], the '''conductor of an abelian variety''' defined over a [[local field|local]] or [[global field]] ''F'' is a measure of how "bad" the [[bad reduction]] at some prime is.  It is connected to the [[ramification]] in the field generated by the [[division point]]s.
+For an [[abelian variety]] ''A'' defined over a field ''F'' with ring of integers ''R'', consider the [[Néron model]] of ''A'', which is a 'best possible' model of ''A'' defined over ''R''. This model may be represented as a [[scheme (mathematics)|scheme]] over
+:Spec(''R'')
+(cf. [[spectrum of a ring]]) for which the [[generic fibre]] constructed by means of the morphism
+:Spec(''F'') &rarr; Spec(''R'')
+gives back ''A''.  Let ''A''<sup>0</sup> denote the open subgroup scheme of the Néron model whose fibres are the connected components.  For a [[residue field]] ''k'', ''A''<sup>0</sup><sub>''k''</sub> is a group variety over ''k'', hence an extension of an abelian variety by a linear group.  This linear group is an extension of a torus by a [[unipotent group]].  Let ''u'' be the dimension of the unipotent group and ''t'' the dimension of the torus.  The order of the conductor is
+:<math> f = 2u + t + \delta , \, </math>
+where &delta; is a measure of wild ramification.
+==Properties==
+* If ''A'' has [[good reduction]] then ''f'' = ''u'' = ''t'' = &delta; = 0.
+* If ''A'' has [[semistable abelian variety|semistable reduction]] or, more generally, acquires semistable reduction over a Galois extension of ''F'' of degree prime to ''p'', the residue characteristic, then &delta; = 0.
+* If ''p'' &gt; 2''d'' + 1, where ''d'' is the dimension of ''A'', then &delta; = 0.
+==References==
+* {{cite book | author=S. Lang | authorlink=Serge Lang | title=Survey of Diophantine geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | pages=70&ndash;71 }}
+* {{cite journal | author=J.-P. Serre | coauthors=J. Tate | title=Good reduction of Abelian varieties | journal=Ann. Math. | volume=88 | year=1968 | pages=492&ndash;517 | doi=10.2307/1970722 | issue=3 | publisher=The Annals of Mathematics, Vol. 88, No. 3 | jstor=1970722 }}
+=Cubic reciprocity=
+Various results connecting the solvability of two related [[cubic equation]]s in [[modular arithmetic]].
+The law of cubic reciprocity is most naturally expressed in terms of the [[Eisenstein integer]]s, that is, the ring $E$ of [[complex number]]s of the form
+$$
+\alpha = a + b\,\omega
+$$
+where and $a$ and $b$ are [[integer]]s and $\omega = \frac{1}{2}(-1 + i\sqrt 3) = e^{2\pi i/3}$ is a complex [[cube root of unity]].  The field norm $\mathbf{N}(\alpha) = \alpha \bar\alpha$.
+If $\pi$ is a [[Eisenstein prime|prime]] element of  $E$ of norm $P$ and $\alpha$ is an element coprime to $\pi$, we define the [[cubic residue]] symbol $\left(\frac{\alpha}{\pi}\right)_3$ to be the cube root of unity (power of $\omega$) satisfying
+$$
+\alpha^{(P-1)/3} \equiv \left(\frac{\alpha}{\pi}\right)_3 \pmod \pi
+$$
+We further define a ''primary'' prime to be one which is congruent to $-1 \pmod 3$.  Then for distinct primary primes $\pi$ and $\theta$ the law of cubic reciprocity is simply
+$$
+\left(\frac{\pi}{\theta}\right)_3 = \left(\frac{\theta}{\pi}\right)_3
+$$
+with the supplementary laws for the units and for the prime $1-\omega$ of norm 3 that
+if $\pi = -1 + 3(m+n\omega)$ then
+$$
+\left(\frac{\omega}{\pi}\right)_3 = \omega^{m+n}\,,
+$$
+$$
+\left(\frac{1-\omega}{\pi}\right)_3 = \omega^{2m} \ .
+$$
+==References==
+* David A. Cox, ''Primes of the form $x^2+ny^2$'', Wiley, 1989, ISBN 0-471-50654-0.
+* K. Ireland and M. Rosen, ''A classical introduction to modern number theory'', 2nd ed, Graduate Texts in Mathematics '''84''', Springer-Verlag, 1990.
+* Franz Lemmermeyer, ''Reciprocity laws: From Euler to Eisenstein'', Springer Verlag, 2000, ISBN 3-540-66957-4.
+=Genus field=
+In [[algebraic number theory]], the '''genus field''' $G$ of a [[number field]] $K$ is the [[maximal]] [[abelian extension|abelian]] extension of $K$ which is obtained by composing an absolutely abelian field with $K$ and which is  [[unramified]] at all finite primes of $K$.  The '''genus number''' of $K$ is the degree $[G:K]$ and the '''genus group''' is the [[Galois group]] of $G$ over $K$.
+If $K$ is itself absolutely abelian, the genus field may be described as the maximal absolutely abelian extension of $K$ unramified at all finite primes: this definition was used by Leopoldt and Hasse.
+If $K = \mathbb{Q}(\sqrt m)$ ($m$ squarefree) is a quadratic field of discriminant $D$, the genus field of $K$ is a composite of quadratic fields.  Let $p_i$ run over the prime factors of  $D$.  For each such prime $p$, define $p^*$ as follows:
+$$p^* = \pm p \equiv 1 \pmod 4 \text{ if } p \text{ is odd} ; $$
+$$2^* = -4, 8, -8 \text{ according as } m \equiv 3 \pmod 4, 2 \pmod 8, -2 \pmod 8 . $$
+Then the genus field is the composite $K(\sqrt{p^*_i})$.
+==See also==
+* [[Hilbert class field]]
+==References==
+* {{cite book | zbl=0353.12001 | last=Ishida | first=Makoto | title=The genus fields of algebraic number fields | series=Lecture Notes in Mathematics | volume=555 | publisher=[[Springer-Verlag]] | year=1976 | isbn=3-540-08000-7 }}
+=Group isomorphism problem=
+In [[abstract algebra]], the '''group isomorphism problem''' is the [[decision problem]] of determining whether two [[Presentation of a group|group presentation]]s present [[Isomorphism|isomorphic]] [[Group (mathematics)|group]]s.
+The isomorphism problem was identified by [[Max Dehn]] in 1911 as one of three fundamental decision problems in group theory; the other two being the [[Word problem for groups|word problem]] and the [[conjugacy problem]].
+==References==
+* {{cite book
+  | last = Magnus
+  | first = Wilhelm
+  | authorlink = Wilhelm Magnus
+  | coauthors = Abraham Karrass, Donald Solitar
+  | title = Combinatorial group theory. Presentations of groups in terms of generators and relations
+  | publisher = [[Dover Publications]]
+  | date = 1976
+  | location =
+  | pages = 24
+  | url =
+  | doi =
+  | id =
+  | isbn = 0-486-63281-4}}
+* {{cite book
+  | last = Johnson
+  | first = D.L.
+  | authorlink =
+  | coauthors =
+  | title = Presentations of groups
+  | publisher = [[Cambridge University Press]]
+  | date = 1990
+  | location =
+  | pages = 49
+  | url =
+  | doi =
+  | id =
+  | isbn = 0-521-37203-8}}
+* {{cite journal
+  | last = Dehn
+  | first = Max
+  | authorlink = Max Dehn
+  | coauthors =
+  | title = Über unendliche diskontinuierliche Gruppen
+  | journal = [[Mathematische Annalen|Math. Ann.]]
+  | volume = 71
+  | issue =
+  | pages = 116-144
+  | publisher =
+  | location =
+  | date = 1911
+  | url =
+  | doi = 10.1007/BF01456932
+  | id =
+  | accessdate = }}
+=Hall algebra=
+The '''Hall polynomials''' in [[mathematics]] were developed by [[Philip Hall]] in the 1950s in the study of [[group representation]]s.
+A [[finite set|finite]] [[abelian group|abelian]] [[p-group|''p''-group]] ''M'' is a direct sum of [[cyclic group|cyclic]] ''p''-power components
+<math>C_{p^\lambda_i}</math> where
+<math>\lambda=(\lambda_1,\lambda_2,\ldots)</math> is a [[Partition_%28number_theory%29|partition]] of <math>n</math> called the ''type'' of ''M''.  Let <math>g^\lambda_{\mu,\nu}(p)</math> be the number of
+subgroups ''N'' of ''M'' such that ''N'' has type <math>\nu</math> and the quotient ''M/N'' has type <math>\mu</math>.  Hall showed that the functions ''g'' are [[polynomial]] functions of ''p'' with integer coefficients: these are the ''Hall polynomials''.
+Hall next constructs an [[Algebra over a field|algebra]] <math>H(p)</math> with symbols <math>u_\lambda</math> a generators and multiplication given by the <math>g^\lambda_{\mu,\nu}</math> as structure constants
+:<math> u_\mu u_\nu = \sum_\lambda g^\lambda_{\mu,\nu} u_\lambda </math>
+which is freely generated by the <math>u_{\mathbf1_n}</math> corresponding to the elementary ''p''-groups.
+The map from <math>H(p)</math> to the algebra of [[symmetric function]]s <math>e_n</math> given
+by <math>u_{\mathbf 1_n} \mapsto p^{-n(n-1)}e_n</math> is a homomorphism and its image may be
+interpreted as the [[Hall-Littlewood polynomial]] functions.  The theory of [[Schur function]]s
+is thus closely connected with the theory of Hall polynomials.
+==References==
+* I.G. Macdonald, ''Symmetric functions and Hall polynomials'', (Oxford University Press, 1979) ISBN 0-19-853530-9
+=Hall–Littlewood polynomials=
+In [[mathematics]], the '''Hall–Littlewood polynomials''' encode combinatorial data relating to the [[Group representation|representation]]s of the [[general linear group]].  They are named for [[Philip Hall]] and [[Dudley E. Littlewood]].
+==See also==
+* [[Littlewood polynomial]]
+==References==
+* {{User:Richard Pinch/sandbox/Ref  | author=I.G. Macdonald | authorlink=Ian G. Macdonald | title=Symmetric Functions and Hall Polynomials | publisher=Oxford University Press | pages=101-104 | year=1979 | isbn=0-19-853530-9 }}
+* {{User:Richard Pinch/sandbox/Ref  | author=D.E. Littlewood | title=On certain symmetric functions | journal=Proc. London Math. Soc. | volume=43 | year=1961 | pages=485–498 | doi=10.1112/plms/s3-11.1.485 }}
+=Hutchinson operator=
+In [[mathematics]], in the study of [[fractal]]s, a '''Hutchinson operator''' is a collection of functions on an underlying space ''E''.  The [[Iterated function|iteration]] on these functions gives rise to an [[iterated function system]], for which the fixed set is [[self-similar]].
 ==Definition==
-A [[subgroup]] <math>S</math> of a (typically [[abelian group|abelian]]) [[group]] <math>G</math> is said to be '''essential''' if whenever ''H'' is a non-trivial subgroup of ''G'', the intersection of ''S'' and ''H'' is non-trivial: here "non-trivial" means "containing an element other than the identity".
+Formally, let ''f''<sub>''i''</sub> be a finite set of ''N'' functions from a set ''X'' to itself.  We may regard this as defining an operator ''H'' on the power set '''P''' ''X'' as
+:<math>H : A \mapsto \bigcup_{i=1}^N f_i[A],\,</math>
+where ''A'' is any subset of ''X''.
+A key question in the theory is to describe the fixed sets of the operator ''H''.  One way of constructing such a fixed set is to start with an initial point or set ''S''<sub>0</sub> and iterate the actions of the ''f''<sub>''i''</sub>, taking ''S''<sub>''n''+1</sub> to be the union of the images of ''S''<sub>n</sub> under the operator ''H''; then taking ''S'' to be the union of the ''S''<sub>''n''</sub>, that is,
+:<math>S_{n+1} = \bigcup_{i=1}^N f_i[S_n] </math>
+and
+:<math>S = \bigcup_{n=0}^\infty S_n . </math>
+==Properties==
+Hutchinson (1981) considered the case when the ''f''<sub>''i''</sub> are [[contraction mapping]]s on a Euclidean space ''X'' = '''R'''<sup>d</sup>.  He showed that such a system of functions has a unique [[Compact space|compact]] (closed and bounded) fixed set ''S''.
+The collection of functions <math>f_i</math> together with composition form a [[monoid]]. With ''N'' functions, then one may visualize the monoid as a full [[k-ary tree|N-ary tree]] or a [[Cayley tree]].
+==References==
+* {{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 }}
+* {{cite book | title=Chaos and Fractals: New Frontiers of Science | author=Heinz-Otto Peitgen | authorlink=Heinz-Otto Peitgen | coauthors=Hartmut Jürgens, Dietmar Saupe | year=2004 | puclisher=[[Springer-Verlag]] | isbn=0387202293 | pages=84,225 }}
+=Kneser theorem=
+A statement about [[Sumset|set addition]] in [[finite group]]s.  It may be regarded as an extension of the [[Cauchy-Davenport theorem]] on sumsets in groups of prime order.
+Let $G$ be a non-trivial [[abelian group]] and $A,B$ finite non-empty subsets.  If $|A| + |B| \le |G|$ then there is a finite subgroup $H$ of $G$ such that
+$$
+ |A+B| \ge |A+H| + |B+H| - |H| \ge |A| + |B| - |H| \ .
+$$
+The subgroup $H$ can be taken to be the ''stabiliser'' of $A+B$
+$$
+H = \lbrace  g \in G : g + (A+B) = (A+B) \rbrace \ .
+$$
+==References==
+* {{User:Richard Pinch/sandbox/Ref | editor1-last=Geroldinger | editor1-first=Alfred | editor2-last=Ruzsa | editor2-first=Imre Z. | others=Elsholtz, C.; Freiman, G.; Hamidoune, Y. O.; Hegyvári, N.; Károlyi, G.; Nathanson, M.; Solymosi, J.; Stanchescu, Y. With a foreword by Javier Cilleruelo, Marc Noy and Oriol Serra (Coordinators of the DocCourse) | title=Combinatorial number theory and additive group theory | series=Advanced Courses in Mathematics CRM Barcelona | location=Basel | publisher=Birkhäuser | year=2009 | isbn=978-3-7643-8961-1 | zbl=1177.11005 }}
+* {{User:Richard Pinch/sandbox/Ref  | first=Martin | last=Kneser | title=Abschätzungen der asymptotischen Dichte von Summenmengen | language=German | journal=Math. Zeitschr. | volume=58 | year=1953 | pages=459–484 | zbl=0051.28104 }}
+* {{User:Richard Pinch/sandbox/Ref  | first=Melvyn B. | last=Nathanson | title=Additive Number Theory: Inverse Problems and the Geometry of Sumsets | volume=165 | series=Graduate Texts in Mathematics | publisher=Springer-Verlag | year=1996 | isbn=0-387-94655-1 | zbl=0859.11003 | pages=109–132 }}
+* {{User:Richard Pinch/sandbox/Ref  | first1=Terence | last1=Tao | author1-link=Terence Tao | first2=Van H. | last2=Vu | title=Additive Combinatorics | year=2010 | publisher=Cambridge University Press | isbn=978-0-521-13656-3 | zbl=1179.11002 }}
+=Linked field=
+A [[field]] for which the [[quadratic form]]s attached to [[quaternion algebra]]s have a common property.
+Let $F$ be a field of [[Characteristic (algebra)|characteristic]] not equal to 2.  Let $A = (a_1,a_2)_F$, $B = (b_1,b_2)_F$ be quaternion algebras over ''F''.  The algebras $A$ and $B$ are '''linked quaternion algebras''' over $F$ if there is $x \in F$ such that $A$ is equivalent to $(x,y)_F$ and $B$ is equivalent to $(x,z)_F$.
+The '''Albert form''' for $A,B$ is
+$$
+q = \left\langle{ -a_1, -a_2, a_1a_2, b_1, b_2, -b_1b_2 }\right\rangle \ .
+$$
+It can be regarded as the difference in the [[Witt ring]] of the ternary forms attached to the imaginary subspaces of $A$ and $B$.  The quaternion algebras are linked if and only if the Albert form is [[isotropic quadratic form|isotropic]].
+The field $F$ is ''linked'' if any two quaternion algebras over $F$ are linked.  Every [[global field|global]] and [[local field]] is linked since all quadratic forms of degree 6 over such fields are isotropic.
+The following properties of ''F'' are equivalent:
+* $F$ is linked.
+* Any two quaternion algebras over ''F'' are linked.
+* Every ''Albert form'' (dimension six form of discriminant −1) is isotropic.
+* The quaternion algebras form a subgroup of the [[Brauer group]] of ''F''.
+* Every dimension five form over ''F'' is a [[Pfister neighbour]].
+* No [[biquaternion algebra]] over ''F'' is a [[division algebra]].
+A nonreal linked field has [[u-invariant]] equal to 1,2,4 or 8.
 ==References==
-* {{cite book | author=Phillip A. Griffith | title=Infinite Abelian group theory | series=Chicago Lectures in Mathematics | publisher=University of Chicago Press | year=1970 | isbn=0-226-30870-7 | page=19}}
+*{{cite book | last=Knus | first=Max-Albert | title=Quadratic and Hermitian forms over rings | series=Grundlehren der Mathematischen Wissenschaften | volume=294 | location=Berlin etc. | publisher=[[Springer-Verlag]] | year=1991 | isbn=3-540-52117-8 | zbl=0756.11008 | page=192 }}
+* {{cite book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | first=Tsit-Yuen | last=Lam | author-link=T. Y. Lam | publisher=[[American Mathematical Society]] | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 | mr = 2104929 }}
+* {{cite journal | last=Gentile | first=Enzo R. | title=On linked fields | journal=Rev. Unión Mat. Argent. | volume=35 | year=1990 | pages=67–81 | year=1989 | issn=0041-6932 | zbl=0823.11010 }}
 =Pinch point=
 A '''pinch point''' or '''cuspidal point''' is a type of [[Singular point of an algebraic variety|singular point]] on an [[algebraic surface]].  It is one of the three types of ordinary singularity of a surface.
@@ Line 18: / Line 252: @@
 ==References==
-* {{cite book | author=P. Griffiths | authorlink=Phillip Griffiths | coauthors=[[Joe Harris (mathematician)|J. Harris]] | title=Principles of Algebraic Geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | page=617 }}
+* {{User:Richard Pinch/sandbox/Ref | author=P. Griffiths | authorlink=Phillip Griffiths | coauthors=[[Joe Harris (mathematician)|J. Harris]] | title=Principles of Algebraic Geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | page=617 }}
-=Residual property=
+=Quaternionic structure=
+''Q-structure''
-In the [[mathematics|mathematical]] field of [[group theory]], a group is '''residually ''X''''' (where ''X'' is some property of groups) if it "can be recovered from groups with property ''X''".
+An axiomatic system that abstracts the concept of a [[quaternion algebra]] over a [[field]].
-Formally, a group ''G'' is residually ''X'' if for every non-trivial element ''g'' there is a [[Group homomorphism|homomorphism]] ''h'' from ''G'' to a group with property ''X'' such that <math>h(g)\neq e</math>.
+A ''quaternionic structure'' is a triple $(G,Q,q)$ where $G$ is an [[elementary abelian group]] of [[Exponent of a group|exponent]] 2 with a distinguished element $−1$, $Q$ is a [[pointed set]] with distinguished element $1$, and $q$ is a symmetric [[surjection]] from $G \times G \rightarrow Q$ satisfying axioms
+# $q(a,(-1)a) = 1$;
+# $q(a,b) = q(a,c) \Leftrightarrow q(q,bc) = 1$;
+# $q(a,b) = q(c,d) \Rightarrow \exists x\in Q\,:\, q(a,b) = q(a,x),\, q(c,d) = q(c,x)$.
-More categorically, a group is residually ''X'' if it embeds into its pro-''X'' completion (see [[profinite group]], [[pro-p group]]), that is, the [[inverse limit]] of <math>\phi\colon G \to H</math> where ''H'' is a group with property ''X''.
+Every field $F$ supports a  quaternionic structure by taking $G$ to be $F^*/(F^*)^2$, $Q$ the set of [[Brauer class]]es of quaternion algebras in the [[Brauer group]] of $F$ with the [[split quaternion algebra]] as distinguished element and $q(a,b)$ the quaternion algebra $(a,b)_F$.
-==Examples==
+==References==
-Important examples include:
+* Tsit-Yuen Lam, ''Introduction to Quadratic Forms over Fields'', Graduate Studies in Mathematics '''67''' , American Mathematical Society (2005) ISBN 0-8218-1095-2 {{ZBL|1068.11023}} {{MR|2104929}}
-* [[Residually finite group|Residually finite]]
-* Residually [[nilpotent group|nilpotent]]
+=Siegel identity=
-* Residually [[solvable group|solvable]]
+One of two formulae that are used in the resolution of [[Diophantine equation]]s.
-* Residually [[free group|free]]
+The first formula is
+$$
+\frac{x_3 - x_1}{x_2 - x_1} + \frac{x_2 - x_3}{x_2 - x_1} = 1 \ .
+$$
+The second is
+$$
+\frac{x_3 - x_1}{x_2 - x_1} \cdot\frac{t - x_2}{t - x_3} + \frac{x_2 - x_3}{x_2 - x_1} \cdot \frac{t - x_1}{t - x_3} = 1 \ .
+$$
+The identities are used in translating Diophantine problems connected with integral points on [[hyperelliptic curve]]s into [[S-unit equation]]s.
+See also: [[Siegel formula]].
 ==References==
-* {{cite book | title=The theory of groups | author=Marshall Hall Jr | authorlink=Marshall Hall (mathematician) | location=New York | publisher=Macmillan | year=1959 | page=16 }}
+* {{cite book | first=Alan | last=Baker | authorlink=Alan Baker (mathematician) | title=Transcendental Number Theory | publisher=[[Cambridge University Press]] | year=1975 | isbn=0-521-20461-5 | zbl=0297.10013 | page=40 }}
+* {{cite book | first1=Alan | last1=Baker | authorlink1=Alan Baker (mathematician)| first2=Gisbert | last2= Wüstholz | authorlink2=Gisbert Wüstholz | title=Logarithmic Forms and Diophantine Geometry | series=New Mathematical Monographs | volume=9 | publisher=[[Cambridge University Press]] | year=2007 | isbn=978-0-521-88268-2 | zbl=1145.11004 | page=53 }}
+* {{cite book | first1=Daniel S. | last1=Kubert | authorlink1=Daniel Kubert | first2=Serge | last2=Lang | authorlink2=Serge Lang | title=Modular Units | series= Grundlehren der Mathematischen Wissenschaften | volume=244 | year=1981 | isbn=0-387-90517-0 }}
+* {{cite book | first=Serge | last=Lang | authorlink=Serge Lang | title=Elliptic Curves: Diophantine Analysis | volume=231 | series=Grundlehren der mathematischen Wissenschaften | publisher=[[Springer-Verlag]] | year=1978 | isbn=0-387-08489-4 }}
+* {{cite book | title=The Algorithmic Resolution of Diophantine Equations | volume=41 | series=London Mathematical Society Student Texts | first=N. P. | last=Smart | authorlink=Nigel Smart (cryptographer) | publisher=[[Cambridge University Press]] | year=1998 | isbn=0-521-64633-2 | pages=36–37 }}
-=Stably free module=
-A [[module]] which is close to being [[free module|free]].
+=Splicing rule=
+A transformation on [[formal language]]s which formalises the action of gene splicing in molecular biology.  A '''splicing language''' is a language generated by iterated application of a splicing rule: the splicing languages form a proper subset of the [[regular language]]s.
-==Definition==
+Let $A$ be an alphabet and $L$ a language over $A$, that is, a subset of the [[free monoid]] $A^*$.  A '''splicing rule''' is a quadruple $r = (a,b,c,d)$ of elements of $A^*$, and the action of the rule $r$ on $L$ is to produce the language
-A module ''M'' over a ring ''R'' is ''stably free'' if there exist free modules ''F'' and ''G'' over ''R'' such that
+$$
+r(L) = \{ xady : xabq, pcdy \in L \} \ .
+$$
+If $R$ is a set of rules then $R(L)$ is the union of the languages produced by the rules of $R$.  We say that $R$ ''respects'' $L$ if $R(L)$ is a subset of $L$.  The $R$-closure of $L$ is the union of $L$ and all iterates of $R$ on$L$: clearly it is respected by $R$.  A '''splicing language''' is an $R$-closure of a finite language.
-:<math> M \oplus F = G . \, </math>
+A rule set $R$ is '''reflexive''' if $(a,b,c,d)$ in $R$ implies that $(a,b,a,b)$ and $(c,d,c,d)$ are in $R$.  A splicing language is reflexive if it is defined by a reflexive rule set.
-==Properties==
+Let $A = \{a,b,c\}$.  The rule $(caba,a,cab,a)$ applied to the finite set $\{cabb,cabab,cabaab\}$ generates the regular language $caba^*b$.
-* A projective module is stably free if and only if it possesses a finite [[free resolution]].
-== See also ==
+====Properties====
-* [[Free object]]
+* All splicing languages are regular.
+* Not all regular languages are splicing.  An example is $(aa)^*$ over $\{a,b\}$.
+* If $L$ is a regular language on the alphabet $A$, and $z$ is a letter not in $A$, then the language $\{ zw : w \in L \}$ is a splicing language.
+* There is an algorithm to determine whether a given regular language is a reflexive splicing language.
+* The set of splicing rules that respect a regular language can be determined from the [[syntactic monoid]] of the language.
-==References==
+==== References ====
-* {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed. | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 | page=840}}
+* James A. Anderson, "Automata theory with modern applications" (with contributions by Tom Head) Cambridge University Press (2006) ISBN 0-521-61324-8 {{ZBL|1127.68049}}

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Difference between revisions of "User:Richard Pinch/sandbox-WP"

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Analytic subgroup theorem

References

Conductor of an abelian variety

Properties

References

Cubic reciprocity

References

Genus field

See also

References

Group isomorphism problem

References

Hall algebra

References

Hall–Littlewood polynomials

See also

References

Hutchinson operator

Definition

Properties

References

Kneser theorem

References

Linked field

References

Pinch point

References

Quaternionic structure

References

Siegel identity

References

Splicing rule

Properties

References