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=Subnormal series=
+
=Radical-inverse function=
==Subnormal series==
+
A function which maps positive integers to real numbers in the unit interval by reversing the expansion in a given base $b$.  Let $n$ have the base $b$ expansion
''of a group $G$''
+
$$
 
+
n = \sum_{i=0}^k d_i b^i
A [[subgroup series]] of $G$,
+
$$
 +
where $0 \le d_i < b$ and $d_k \ne 0$; then the radical inverse
 
$$
 
$$
E = G_0 \le G_1 \le \cdots \le G_n = G
+
\phi_b(n) = \sum_{i=0}^k d_i b^{-1-i} \ .
 
$$
 
$$
where each subgroup $G_i$ is a normal subgroup of $G_{i+1}$. The quotient groups $G_{i+1}/G_i$ are called ''factors'', and the number $n$ is called the length of the subnormal series. Infinite subnormal series have also been studied (see [[Subgroup system]]). A subnormal series that cannot be refined further is called a ''[[composition series]]'', and its factors are called ''composition factors''.
 
  
====Comments====
+
The radical inverse function is used to construct sequences with low [[discrepancy]]: these are of use in quasi-Monte-Carlo methods.
A subnormal series is also called a subinvariant series.
 
  
====References====
+
The '''van der Corput sequence''' is the sequence $\left({\phi_2(n)}\right)$.  The discrepancy $D_N$ of this sequence satisfies
<table>
+
$$
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Hall jr.,  "The theory of groups" , Macmillan  (1959)  pp. Sect. 8.4</TD></TR>
+
N D_N \le \frac{\log(N+1)}{\log 2} \ .
</table>
+
$$
 
 
==Subnormal subgroup==
 
''attainable subgroup''
 
 
 
Any member of any [[subnormal series]] of a group. To indicate the subnormality of a subgroup $H$ in a group $G$, the notation $H \lhd\!\lhd G$ is used.
 
  
 
====References====
 
====References====
<table>
+
* L. KuipersH. Niederreiter,  "Uniform distribution of sequences" , Wiley (1974) {{ZBL|0281.10001}}; repr. Dover (2006) ISBN 0-486-45019-8
<TR><TD valign="top">[1]</TD> <TD valign="top">  M.I. KargapolovJ.I. [Yu.I. Merzlyakov] Merzljakov,  "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)</TD></TR>
 
</table>
 
  
====Comments====
+
=Dispersion=
A subnormal subgroup is also called a subinvariant subgroup.
+
The dispersion of a sequence $x_n$ in a [[metric space]] $(X,d)$ is defined as the function
 +
$$
 +
D_N = \sup_{x \in X} \min_{n=1,\ldots,N} d(x,x_n) \ .
 +
$$
 +
The associated ''dispersion constant'' is $\limsup_{N\rightarrow\infty} N d_N$.
  
A subnormal subgroup of $G$ that coincides with its commutator subgroup and whose quotient by its centre is simple is called a component of $G$. The product of all components of $G$ is known as the layer of $G$. It is an important [[characteristic subgroup]] of $G$ in the theory of finite simple groups, see e.g. [[#References|[a1]]].
+
If a sequence has low [[discrepancy]] then its dispersion must also be low, but the converse does not hold.
  
====References====
+
Sequences with low dispersion play a part in quasi-Monte-Carlo methods for global optimisation problems.
<table>
 
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Suzuki,  "Group theory" , '''1–2''' , Springer  (1986)</TD></TR>
 
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J.C. Lennox,  S.E. Stonehewer,  "Subnormal subgroups of groups" , Clarendon Press  (1987)</TD></TR>
 
<TR><TD valign="top">[a3]</TD> <TD valign="top">  D.J.S. Robinson,  "A course in the theory of groups" , Springer  (1982)</TD></TR>
 
</table>
 
 
 
==Attainable subgroup==
 
A subgroup $H$ that can be included in a finite normal series of a group $G$, i.e. in a series
 
 
 
$$\{1\}\subset H=H_0\subset H_1\subset\ldots\subset H_n=G$$
 
 
 
in which each subgroup $H_i$ is a normal subgroup in $H_{i+1}$. The property of a subgroup to be attainable is transitive. An intersection of attainable subgroups is an attainable subgroup. The subgroup generated by two attainable subgroups need not be an attainable subgroup. A group $G$ all subgroups of which are attainable satisfies the normalizer condition, i.e. all subgroups differ from their normalizers (cf. [[Normalizer of a subset|Normalizer of a subset]]). Such a group is therefore locally nilpotent.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,   "The theory of groups" , '''1–2''' , Chelsea  (1955–1956)  (Translated from Russian)</TD></TR></table>
+
* Harald Niederreiter, "On a measure of denseness for sequences", ''Topics in classical number theory, Colloq. Budapest 1981, Vol. II'', Colloq. Math. Soc. János Bolyai '''34''' (1984) 1163-1208 {{ZBL|0547.10045}}
 
 
 
 
 
 
====Comments====
 
Instead of attainable subgroup, the term accessible subgroup is used in [[#References|[1]]]. In the Western literature the term subnormal subgroup is standard for this kind of subgroup.
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Suzuki,  "Group theory" , '''2''' , Springer  (1986)</TD></TR></table>
 
 
 
 
 
=Regular graph=
 
An unoriented [[graph]] in which each vertex has the same degree.
 
 
 
A '''strongly regular graph''' is a regular graph in which any twp adjacent vertices have the same number of neighbours in common, and any two non-adjacent vertices have the same number of neighbours in common. The [[Graph complement|complement]] of a strongly regular graph is again strongly regular.
 
 
 
A '''distance regular graph''' is one with the property that for any two vertices $x,y$ the number of vertices at distance $i$ from $x$ and $j$ from $y$ depends only on $i$, $j$ and the distance $d(x,y)$.
 
 
 
====References====
 
* Richard A Brualdi, Herbert J. Ryser, "Combinatorial matrix theory", Cambridge University Press (2014) ISBN 978-0-521-32265-2  {{ZBL|0746.05002}} {{ZBL|1286.05001}}
 
* Andries E. Brouwer, Arjeh M. Cohen, Arnold Neumaier, "Distance-regular graphs" Springer (1989) ISBN 3-642-74343-6 {{ZBL|0747.05073}}
 
  
 
=Dyck path=
 
=Dyck path=
Line 152: Line 116:
 
====References====
 
====References====
 
* J. G. Oxley, "Matroid Theory" (2 ed) Oxford University Press (2011) ISBN 978-0-19-856694-6  {{ZBL|1254.05002}}
 
* J. G. Oxley, "Matroid Theory" (2 ed) Oxford University Press (2011) ISBN 978-0-19-856694-6  {{ZBL|1254.05002}}
* D. J. A. Welsh, "Matroid Theory", Dover (2010) [1976] ISBN 0486474399 {{ZBL|}}0343.05002
+
* D. J. A. Welsh, "Matroid Theory", Dover (2010) [1976] ISBN 0486474399 {{ZBL|0343.05002}}
  
 
=Ordered topological space=
 
=Ordered topological space=
Line 164: Line 128:
 
* Samuel Eilenberg, "Ordered Topological Spaces", ''American Journal of Mathematics'' '''63''' (1941) 39-45  {{DOI|10.2307/2371274}}  {{ZBL|0024.19203}}
 
* Samuel Eilenberg, "Ordered Topological Spaces", ''American Journal of Mathematics'' '''63''' (1941) 39-45  {{DOI|10.2307/2371274}}  {{ZBL|0024.19203}}
 
* T.S. Blyth, "Lattices and ordered algebraic structures", Springer (2005) ISBN 1-85233-905-5  {{ZBL|1073.06001}}
 
* T.S. Blyth, "Lattices and ordered algebraic structures", Springer (2005) ISBN 1-85233-905-5  {{ZBL|1073.06001}}
 
=Reed–Solomon code=
 
A family of codes defined over finite fields.  Let $k = \mathbf{F}_q$ and put $n = q-1$.  Let $\beta$ be a primitive element of $k^*$.  For an integer $k \le n$, let $L_q$ denote the vector space of polynomials over $k$ of degree $\le k-1$, and let $E$ be the evaluation map $e : L_k \rightarrow k^n$ given by
 
$$
 
E : f \mapsto \left({ f(\beta), f(\beta^2), \ldots, f(\beta^n) }\right) \ .
 
$$
 
 
The image $E[L_k]$ is a subspace of $k^n$ and is the ''Reed–Solomon code'' $\mathrm{RS}(q,k)$.
 
 
The map $E$ is injective, as a non-zero polynomial of degree $k<n$ cannot be zero at $n$ distinct values.  The rank of the code is thus $k$.  The minimum weight is $n-k+1$, corresponding to a polynomial with $k-1$ zeroes all in $k$.  The Reed–Solomon code thus meets the [[singleton bound]] and is an [[MDS code]].
 
 
====References====
 
* C.M. Goldie,  R.G.E. Pinch, ''Communication theory'', London Mathematical Society Student Texts. '''20''' Cambridge University Press (1991) ISBN 0-521-40456-8 {{ZBL|0746.94001}}
 
* J.H. van Lint, "Introduction to coding theory" (2nd ed.) Graduate Texts in Mathematics '''86''' Springer (1992) ISBN 3-540-54894-7 {{ZBL|0747.94018}}
 
* H. Stichtenoth,  "Algebraic function fields and codes", Universitext, Springer  (1993)  ISBN 3-540-58469-6 {{ZBL|0816.14011}}
 
 
 
=Singleton bound=
 
A constraint on the parameters of a linear block code.  If a code has length $n$, rank $k$ and minimum distance $d$, then
 
$$
 
k+d \le n+1 \ .
 
$$
 
The bound is obtained by ''puncturing'' a code $C$ by selecting all the words with the symbol $0$ in a specific location and then deleting that symbol from all words.
 
 
A code for which equality holds is a '''maximum distance separable''' or '''MDS code'''.  Examples of MDS codes are the [[Reed–Solomon code]]s, which show that
 
if $n \le q+1$ there are MDS codes over $\mathbf{F}_q$ of rank $k$ for all $k< n$.
 
 
 
====References====
 
* C.M. Goldie,  R.G.E. Pinch, ''Communication theory'', London Mathematical Society Student Texts. '''20''' Cambridge University Press (1991) ISBN 0-521-40456-8 {{ZBL|0746.94001}}
 
* J.H. van Lint, "Introduction to coding theory" (2nd ed.) Graduate Texts in Mathematics '''86''' Springer (1992) ISBN 3-540-54894-7 {{ZBL|0747.94018}}
 
* F.J. MacWilliams, N.J.A. Sloane, "The theory of error-correcting codes. Parts I, II" (3rd repr.) North-Holland Mathematical Library '''16'''  Elsevier (1985) ISBN 0-444-85193-3  {{ZBL|0657.94010}}
 
* H. Stichtenoth,  "Algebraic function fields and codes", Universitext, Springer  (1993)  ISBN 3-540-58469-6 {{ZBL|0816.14011}}
 
 
=BCH code=
 
A [[cyclic code]] over a finite field.  Fix length $n$ and ground field $\mathbf{F}_q$ and a design distance parameter $\delta$.  Let $\beta$ be a primitive $n$-th root of unity in a suitable extension of $\mathbf{F}_q$.  The generator of the cyclic code is the least common multiple $g$ of the minimal polynomials (over $\mathbf{F}_q$) of the elements $\beta^1, \beta^2, \ldots, \beta^{\delta-1}$. 
 
 
The minimum distance of the BCH code generated by $g$ is at least $\delta$: this is the ''BCH bound''.
 
 
As an example, let $q=2$ and $\beta$ be a primitive $7$-th root of unity in $\mathrm{F}_{8}$: we may take $\beta$ to satisfy the polynomial $x^3 + x + 1$.  Choose $\delta = 3$.  The minimal polynomial for $\beta^2$ is the same as that of $\beta$, so that the cyclic code is generated by the word $1101000$.  This is in fact the  Hamming [7,4] code.
 
 
====References====
 
* C.M. Goldie,  R.G.E. Pinch, ''Communication theory'', London Mathematical Society Student Texts. '''20''' Cambridge University Press (1991) ISBN 0-521-40456-8 {{ZBL|0746.94001}}
 
 
=Hamming code=
 
A binary block code capable of error correction: see [[Error-correcting code]].
 
 
The Hamming $[7,4]$ code has generator matrix with first row $1101000$ and the others being its three right shifts.  It is a [[cyclic code]], and indeed a [[BCH code]] with design distance $3$.  It is perfect, because the balls of radius $1$ about the codewords have eight elements and the 16 balls precisely fill out the 7-dimensional space.
 
 
====References====
 
* C.M. Goldie,  R.G.E. Pinch, ''Communication theory'', London Mathematical Society Student Texts. '''20''' Cambridge University Press (1991) ISBN 0-521-40456-8 {{ZBL|0746.94001}}
 
 
 
=Cyclic code=
 
A block code over a finite field for which the code words have cyclic symmetric: any cyclic permutation of a code word is again a code word.  A linear code of length $n$ over the field $k$ may be identified with
 
polynomials of degree $< n$ over $k$ in an indeterminate $X$: the cyclic symmetry condition states that the code is invariant under multiplication by $X$ modulo $X^n-1$, so that the code may be identified with an
 
ideal of the quotient ring $k[X]/\langle X^n-1 \rangle$.  Since the polynomial ring $k[X]$ is a [[principal ideal domain]], the codes are all principal ideals, and hence are determined by their generators, which must be factors of $X^n-1$. 
 
 
 
====References====
 
* C.M. Goldie,  R.G.E. Pinch, ''Communication theory'', London Mathematical Society Student Texts. '''20''' Cambridge University Press (1991) ISBN 0-521-40456-8 {{ZBL|0746.94001}}
 

Latest revision as of 09:27, 19 January 2021

Radical-inverse function

A function which maps positive integers to real numbers in the unit interval by reversing the expansion in a given base $b$. Let $n$ have the base $b$ expansion $$ n = \sum_{i=0}^k d_i b^i $$ where $0 \le d_i < b$ and $d_k \ne 0$; then the radical inverse $$ \phi_b(n) = \sum_{i=0}^k d_i b^{-1-i} \ . $$

The radical inverse function is used to construct sequences with low discrepancy: these are of use in quasi-Monte-Carlo methods.

The van der Corput sequence is the sequence $\left({\phi_2(n)}\right)$. The discrepancy $D_N$ of this sequence satisfies $$ N D_N \le \frac{\log(N+1)}{\log 2} \ . $$

References

  • L. Kuipers, H. Niederreiter, "Uniform distribution of sequences" , Wiley (1974) Zbl 0281.10001; repr. Dover (2006) ISBN 0-486-45019-8

Dispersion

The dispersion of a sequence $x_n$ in a metric space $(X,d)$ is defined as the function $$ D_N = \sup_{x \in X} \min_{n=1,\ldots,N} d(x,x_n) \ . $$ The associated dispersion constant is $\limsup_{N\rightarrow\infty} N d_N$.

If a sequence has low discrepancy then its dispersion must also be low, but the converse does not hold.

Sequences with low dispersion play a part in quasi-Monte-Carlo methods for global optimisation problems.

References

  • Harald Niederreiter, "On a measure of denseness for sequences", Topics in classical number theory, Colloq. Budapest 1981, Vol. II, Colloq. Math. Soc. János Bolyai 34 (1984) 1163-1208 Zbl 0547.10045

Dyck path

A lattice path on the square lattice from the origin $(0,0)$ to some point $(n,n)$ consisting of $2n$ steps of the form $N : (x,y) \rightarrow (x,y+1)$ and $E : (x,y) \rightarrow (x+1,y)$ with the property that the path never passes below the line $y=x$.

The number of Dyck paths of length $2n$ is given by the $n$-th Catalan number $$ C_n = \frac{1}{n+1}\binom{2n}{n} \ . $$

References

Catalan number

The $n$-th Catalan number $$ C_n = \frac{1}{n+1}\binom{2n}{n} \ . $$ The generating function is given by $$ \sum_{n=1}^\infty C_n z^n = \frac{1-\sqrt{1-4z}}{2z} \ . $$ The Catalan numbers appear in the enumeration of a number of combinatorially defined object:

References

Poisson ratio

The ratio of longitudinal extension to lateral compression when an elastic substance is put under tension.

See: Elasticity, mathematical theory of; Lamé constants.

References

  • Horace Lamb, "Statics", Cambridge University Press (1960)

Elastic modulus

Young's modulus

The ratio of longitudinal extension to force applied per unit area when an elastic substance is put under tension.

See: Elasticity, mathematical theory of; Lamé constants.

References

  • Horace Lamb, "Statics", Cambridge University Press (1960)

Partition symbol

A notation used to compactly express propositions of partition calculus. The symbol $$ \alpha \rightarrow (\beta)_\gamma^r $$ for cardinals $\alpha,\beta,\gamma$ and natural number $r$, denotes the following proposition.

Given a set $S$ and a colouring of $S^r$ into a set of $\gamma$ colours, there exists a subset $T$ of $S$ of cardinality $|T|=\beta$ such that the colouring restricted to $T^r$ is monochrome.

Here a colouring of a set $X$ by a set of colours $C$ is simply a partition of $X$ into parts indexed by the set $C$.

The symbol $$ \alpha \rightarrow (\beta_1,\ldots,\beta_j)^r $$ denotes the following proposition:

Given a set $S$ of cardinality $\alpha$ and a colouring of $S^r$ by $j$ colours, there exists an index $i$ subset $T$ of $S$ of cardinality $|T|=\beta_i$ such that the colouring restricted to $T^r$ is monochrome.

Examples.

  • Ramsey's theorem: $\omega \rightarrow (\omega)_n^r$.
  • Sierpinski's theorem: $c \not\rightarrow (\omega_1,\omega_2)^2$.


References

  • M.E. Rudin, "Lectures on set theoretic topology", Amer. Math. Soc. (1975) ISBN 0-8218-1673-X Zbl 0318.54001

Isthmus

bridge, co-loop

An isthmus of a graph is an edge for which deletion increases the number of connected components of the graph.

An isthmus of a matroid $M$ on a set $E$ is an element of $E$ which is in every basis for $M$. An element of $E$ is a co-loop of $M$ if and only if it is a loop of the dual matroid $M^*$, that is, does not belong to any base of $M^*$. If $M$ is a graphic matroid, then the definitions coincide.

References

  • J. G. Oxley, "Matroid Theory" (2 ed) Oxford University Press (2011) ISBN 978-0-19-856694-6 Zbl 1254.05002
  • D. J. A. Welsh, "Matroid Theory", Dover (2010) [1976] ISBN 0486474399 Zbl 0343.05002

Ordered topological space

A topological space $X$ with a partial order ${\le}$ related to the topology by the condition that if $x < y$ then there are neighbourhoods $N_x$, $N_y$ such that $x < y'$ for all $y' \in N_y$ and $x' < y$ for all $x' \in N_x$. An ordered topological space is necessarily a Hausdorff space.

An ordered topological space is totally order-disconnected if whenever $x \not\le y$ there exists a clopen down-set $D$ such that $x \not\in D$ and $y \in D$.

A Priestley space is a compact totally order-disconnected space. An Ockham space is a Priestley space equipped with an order-reversing continuous mapping $g$: see also Ockham algebra.

References

  • Samuel Eilenberg, "Ordered Topological Spaces", American Journal of Mathematics 63 (1941) 39-45 DOI 10.2307/2371274 Zbl 0024.19203
  • T.S. Blyth, "Lattices and ordered algebraic structures", Springer (2005) ISBN 1-85233-905-5 Zbl 1073.06001
How to Cite This Entry:
Richard Pinch/sandbox-12. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-12&oldid=42879