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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

Subnormal series

A subnormal series (or subinvariant series) of a group $G$ is a subgroup series $$ E = G_0 \le G_1 \le \cdots \le G_n = G $$ in which 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.

A subnormal subgroup (also subinvariant, attainable or accessible) of $G$ is a subgroup that appears in some subnormal series of $G$. To indicate the subnormality of a subgroup $H$ in a group $G$, the notation $H \lhd\!\lhd G$ is used.

The property of a subgroup to be subnormal is transitive. An intersection of subnormal subgroups is again a subnormal subgroup. The subgroup generated by two subnormal subgroups need not be subnormal. A group $G$ all subgroups of which are subnormal satisfies the normalizer condition, i.e. all subgroups differ from their normalizers (cf. Normalizer of a subset). Such a group is therefore locally nilpotent.

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. [6].

References

[1]	M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)
[2]	A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[3]	M. Hall jr., "The theory of groups" , Macmillan (1959) pp. Sect. 8.4
[4]	J.C. Lennox, S.E. Stonehewer, "Subnormal subgroups of groups" , Clarendon Press (1987)
[5]	D.J.S. Robinson, "A course in the theory of groups" , Springer (1982)
[6]	M. Suzuki, "Group theory" , 1–2 , Springer (1986)

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 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

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 $$ C_n = \frac{1}{n+1}\binom{2n}{n} \ . $$

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:

• Bernoulli excursion
• Dyck paths
• Parenthesised sequences; words of the Dyck language
• Complete binary rooted plane trees

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