User:Richard Pinch/sandbox-7

Identity

An equality that holds true for all values of the variables involved within some domain of valididy.

A condition that holds true for all elements of some algebraic structures.

A neutral element for a binary operation.

A map from a set to itself which maps each element to itself.

Dense ordered set

A totally ordered set $(X,{<})$ with the property that if $x < y$ then there exists $z \in X$ with $x < z < y$.

Cantor showed that any countable dense unbounded linearly ordered sets are isomorphic.

References

• T. Jech, "Set theory. The third millennium edition, revised and expanded" Springer Monographs in Mathematics (2003). Zbl 1007.03002

Order isomorphism

between partially ordered sets

A bijection that is also an order-preserving mapping. Order isomorphic sets are said to have the same order type, although this term is often restricted to linearly ordered sets.

Another term is similarity.

References

• Ciesielski, Krzysztof. "Set theory for the working mathematician" London Mathematical Society Student Texts 39 Cambridge University Press (1997) Zbl 0938.03067
• Halmos, Paul R. "Naive Set Theory", Springer (1960, repr. 1974) ISBN 0-387-90092-6 Zbl 0287.04001

Sign of a permutation

The sign of a permutation of a finite set, which we can identify with $\{1,2,\ldots,n\}$ for some $n$, is a multiplicative map $\epsilon$ from the groups of permutations to $\pm 1$. Permutations with sign $+1$ are even and those with sign $-1$ are odd. The sign may be defined in a number of ways. A simple formula is $$ \epsilon(\pi) = \frac{\prod_{1 \le i < j \le n} (x^{\pi(i)}-x^{\pi(j)})}{\prod_{1 \le i < j \le n} (x^i-x^j)} \ .\label{1} $$

The sign of $\pi$ may also be defined by the parity of the number of transpositions which compose $\pi$: this is well-defined since an odd number of transpositions cannot give the identity. A related definition is that the sign is the parity of $n - c$ where $c = c(\pi)$ is the number of cycles (orbits) of $\pi$.

Since $\epsilon$ is a homomorphism from $S_n$ to $C_2 = \{\pm1\}$, the alternating group, or group of even permutations, $A_n$ is the kernel of $\epsilon$ and so a normal subgroup of the symmetric group.

The definition of $\epsilon$ may be extended to maps which are not permutations by defining it to be zero. This is consistent with (1) and also with the use of the Kronecker symbol in tensor notation.

Baer radical

of a ring $R$

The intersection of the prime ideals of the ring $R$. It is an instance of a radical: it is the lower radical determined by the class of all nilpotent rings; and the upper radical determined by the class of all primary rings.

References

• Sapir, Mark V. "Combinatorial algebra: syntax and semantics" with contributions by Victor S. Guba and Mikhail V. Volkov. Springer Monographs in Mathematics. Springer (2014) SBN 978-3-319-08030-7 Zbl 1319.05001

Krawtchouk polynomials

Polynomials orthogonal on the finite system of $N+1$ integer points whose distribution function $\sigma(z)$ is a step function with discontinuities: $$ \sigma(x+) - \sigma(x-) = \binom{N}{x} p^x q^{N-x} \,,\ \ \ x=0,\ldots,N $$ where $\binom{\cdot}{\cdot}$ is the binomial coefficient, $p,q > 0$ and $p+q = 1$. The Krawtchouk polynomials are given by the formulas $$ P_n(x) = \left[ \binom{N}{x} \right]^{-1/2} (pq)^{-n/2} \sum_{k=0}^n (-1)^{n-k} \binom{N-x}{n-k} \binom{x}{k} p^{n-k} q^k \ . $$ Here $\binom{x}{k}$ denotes the polynomial $$ \binom{x}{k} = \frac{x(x-1)\cdots(x-k+1)}{k!} $$ of degree $k$ in $x$

The concept is due to M.F. Krawtchouk [1].

References

[1]	M.F. Krawtchouk, "Sur une généralisation des polynômes d'Hermite" C.R. Acad. Sci. Paris , 189 (1929) pp. 620–622
[2]	G. Szegö, "Orthogonal polynomials" , Amer. Math. Soc. (1975)

Comments

Krawtchouk polynomials can be written as hypergeometric functions of type ${}_2F_1$. The unitarity relations for the matrix elements of the irreducible unitary representations of the group $SU(2)$ can be rewritten as the orthogonality relations for the Krawtchouk polynomials, cf. [a2], [a3]. These polynomials have also an interpretation as spherical functions on wreath products of symmetric groups, cf. [a4], where $q$-Krawtchouk polynomials are also treated. Coding theorists rather (but equivalently) relate them to Hamming schemes, where Krawtchouk polynomials are used for dealing with problems about perfect codes, cf. [a1].

References

[a1]	J.H. van Lint, "Introduction to coding theory" , Springer (1982)
[a2]	T.H. Koornwinder, "Krawtchouk polynomials, a unification of two different group theoretic interpretations" SIAM J. Math. Anal. , 13 (1982) pp. 1011–1023
[a3]	V.B. Uvarov, "Special functions of mathematical physics" , Birkhäuser (1988) (Translated from Russian)
[a4]	D. Stanton, "Orthogonal polynomials and Chevalley groups" R.A. Askey (ed.) T.H. Koornwinder (ed.) W. Schempp (ed.) , Special functions: group theoretical aspects and applications , Reidel (1984) pp. 87–128

Comments

A simpler version of the polynomials may be written as $$ K_n(x) = \sum_{k=0}^n (-1)^{n-k} \binom{N-x}{n-k} \binom{x}{k} p^{n-k} q^k \ . $$ The orthogonality relation is then $$ \sum_{i=0}^n \binom{n}{i} (q-1)^i K_r(i)K_s(i) = \delta_{rs} \binom{n}{r} (q-1)^r q^n \ . $$

There is a generating function $$ \sum_{k=0}^\infty K_r(x) z^k = (1-z)^x (1 + (q-1)z)^{n-x} \ . $$

Distance enumerator

The distribution of Hamming distances between elements of a code, expressed as a polynomial. Let $C$ be a code of length $n$ over an alphabet $F$ and let $A_k$ be the number of pairs $x,y$ of words of $C$ of at Hamming distance $d(x,y) = k$. The weight enumerator $$ W_C(z) = \sum_{k=0}^n A_k z^k = \sum_{x,y \in C} z^{d(x,y)} \ . $$ It is also common to express the weight enumerator as a homogeneous binary form $$ W_C(x,y) = \sum_{k=0}^n A_k x^k y^{n-k} \ . $$

We have $W_C(0) = |C|$ and $W_C(1) = |C|^2$, where $|C|$ is the number of words in $C$.

The weight enumerator similarly expresses the distribution of Hamming weightss of elements of a code, expressed as a polynomial. Let $C$ be a code of length $n$ over an alphabet $F$ and let $A_k$ be the number of of words of $C$ of weight $k$. The weight enumerator $$ W_C(z) = \sum_{k=0}^n A_k z^k = \sum_{x \in C} z^{w(x)} $$ where $w(x)$ is the weight of the word $x$. It is also common to express the weight enumerator as a homogeneous binary form $$ W_C(x,y) = \sum_{k=0}^n A_k x^k y^{n-k} \ . $$

We have $W_C(0) = 1$ or $0$, depending on whether the zero word is in $C$ or not, and $W_C(1) = |C|$, the number of words in $C$.

The MacWilliams identities relate the weight enumerator of a linear code over a finite field $\mathbf{F}_q$ to the enumerator of the dual code $C^\perp$: $$ W_{C^\perp}(x,y) = \frac{1}{|C|} W_C(x + (q-1)y, x-y) \ . $$

References

• Goldie, Charles M.; Pinch, Richard G.E. Communication theory, London Mathematical Society Student Texts. 20 Cambridge University Press (1991) iSBN 0-521-40456-8 Zbl 0746.94001
• van Lint, J.H., "Introduction to coding theory" (2nd ed.) Graduate Texts in Mathematics 86 Springer (1992) ISBN 3-540-54894-7 Zbl 0747.94018