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=Partition function (number theory)=
 
=Partition function (number theory)=
The function $p(n)$ that counts the number of [[partition]]s of a positive integer $n$, that is, the number of ways of expressing $n$ as a sum of positive integers (where order is not significant).
+
A partition of a positive integer $n$ is a decomposition of $n$ as a sum of positive integers. For example, the partitions of 4 read: $4, 3+1, 2+2, 2+1+1, 1+1+1+1$. The partition function $p(n)$ counts the number of different partitions of $n$, so that $p(4) = 5$. L. Euler gave a non-trivial recurrence relation for $p(n)$ (see [[#References|[a1]]]) and Ramanujan discovered the surprising congruences $p(5m+4) \equiv 0 \pmod 5$, $p(7m+5) \equiv 0 \pmod 7$, $p(11m+6) \equiv 0 \pmod 11$, and others. He also found the asymptotic relation
 
 
Thus ''p''(3) = 3, since the number 3 has 3 partitions:
 
* $3$
 
* $2+1$
 
* $1+1+1$
 
 
 
The partition function satisfies an asymptotic relation
 
$$
 
p(n) \sim \frac{\exp\left(\pi\sqrt{2/3}\sqrt n\right)}{4n\sqrt3} \ .
 
$$
 
 
 
==References==
 
* Tom M. Apostol; ''Modular functions and Dirichlet Series in Number Theory'' Graduate Texts in Mathematics '''41''' Springer-Verlag (1990) ISBN 0-387-97127-0  pp. 94-112,
 
* G.H. Hardy; E. M. Wright; ''An Introduction to the Theory of Numbers'' Oxford University Press (2008) ISBN 0-19-921986-5 pp. 361-392,
 
 
 
==Of a positive integer==
 
A partition of a positive integer $n$ is a decomposition of $n$ as a sum of positive integers. For example, the partitions of 4 read: $4, 3+1, 2+2, 2+1+1, 1+1+1+1$. The number of different partitions of $n$ is denoted by $p(n)$. So, $p(4) = 5$. L. Euler gave a non-trivial recurrence relation for $p(n)$ (see [[#References|[a1]]]) and Ramanujan discovered the surprising congruences $p(5m+4) \equiv 0 \pmod 5$, $p(7m+5) \equiv 0 \pmod 7$, $p(11m+6) \equiv 0 \pmod 11$, and others. He also found the asymptotic relation
 
 
$$
 
$$
 
p(n) \sim \frac{e^{K \sqrt{n}}}{4n\sqrt{3}}\ \ \text{as}\ \ n \rightarrow \infty \ ,
 
p(n) \sim \frac{e^{K \sqrt{n}}}{4n\sqrt{3}}\ \ \text{as}\ \ n \rightarrow \infty \ ,
Line 44: Line 27:
 
where $K = \pi\sqrt{2/3}$. Later this was completed to an exact series expansion by H. Rademacher (see [[#References|[a2]]]).
 
where $K = \pi\sqrt{2/3}$. Later this was completed to an exact series expansion by H. Rademacher (see [[#References|[a2]]]).
  
One can also distinguish other partitions, having particular properties, such as the numbers in the decomposition being distinct (see [[#References|[a3]]]). See also [[Additive number theory|Additive number theory]]; [[Additive problems|Additive problems]].
+
One can also distinguish other partitions, having particular properties, such as the numbers in the decomposition being distinct (see [[#References|[a3]]]). See also [[Additive number theory]]; [[Additive problems]].
  
 
====References====
 
====References====
 
<table>
 
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. HardyE.M. Wright,  "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XVI</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 University Press (2008) ISBN 0-19-921986-5</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  T.M. Apostol,  "Modular functions and Dirichlet series in number theory" , Springer (1976)</TD></TR>
+
<TR><TD valign="top">[a2]</TD> <TD valign="top">  Tom M. Apostol; ''Modular functions and Dirichlet Series in Number Theory'' Graduate Texts in Mathematics '''41''' Springer-Verlag (1990) ISBN 0-387-97127-0</TD></TR>
 
<TR><TD valign="top">[a3]</TD> <TD valign="top">  G.E. Andrews,  "The theory of partitions" , Addison-Wesley  (1976)</TD></TR>
 
<TR><TD valign="top">[a3]</TD> <TD valign="top">  G.E. Andrews,  "The theory of partitions" , Addison-Wesley  (1976)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Engelking,  "Dimension theory" , North-Holland &amp; PWN  (1978)  pp. 19; 50</TD></TR></table>
+
<TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Engelking,  "Dimension theory" , North-Holland &amp; PWN  (1978)  pp. 19; 50</TD></TR>
 +
</table>
  
 
=Selberg sieve=
 
=Selberg sieve=

Revision as of 17:18, 6 February 2016


Justesen code

A class of error-correcting codes which are derived from Reed-Solomon codes and have good error-control properties.

Let $R$ be a Reed-Solomon code of length $N = 2^m-1$, rank $K$ and minimum weight $N-K+1$. The symbols of $R$ are elements of $F=GF(2^m)$ and the codewords are obtained by taking every polynomial $f$ over $F$ of degree less than $K$ and listing the values of $f$ on the non-zero elements of $F$ in some predetermined order. Let $\alpha$ be a primitive element of $F$. For a codeword $\mathbf{a} = (a_1,\ldots,a_N)$ from $R$, let $\mathbf{b}$ be the vector of length $2N$ over $F$ given by $$ \mathbf{b} = \left( a_1, a_1, a_2, \alpha^1 a_2, \ldots, a_N, \alpha^{N-1} a_N \right) $$ and let $\mathbf{c}$ be the vector of length $2Nm$ obtained from $\mathbf{b}$ by expressing each element of $F$ as a binary vector of length $m$. The Justesen code is the linear code containing all such $\mathbf{c}$.

The parameters of this code are length $2mN$, dimension $mK$ and minimum distance at least $$ \sum_{i=1}^l i \binom{2m}{i} \ . $$ The Justesen codes are examples of concatenated codes.

References

  • J. Justesen; A class of constructive asymptotically good algebraic codes, IEEE Trans. Info. Theory, 18 (1972), pp. 652-656
  • F.J. MacWilliams; N.J.A. Sloane; The Theory of Error-Correcting Codes, , pp. 306-316, North-Holland ISBN: 0-444-85193-3

Partition function (number theory)

A partition of a positive integer $n$ is a decomposition of $n$ as a sum of positive integers. For example, the partitions of 4 read: $4, 3+1, 2+2, 2+1+1, 1+1+1+1$. The partition function $p(n)$ counts the number of different partitions of $n$, so that $p(4) = 5$. L. Euler gave a non-trivial recurrence relation for $p(n)$ (see [a1]) and Ramanujan discovered the surprising congruences $p(5m+4) \equiv 0 \pmod 5$, $p(7m+5) \equiv 0 \pmod 7$, $p(11m+6) \equiv 0 \pmod 11$, and others. He also found the asymptotic relation $$ p(n) \sim \frac{e^{K \sqrt{n}}}{4n\sqrt{3}}\ \ \text{as}\ \ n \rightarrow \infty \ , $$ where $K = \pi\sqrt{2/3}$. Later this was completed to an exact series expansion by H. Rademacher (see [a2]).

One can also distinguish other partitions, having particular properties, such as the numbers in the decomposition being distinct (see [a3]). See also Additive number theory; Additive problems.

References

[a1] G.H. Hardy; E. M. Wright; An Introduction to the Theory of Numbers Oxford University Press (2008) ISBN 0-19-921986-5
[a2] Tom M. Apostol; Modular functions and Dirichlet Series in Number Theory Graduate Texts in Mathematics 41 Springer-Verlag (1990) ISBN 0-387-97127-0
[a3] G.E. Andrews, "The theory of partitions" , Addison-Wesley (1976)
[a4] R. Engelking, "Dimension theory" , North-Holland & PWN (1978) pp. 19; 50

Selberg sieve

A technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Atle Selberg in the 1940s.

Description

In terms of sieve theory the Selberg sieve is of combinatorial type: that is, derives from a careful use of the inclusion-exclusion principle. Selberg replaced the values of the Möbius function which arise in this by a system of weights which are then optimised to fit the given problem. The result gives an upper bound for the size of the sifted set.

Let A be a set of positive integers ≤ x and let P be a set of primes. For each p in P, let Ap denote the set of elements of A divisible by p and extend this to let Ad the intersection of the Ap for p dividing d, when d is a product of distinct primes from P. Further let A1 denote A itself. Let z be a positive real number and P(z) denote the product of the primes in P which are ≤ z. The object of the sieve is to estimate

\[S(A,P,z) = \left\vert A \setminus \bigcup_{p \in P(z)} A_p \right\vert . \]

We assume that |Ad| may be estimated by

\[ \left\vert A_d \right\vert = \frac{1}{f(d)} X + R_d . \]

where f is a multiplicative function and X   =   |A|. Let the function g be obtained from f by Möbius inversion, that is

\[ g(n) = \sum_{d \mid n} \mu(d) f(n/d) \] \[ f(n) = \sum_{d \mid n} g(d) \]

where μ is the Möbius function. Put

\[ V(z) = \sum_{\begin{smallmatrix}d < z \\ d \mid P(z)\end{smallmatrix}} \frac{\mu^2(d)}{g(d)} . \]

Then

\[ S(A,P,z) \le \frac{X}{V(z)} + O\left({\sum_{\begin{smallmatrix} d_1,d_2 < z \\ d_1,d_2 \mid P(z)\end{smallmatrix}} \left\vert R_{[d_1,d_2]} \right\vert} \right) .\]

It is often useful to estimate V(z) by the bound

\[ V(z) \ge \sum_{d \le z} \frac{1}{f(d)} . \, \]

Applications

  • The Brun-Titchmarsh theorem on the number of primes in an arithmetic progression;
  • The number of nx such that n is coprime to φ(n) is asymptotic to e x / log log log (x) .

References

  • Alina Carmen Cojocaru; M. Ram Murty; An introduction to sieve methods and their applications, ser. London Mathematical Society Student Texts 66 , pp. 113-134, Cambridge University Press ISBN: 0-521-61275-6
  • George Greaves; Sieves in number theory, , Springer-Verlag ISBN: 3-540-41647-1
  • Heini Halberstam; H.E. Richert; Sieve Methods, , Academic Press ISBN: 0-12-318250-6
  • Christopher Hooley; Applications of sieve methods to the theory of numbers, , pp. 7-12, Cambridge University Press ISBN: 0-521-20915-3
  • Atle Selberg; On an elementary method in the theory of primes, Norske Vid. Selsk. Forh. Trondheim, 19 (1947), pp. 64-67

Separation axioms

In topology, separation axioms describe classes of topological space according to how well the open sets of the topology distinguish between distinct points.


Terminology

A neighbourhood of a point x in a topological space X is a set N such that x is in the interior of N; that is, there is an open set U such that \(x \in U \subseteq N\). A neighbourhood of a set A in X is a set N such that A is contained in the interior of N; that is, there is an open set U such that \(A \subseteq U \subseteq N\).

Subsets U and V are separated in X if U is disjoint from the closure of V and V is disjoint from the closure of U.

A Urysohn function for subsets A and B of X is a continuous function f from X to the real unit interval such that f is 0 on A and 1 on B.

Axioms

A topological space X is

  • T0 if for any two distinct points there is an open set which contains just one
  • T1 if for any two points x, y there are open sets U and V such that U contains x but not y, and V contains y but not x
  • T2 if any two distinct points have disjoint neighbourhoods
  • T2½ if distinct points have disjoint closed neighbourhoods
  • T3 if a closed set A and a point x not in A have disjoint neighbourhoods
  • T3½ if for any closed set A and point x not in A there is a Urysohn function for A and {x}
  • T4 if disjoint closed sets have disjoint neighbourhoods
  • T5 if separated sets have disjoint neighbourhoods
  • Hausdorff is a synonym for T2
  • completely Hausdorff is a synonym for T2½
  • regular if T0 and T3
  • completely regular if T0 and T3½
  • Tychonoff is completely regular and T1
  • normal if T0 and T4
  • completely normal if T1 and T5
  • perfectly normal if normal and every closed set is a Gδ

Properties

  • A space is T1 if and only if each point (singleton) forms a closed set.
  • Urysohn's Lemma: if A and B are disjoint closed subsets of a T4 space X, there is a Urysohn function for A and B'.

References

  • Steen, Lynn Arthur; Seebach, J. Arthur Jr.; Counterexamples in Topology, (1978), Springer-Verlag ISBN: 0-387-90312-7

Turan sieve

In mathematics, in the field of number theory, the Turán sieve is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. It was developed by Pál Turán in 1934.

Description

In terms of sieve theory the Turán sieve is of combinatorial type: deriving from a rudimentary form of the inclusion-exclusion principle. The result gives an upper bound for the size of the sifted set.

Let A be a set of positive integers ≤ x and let P be a set of primes. For each p in P, let Ap denote the set of elements of A divisible by p and extend this to let Ad the intersection of the Ap for p dividing d, when d is a product of distinct primes from P. Further let A1 denote A itself. Let z be a positive real number and P(z) denote the product of the primes in P which are ≤ z. The object of the sieve is to estimate

\[S(A,P,z) = \left\vert A \setminus \bigcup_{p \in P(z)} A_p \right\vert . \]

We assume that |Ad| may be estimated, when d is a prime p by

\[ \left\vert A_p \right\vert = \frac{1}{f(p)} X + R_p \]

and when d is a product of two distinct primes d = p q by

\[ \left\vert A_{pq} \right\vert = \frac{1}{f(p)f(q)} X + R_{p,q} \]

where X   =   |A| and f is a function with the property that 0 ≤ f(d) ≤ 1. Put

\[ U(z) = \sum_{p \mid P(z)} f(p) . \]

Then

\[ S(A,P,z) \le \frac{X}{U(z)} + \frac{2}{U(z)} \sum_{p \mid P(z)} \left\vert R_p \right\vert + \frac{1}{U(z)^2} \sum_{p,q \mid P(z)} \left\vert R_{p,q} \right\vert . \]

Applications

References

Weierstrass preparation theorem

In algebra, the Weierstrass preparation theorem describes a canonical form for formal power series over a complete local ring.

Let O be a complete local ring and f a formal power series in O''X''. Then f can be written uniquely in the form

\[f = (X^n + b_{n-1}X^{n-1} + \cdots + b_0) u(x) , \,\]

where the bi are in the maximal ideal m of O and u is a unit of O''X''.

The integer n defined by the theorem is the Weierstrass degree of f.

References

  • Serge Lang; Algebra, (1993), pp. 208-209, Addison-Wesley ISBN: 0-201-55540-9


Zipf distribution

In probability theory and statistics, the Zipf distribution and zeta distribution refer to a class of discrete probability distributions. They have been used to model the distribution of words in words in a text , of text strings and keys in databases, and of the sizes of businesses and towns.

The Zipf distribution with parameter n assigns probability proportional to 1/r to an integer rn and zero otherwise, with normalization factor Hn, the n-th harmonic number.

A Zipf-like distribution with parameters n and s assigns probability proportional to 1/rs to an integer rn and zero otherwise, with normalization factor \(\sum_{r=1}^n 1/r^s\).

The zeta distribution with parameter s assigns probability proportional to 1/rs to all integers r with normalization factor given by the Riemann zeta function 1/ζ(s).

References

How to Cite This Entry:
Richard Pinch/sandbox-CZ. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-CZ&oldid=37677