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=Srivastava code=
 
  
{{subpages}}
 
In [[coding theory]], '''Srivastava codes''' form a class of parameterised [[Error detection and correction|error-correcting codes]] which are a special case of [[alternant code]]s.
 
  
==Definition==
+
=Selberg sieve=
The original ''Srivastava code'' over GF(''q'') of length ''n'' is defined by a parity check matrix ''H'' of [[alternant matrix|alternant]] form
+
A technique for estimating the size of "sifted sets" of [[positive integer]]s which satisfy a set of conditions which are expressed by [[Congruence relation#Modular arithmetic|congruence]]s.  It was developed by [[Atle Selberg]] in the 1940s.
  
:<math>\begin{bmatrix}
+
==Description==
\frac{\alpha_1^\mu}{\alpha_1-w_1} & \cdots & \frac{\alpha_n^\mu}{\alpha_n-w_1} \\
+
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.
\vdots & \ddots & \vdots \\
+
 
\frac{\alpha_1^\mu}{\alpha_1-w_s} & \cdots & \frac{\alpha_n^\mu}{\alpha_n-w_s} \\
+
Let ''A'' be a set of positive integers &le; ''x'' and let ''P'' be a set of primes.  For each ''p'' in ''P'', let ''A''<sub>''p''</sub> denote the set of elements of ''A'' divisible by ''p'' and extend this to let ''A''<sub>''d''</sub> the intersection of the ''A''<sub>''p''</sub> for ''p'' dividing ''d'', when ''d'' is a product of distinct primes from ''P''.  Further let A<sub>1</sub> denote ''A'' itself.  Let ''z'' be a positive real number and ''P''(''z'') denote the product of the primes in ''P'' which are &le; ''z''.  The object of the sieve is to estimate
\end{bmatrix}
+
 
</math>
+
:<math>S(A,P,z) = \left\vert A \setminus \bigcup_{p \in P(z)} A_p \right\vert . </math>
where the α<sub>''i''</sub> and ''z''<sub>''i''</sub> are elements of GF(''q''<sup>''m''</sup>)
+
 
 +
We assume that |''A''<sub>''d''</sub>| may be estimated by
 +
 
 +
:<math> \left\vert A_d \right\vert = \frac{1}{f(d)} X + R_d . </math>
  
==Properties==
+
where ''f'' is a [[multiplicative function]] and ''X'' &nbsp; = &nbsp; |''A''|. Let the function ''g'' be obtained from ''f'' by [[Möbius inversion formula|Möbius inversion]], that is
The parameters of this code are length ''n'', dimension ≥ ''n''&nbsp;&minus;&nbsp;''m''s and minimum distance ≥&nbsp;s&nbsp;+&nbsp;1.
 
  
== References ==
+
:<math> g(n) = \sum_{d \mid n} \mu(d) f(n/d) </math>
* {{cite book | author=F.J. MacWilliams | authorlink=Jessie MacWilliams | coauthors=N.J.A. Sloane | title=The Theory of Error-Correcting Codes | publisher=North-Holland | date=1977 | isbn=0-444-85193-3 | pages=357-360 }}
+
:<math> f(n) = \sum_{d \mid n} g(d) </math>
=Stably free module=
 
In [[mathematics]], a '''stably free module''' is a [[module (mathematics)|module]] which is close to being [[free module|free]].
 
  
==Definition==
+
where &mu; is the [[Möbius function]]. 
A module ''M'' over a ring ''R'' is ''stably free'' if there exist free modules ''F'' and ''G'' over ''R'' such that
+
Put
  
:<math> M \oplus F = G . \, </math>
+
:<math> V(z) = \sum_{\begin{smallmatrix}d < z \\ d \mid P(z)\end{smallmatrix}} \frac{\mu^2(d)}{g(d)} . </math>
  
==Properties==
+
Then
* A module is stably free if and only if it possesses a finite [[free resolution]].
 
  
== See also ==
+
:<math> 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) .</math>
* [[Free object]]
 
  
==References==
+
It is often useful to estimate ''V''(''z'') by the bound
* {{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}}
 
  
 +
:<math> V(z) \ge \sum_{d \le z} \frac{1}{f(d)} . \, </math>
  
=Stirling numbers=
+
==Applications==
 +
* The [[Brun-Titchmarsh theorem]] on the number of primes in an arithmetic progression;
 +
* The number of ''n'' &le; ''x'' such that ''n'' is coprime to &phi;(''n'') is asymptotic to e<sup>-&gamma;</sup> ''x'' / log log log (''x'') .
  
In [[combinatorics]], the '''Stirling numbers''' count certain arrangements of objects into a given number of structures. There are two kinds of Stirling number, depending on the nature of the structure being counted.
+
==References==
 +
* {{User:Richard Pinch/sandbox/Ref | author=Alina Carmen Cojocaru | coauthors=M. Ram Murty | title=An introduction to sieve methods and their applications | series=London Mathematical Society Student Texts | volume=66 | publisher=[[Cambridge University Press]] | isbn=0-521-61275-6 | pages=113-134 }}
 +
* {{User:Richard Pinch/sandbox/Ref | author=George Greaves | title=Sieves in number theory | publisher=[[Springer-Verlag]] | date=2001 | isbn=3-540-41647-1}}
 +
* {{User:Richard Pinch/sandbox/Ref | author=Heini Halberstam | coauthors=H.E. Richert | title=Sieve Methods | publisher=[[Academic Press]] | date=1974 | isbn=0-12-318250-6}}
 +
* {{User:Richard Pinch/sandbox/Ref | author= Christopher Hooley | authorlink=Christopher Hooley | title=Applications of sieve methods to the theory of numbers | publisher=Cambridge University Press | date=1976 | isbn=0-521-20915-3| pages=7-12}}
 +
* {{User:Richard Pinch/sandbox/Ref | author=Atle Selberg | authorlink=Atle Selberg | title=On an elementary method in the theory of primes | journal=Norske Vid. Selsk. Forh. Trondheim | volume=19 | year=1947 | pages=64-67 }}
  
The Stirling number of the first kind ''S''(''n'',''k'') counts the number of ways ''n'' labelled objects can be arranged into ''k'' cycles: cycles are regarded as equivalent, and counted only once, if they differ by a [[cyclic permutation]], thus [ABC] = [BCA] = [CAB] but is counted as different from [CBA] = [BAC] = [ACB].  The order of the cycles in the list is irrelevant.
+
=Separation axioms=
 +
In [[topology]], '''separation axioms''' describe classes of [[topological space]] according to how well the [[open set]]s of the topology distinguish between distinct points.
  
For example, 4 objects can be arranged into 2 cycles in eleven ways, so ''S''(4,2) = 11:
 
  
* [ABC],[D]
+
==Terminology==
* [ACB],[D]
+
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 <math>x \in U \subseteq N</math>.
* [ABD],[C]
+
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 <math>A \subseteq U \subseteq N</math>.
* [ADB],[C]
 
* [ACD],[B]
 
* [ADC],[B]
 
* [BCD],[A]
 
* [BDC],[A]
 
* [AB],[CD]
 
* [AC],[BD]
 
* [AD],[BC]
 
  
The Stirling number of the second kind ''s''(''n'',''k'') counts the number of ways ''n'' labelled objects can be arranged into ''k'' subsets: cycles are regarded as equivalent, and counted only once, if they have the same elements, thus {ABC} = {BCA} = {CAB} = {CBA} = {BAC} = {ACB}.  The order of the subsets in the list is irrelevant.
+
Subsets ''U'' and ''V'' are ''separated'' in ''X'' if ''U'' is disjoint from the [[Closure (topology)|closure]] of ''V'' and ''V'' is disjoint from the closure of ''U''.
  
For example, 4 objects can be arranged into 2 subsets in seven ways, so ''s''(4,2) = 7:
+
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''.
  
* {ABC},{D}
+
==Axioms==
* {ABD},{C}
+
A topological space ''X'' is
* {ACD},{B}
+
* '''T0''' if for any two distinct points there is an open set which contains just one
* {BCD},{A}
+
* '''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''
* {AB},{CD}
+
* '''T2''' if any two distinct points have disjoint neighbourhoods
* {AC},{BD}
+
* '''T2½''' if distinct points have disjoint closed neighbourhoods
* {AD},{BC}
+
* '''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
  
==References==
+
* '''Hausdorff''' is a synonym for T2
* {{cite book | author=Ronald L. Graham | coauthors=Donald E. Knuth, Oren Patashnik | title=Concrete Mathematics | publisher=[[Addison Wesley]] | year=1989 | isbn=0-201-14236-8 | pages=243-253 }}
+
* '''completely Hausdorff''' is a synonym for T2½
=Tau function=
 
In [[mathematics]], Ramanujan's '''tau function''' is an [[arithmetic function]] which may defined in terms of the [[Delta form]] by the formal infinite product
 
  
:<math>q \prod_{n=1}^\infty \left(1-q^n\right)^{24} = \sum_n \tau(n) q^n .\,</math>
+
* '''regular''' if T0 and T3
 +
* '''completely regular''' if T0 and T3½
 +
* '''Tychonoff''' is completely regular and T1
  
Since Δ is a [[Hecke operator|Hecke]] [[eigenform]], the tau function is multiplicative, with [[formal Dirichlet series]] and Euler product
+
* '''normal''' if T0 and T4
 +
* '''completely normal''' if T1 and T5
 +
* '''perfectly normal''' if normal and every closed set is a [[G-delta set|G<sub>δ</sub>]]
  
:<math> \sum_n \tau(n) n^{-s} = \prod_p \left(1 - \tau(p) p^{-s} + p^{11-2s} \right)^{-1} .\,</math>
+
==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==
 +
* {{User:Richard Pinch/sandbox/Ref | last1=Steen | first1=Lynn Arthur | last2=Seebach | first2=J. Arthur Jr. | title=Counterexamples in Topology | year=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=0-387-90312-7 }}
  
 
=Turan sieve=
 
=Turan sieve=
Line 116: Line 119:
 
==References==
 
==References==
 
* {{cite book | author=Alina Carmen Cojocaru | coauthors=M. Ram Murty | title=An introduction to sieve methods and their applications | series=London Mathematical Society Student Texts | volume=66 | publisher=[[Cambridge University Press]] | isbn=0-521-61275-6 | pages=47-62 }}
 
* {{cite book | author=Alina Carmen Cojocaru | coauthors=M. Ram Murty | title=An introduction to sieve methods and their applications | series=London Mathematical Society Student Texts | volume=66 | publisher=[[Cambridge University Press]] | isbn=0-521-61275-6 | pages=47-62 }}
=Tutte matrix=
 
In [[graph theory]], the '''Tutte matrix''' <math>A</math> of a [[Graph (mathematics)|graph]] ''G'' = (''V'',''E'') is a matrix used to determine the existence of a [[perfect matching]]: that is, a set of edges which is incident with each vertex exactly once.
 
 
If the set of vertices ''V'' has 2''n'' elements then the Tutte matrix is a 2''n'' × 2''n'' matrix A with entries
 
 
::: <math>A_{ij} = \begin{cases} x_{ij}\;\;\mbox{if}\;(i,j) \in E \mbox{ and } i<j\\
 
-x_{ji}\;\;\mbox{if}\;(i,j) \in E \mbox{ and } i>j\\
 
0\;\;\;\;\mbox{otherwise} \end{cases}</math>
 
 
where the ''x''<sub>''ij''</sub> are indeterminates.  The [[determinant]] of this [[skew-symmetric]] matrix is then a polynomial (in the variables ''x<sub>ij</sub>'', ''i<j'' ): this coincides with the square of the [[pfaffian]] of the matrix ''A'' and is non-zero (as a polynomial) if and only if a perfect matching exists.  (It should be noted that this is not the [[Tutte polynomial]] of ''G''.)
 
 
The Tutte matrix is a generalisation of the [[Edmonds matrix]] for a balanced [[bipartite graph]].
 
 
==References==
 
*{{cite book|author=R. Motwani, P. Raghavan |title=Randomized Algorithms|publisher=Cambridge University Press|year=1995|page=167}}
 
*{{cite book|author=Allen B. Tucker|title=Computer Science Handbook|publisher=CRC Press|date=2004|isbn=158488360X|page=12.19}}
 
* {{cite journal|url= http://jlms.oxfordjournals.org/cgi/reprint/s1-22/2/107.pdf|title= The factorization of linear graphs.
 
|accessdate= 2008-06-15|author= W.T. Tutte|authorlink=W. T. Tutte|year= 1947|month= April|volume=22|journal=J. London Math. Soc.|pages=107-111|doi=10.1112/jlms/s1-22.2.107}}
 
 
 
 
=Weierstrass preparation theorem=
 
=Weierstrass preparation theorem=
 
In [[algebra]], the '''Weierstrass preparation theorem''' describes a canonical form for [[formal power series]] over a [[complete local ring]].
 
In [[algebra]], the '''Weierstrass preparation theorem''' describes a canonical form for [[formal power series]] over a [[complete local ring]].
Line 148: Line 131:
  
 
==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 | pages=208-209 }}
+
* {{User:Richard Pinch/sandbox/Ref | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 | pages=208-209 }}
  
  

Latest revision as of 19:14, 2 May 2020


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=30422