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=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
+
=Selberg sieve=
 +
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>A_{ij} = \begin{cases} x_{ij}\;\;\mbox{if}\;(i,j) \in E \mbox{ and } i<j\\
+
==Description==
-x_{ji}\;\;\mbox{if}\;(i,j) \in E \mbox{ and } i>j\\
+
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.
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''.)
+
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
  
The Tutte matrix is a generalisation of the [[Edmonds matrix]] for a balanced [[bipartite graph]].
+
:<math>S(A,P,z) = \left\vert A \setminus \bigcup_{p \in P(z)} A_p \right\vert . </math>
 +
 
 +
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>
 +
 
 +
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
 +
 
 +
:<math> g(n) = \sum_{d \mid n} \mu(d) f(n/d) </math>
 +
:<math> f(n) = \sum_{d \mid n} g(d) </math>
 +
 
 +
where &mu; is the [[Möbius function]]
 +
Put
 +
 
 +
:<math> V(z) = \sum_{\begin{smallmatrix}d < z \\ d \mid P(z)\end{smallmatrix}} \frac{\mu^2(d)}{g(d)} . </math>
 +
 
 +
Then
 +
 
 +
:<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>
 +
 
 +
It is often useful to estimate ''V''(''z'') by the bound
 +
 
 +
:<math> V(z) \ge \sum_{d \le z} \frac{1}{f(d)} . \, </math>
 +
 
 +
==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'') .
  
 
==References==
 
==References==
*{{cite book|author=R. Motwani, P. Raghavan |title=Randomized Algorithms|publisher=Cambridge University Press|year=1995|page=167}}
+
* {{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 }}
*{{cite book|author=Allen B. Tucker|title=Computer Science Handbook|publisher=CRC Press|date=2004|isbn=158488360X|page=12.19}}
+
* {{User:Richard Pinch/sandbox/Ref | author=George Greaves | title=Sieves in number theory | publisher=[[Springer-Verlag]] | date=2001 | isbn=3-540-41647-1}}
* {{cite journal|url= http://jlms.oxfordjournals.org/cgi/reprint/s1-22/2/107.pdf|title= The factorization of linear graphs.
+
* {{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}}
|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}}
+
* {{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 }}
  
 +
=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.
  
 +
 +
==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 <math>x \in U \subseteq N</math>.
 +
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>.
 +
 +
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''.
 +
 +
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 [[G-delta set|G<sub>δ</sub>]]
 +
 +
==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=
 +
 +
In [[mathematics]], in the field of [[number theory]], the '''Turán sieve''' is 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 [[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 &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
 +
 +
:<math>S(A,P,z) = \left\vert A \setminus \bigcup_{p \in P(z)} A_p \right\vert . </math>
 +
 +
We assume that |''A''<sub>''d''</sub>| may be estimated, when ''d'' is a prime ''p'' by
 +
 +
:<math> \left\vert A_p \right\vert = \frac{1}{f(p)} X + R_p  </math>
 +
 +
and when ''d'' is a product of two distinct primes ''d'' = ''p'' ''q'' by
 +
 +
:<math> \left\vert A_{pq} \right\vert = \frac{1}{f(p)f(q)} X + R_{p,q}  </math>
 +
 +
where ''X'' &nbsp; = &nbsp; |''A''| and ''f'' is a function with the property that 0 &le; ''f''(''d'') &le; 1.  Put
 +
 +
:<math> U(z) = \sum_{p \mid P(z)} f(p) . </math>
 +
 +
Then
 +
 +
:<math> 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 . </math>
 +
 +
==Applications==
 +
* The [[Hardy–Ramanujan theorem]] that the [[normal order of an arithmetic function|normal order]] of &omega;(''n''), the number of distinct [[prime factor]]s of a number ''n'', is log(log(''n''));
 +
* Almost all integer polynomials (taken in order of height) are irreducible.
 +
 +
==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 }}
 
=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 32: 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=30417