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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

\[q \prod_{n=1}^\infty \left(1-q^n\right)^{24} = \sum_n \tau(n) q^n .\,\]

Since Δ is a Hecke eigenform, the tau function is multiplicative, with formal Dirichlet series and Euler product

\[ \sum_n \tau(n) n^{-s} = \prod_p \left(1 - \tau(p) p^{-s} + p^{11-2s} \right)^{-1} .\,\]

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 A_p denote the set of elements of A divisible by p and extend this to let A_d the intersection of the A_p 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 |A_d| 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

• The Hardy–Ramanujan theorem that the normal order of ω(n), the number of distinct prime factors of a number n, is log(log(n));
• Almost all integer polynomials (taken in order of height) are irreducible.

References

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Tutte matrix

In graph theory, the Tutte matrix \(A\) of a 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 2n elements then the Tutte matrix is a 2n × 2n matrix A with entries

\[A_{ij} = \begin{cases} x_{ij}\;\;\mbox{if}\;(i,j) \in E \mbox{ and } i<j\\ -x_{ji}\;\;\mbox{if}\;(i,j) \in E \mbox{ and } i>j\\ 0\;\;\;\;\mbox{otherwise} \end{cases}\]

where the x_ij are indeterminates. The determinant of this skew-symmetric matrix is then a polynomial (in the variables x_ij, 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

• Template:Cite book
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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 b_i 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

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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 r ≤ n and zero otherwise, with normalization factor H_n, the n-th harmonic number.

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

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

References

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