Tollmien–Schlichting wave

In the field of number theory, the Brun sieve (also called Brun's pure 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 Viggo Brun in 1915.

Description

In terms of sieve theory the Brun sieve is of combinatorial type: that is, derives from a careful use of the inclusion-exclusion principle.

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 primes in P ≤ z. The object of the sieve is to estimate

$S(A,P,z)=|A\setminus \bigcup _{p\in P(z)}{A}_p|.$

We assume that |A_d| may be estimated by

$\left|A_d\right|=\frac{w(d)}{d}X+R_d$

where w is a multiplicative function and X   =   |A|. Let

$W(z)=\prod _{p\in P(z)}(1-\frac{w(p)}{p}).$

Brun's pure sieve

This formulation is from Cojocaru & Murty, Theorem 6.1.2. With the notation as above, assume that

• |R_d| ≤ w(d) for any squarefree d composed of primes in P ;
• w(p) < C for all p in P ;
• $\sum _{p\in P_z}\frac{w(p)}{p}<D\log \log z+E;$

where C, D, E are constants.

Then

$S(A,P,z)=X\cdot W(z)\cdot \left(1+O((\log z{)}^{-b\log b})\right)+O\left(z^{b\log \log z}\right).$

In particular, if log z < c log x / log log x for a suitably small c, then

$S(A,P,z)=X\cdot W(z)(1+o(1)).$

Applications

• Brun's theorem: the sum of the reciprocals of the twin primes converges;
• Schnirelmann's theorem: every even number is a sum of at most C primes (where C can be taken to be 6);
• There are infinitely many pairs of integers differing by 2, where each of the member of the pair is the product of at most 9 primes;
• Every even number is the sum of two numbers each of which is the product of at most 9 primes.

The last two results were superseded by Chen's theorem.

References

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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
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  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
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