BOP tuning: Difference between revisions

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where the notation on the left will be explained shortly.

Now, we note that this unusually weighted L1 norm does not provide the weighting we want on all rationals, which is <math>(nd)^s</math>. Rather, when <math>s=1</math>, this is called the [[Wilson Height|Wilson norm]], and is equivalent to the [http://mathworld.wolfram.com/SumofPrimeFactors.html sum of prime factors with repetition] of the ratio <math>n/d</math> in question, often written <math>\text{sopfr}(n/d)</math>. For arbitrary <math>s\geq 1</math>, this is the sum of the <math>s</math>'th ~~power~~ of the prime factors with repetition of the ratio, which we will denote <math>\text{sopfr}^s(n/d)</math>. However, we note that this strange weighting ''does'' equal the weighting we want on the primes, where we have <math>\text{sopfr}^s(n/d) = (n/d)^s</math> - it only diverges on the composite numbers.

Now, we note that this unusually weighted L1 norm does not provide the weighting we want on all rationals, which is <math>(nd)^s</math>. Rather, when <math>s=1</math>, this is called the [[Wilson Height|Wilson norm]], and is equivalent to the [http://mathworld.wolfram.com/SumofPrimeFactors.html sum of prime factors with repetition] of the ratio <math>n/d</math> in question, often written <math>\text{sopfr}(n/d)</math>. For arbitrary <math>s\geq 1</math>, this is the sum of the <math>s</math>'th powers of the prime factors with repetition of the ratio, which we will denote <math>\text{sopfr}^s(n/d)</math>. However, we note that this strange weighting ''does'' equal the weighting we want on the primes, where we have <math>\text{sopfr}^s(n/d) = (n/d)^s</math> - it only diverges on the composite numbers.

Revision as of 07:22, 4 May 2019 view source Mike Battaglia (talk \| contribs) autopatrolled, Bureaucrats, confirmaccount-notify, emailconfirmed, Interface administrators, Sysops 1,399 edits →Proof of Benedetti-Optimality On All Rationals ← Older edit		Revision as of 07:24, 4 May 2019 view source Mike Battaglia (talk \| contribs) autopatrolled, Bureaucrats, confirmaccount-notify, emailconfirmed, Interface administrators, Sysops 1,399 edits →Proof of Benedetti-Optimality On All Rationals: typo Newer edit →
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	where the notation on the left will be explained shortly.		where the notation on the left will be explained shortly.

	Now, we note that this unusually weighted L1 norm does not provide the weighting we want on all rationals, which is <math>(nd)^s</math>. Rather, when <math>s=1</math>, this is called the [[Wilson Height\|Wilson norm]], and is equivalent to the [http://mathworld.wolfram.com/SumofPrimeFactors.html sum of prime factors with repetition] of the ratio <math>n/d</math> in question, often written <math>\text{sopfr}(n/d)</math>. For arbitrary <math>s\geq 1</math>, this is the sum of the <math>s</math>'th ~~power~~ of the prime factors with repetition of the ratio, which we will denote <math>\text{sopfr}^s(n/d)</math>. However, we note that this strange weighting ''does'' equal the weighting we want on the primes, where we have <math>\text{sopfr}^s(n/d) = (n/d)^s</math> - it only diverges on the composite numbers.		Now, we note that this unusually weighted L1 norm does not provide the weighting we want on all rationals, which is <math>(nd)^s</math>. Rather, when <math>s=1</math>, this is called the [[Wilson Height\|Wilson norm]], and is equivalent to the [http://mathworld.wolfram.com/SumofPrimeFactors.html sum of prime factors with repetition] of the ratio <math>n/d</math> in question, often written <math>\text{sopfr}(n/d)</math>. For arbitrary <math>s\geq 1</math>, this is the sum of the <math>s</math>'th powers of the prime factors with repetition of the ratio, which we will denote <math>\text{sopfr}^s(n/d)</math>. However, we note that this strange weighting ''does'' equal the weighting we want on the primes, where we have <math>\text{sopfr}^s(n/d) = (n/d)^s</math> - it only diverges on the composite numbers.