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[March for reflection |Maynard] mindset and method

已有 394 次阅读 2022-7-10 12:07 |个人分类:牛津大学|系统分类:科研笔记

[注:下文是群邮件的内容,标题是原有的。内容是学习一篇数学文章的笔记。]

["Terms of awareness /use" folded below] On going is to read a paper of primes to increase generic understanding on mathematics.

How to measure the mindset of a group of people anyway? 

 ℂ ℍ ℕ ℙ ℚ ℝ ℤ ℭ ℜ I|φ∪∩∈ ⊆ ⊂ ⊇ ⊃ ⊄ ⊅ ≤ ≥ Γ Θ Λ α Δ δ μ ≠ ⌊ ⌋ ∨∧∞Φ⁻⁰ 1

To give a brief indication as to why such a principle might hold, we see that by inclusion-exclution on the smallest prime factor P⁻(n) of n we have

#{ ∈ Ap > x^1/2} = #{ n ∈ A: P⁻(n ) > x^1/4} - Σ(p) #{ n ∈ A: P⁻(n) = }.

---- On the left, it is a counting of primes p in A, such that p is greater than x^1/2.

---- To see this more clearly, one can break the left side into parts ——

---- #{ · } denotes the number of elements in the set of { · }.

---- Especially, `#' denotes the total number of something.

---- That is, `#' is a sign of counting.

---- Then, one may fill the member part of the set ——

---- #{ ∈ A: · }, meaning to count the primes  p in A.

---- Here, p is a representative prime in A.

---- Next, one may focus on the condition part of the set ——

---- #{ · : p > x^1/2}, meaning only the primes p such that p > x^1/2 are counted.

---- In the realm of analysis, the ultimately foundamental object is featured by an inequality.(TOM)

---- Inequality is the symbol of method in analysis. (TOM)  

---- Recall that, the definition of `limit' in real analysis is featured by two inequalities.

---- Recall that, the definition of `probability' is featured by an equality. 

---- That's why one sees an inequality here.

---- Question: why the inequality here takes the form of p > x^1/2 ?

---- Partial answer —— 

---- If an inequality is needed here, the primes have to be involved.

---- As the counting is conducted in A ⊆ [1, x], it appears natural to have x involved.

---- Why x^1/2?

---- Why p > · (instead of p < · ) ?

---- The two specific aspects appear related.

---- If one devides A into two parts by some reference value ξ, it appears the larger part (p > ξ) of A is harder to count.

---- By taking ξ = x^1/2, one removes the minor part of A.

---- Say, for x = 10,000, one has x^1/2 = 100.

---- Primes smaller than 100 are not difficult to count.

---- Question: is there any deep meaning in the inequality p > x^1/2 ?

---- Guess no deep (or special) meaning is endowed in this inequality.

---- The deep thing here is of the introduction of an proper inequality.

---- The specific form might be just out of convenience or convention.

---- Perhaps, ξ = x^1/2 is taken for the sake of convenience.

(to be continued).

.

Exercise: write and read aloud the expression #{ ∈ Ap > x^1/2} thirty times.

 ℂ ℍ ℕ ℙ ℚ ℝ ℤ ℭ ℜ I|φ∪∩∈ ⊆ ⊂ ⊇ ⊃ ⊄ ⊅ ≤ ≥ Γ Θ Λ α Δ δ μ ≠ ⌊ ⌋ ∨∧∞Φ⁻⁰ 1

.

Terms of awareness/ use



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