# 墨子现代数学

https://v.ixigua.com/RsbM1yc/

Archimedes Fundamental Theorem of Arithmetic

Equivalence:

Proof: 墨子 = Archimedes

From 墨子， we have：

Let N = n+1 , a< N (Archimedes)

# Blockchain Revolution in “Digital Assets” : What is NFT (Non Fungible Token)?

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Proof: Father (God) = Jesus “Father in Me” : <=> Father ⊂ Jesus and “I am in the Father” : <=> Jesus ⊂ Father Hence, Jesus = Father (God) [QED. ] Merry Xmas! 圣诞快乐！ # The amazing power of word vectors For today’s post, I’ve drawn material not just from one paper, but from five! The subject matter is ‘word2vec’ – the work of Mikolov et al. at Google on efficient vector representations of words (and what you can do with them). The papers are: From the first of these papers (‘Efficient estimation…’) we get a description of the Continuous Bag-of-Words and Continuous Skip-gram models for learning word vectors (we’ll talk about what a word vector is in a moment…). 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So far no one seems to have solved the question on the internet yet!

I have given it a try, and will post the solution below!

If you are interested in Math Olympiad, it is a good idea to invest in a good book to learn more tips and tricks about Math Olympiad. One excellent Math Olympiad author is Titu Andreescu, trainer of the USA IMO team. His book 104 Number Theory Problems: From the Training of the USA IMO Team is…

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# How to type Chinese characters in LaTex (on Mac, using TeXShop)

This is one of the easiest ways to type Chinese characters in LaTeX on Mac, using the default TeXShop editor. (If you know of an easier way, please let me know in the comments below!)

I have tried for hours, experimenting with different packages, before “discovering” the following steps. Hope it helps!

Step 1) Add “usepackage{ctex}” to the beginning of the document. This will load the main package ctex.

Step 2) It is very important to save the LaTeX file in UTF-8 format, otherwise all Chinese characters will appear as question marks. The preferred way to do this is via:

TeXShop > Preferences > Encoding = Unicode (UTF-8). (see image below)

This will “permanently” set the format as UTF-8 by default. If you don’t do this, an annoying thing that can happen is that your TeX file reverts to “non-UTF8” upon saving. That means, the Chinese characters may appear correctly…

View original post 160 more words

China 南京航空航天大学 Nanjing University of Aeronautics and Astronautics set the WiFi password as the answer of this integral (first 6 digits).

Can you solve it?

(If can’t, please revise GCE “A-level” / Baccalaureate / 高考 Calculus 微积分)

Answer : Break the integral (I) into 2 parts:

I = A(x) + B(x)

$\displaystyle A(x) = \int_{-2 }^ {2} x^{3}. \cos \frac{x}{2}.\sqrt{4-x^2}dx$

$\displaystyle B(x) = \int_{-2 }^ {2} \frac{1}{2}\sqrt{4-x^2}dx$

A(x) = – A(-x) => Odd function
=> A(x) = 0 since its area canceled out over [-2, 2]

B(x) = B(-x) => Even function
$\displaystyle\implies B(x) = 2\int_{0 }^ {2} \frac{1}{2}\sqrt{4 - x^2}dx$
$\displaystyle\implies B(x) = \int_{0 }^ {2} \sqrt{4 - x^2}dx$

Let x = 2 sin t => dx = 2 cos t. dt

x = 2 = 2 sin t => sin t = 1 => t = π / 2

x = 0 = 2 sin t => sin t = 0 => t = 0

$\displaystyle B(x) = \int_{0 }^ {\pi/2} \sqrt{4 - 4.\sin^{2} {t} }. (2 \cos t. dt)$

$\displaystyle \implies B(x) = \int_{0 }^ {\pi/2} 2.\cos t. (2 \cos t. dt)$

$\displaystyle\implies B(x) = \int_{0 }^ {\pi/2} 4 \cos^{2} t. dt$

$\displaystyle \cos ^{2} t = \frac {1 + \cos 2t} {2 }$

$\displaystyle\implies B(x) = \int_{0 }^ {\pi/2} (2 + 2\cos 2t) . dt$

$\displaystyle\implies B(x) = (2 t) \Bigr|_{0 }^ {\pi/2} + (2. \frac{1}{2} \sin 2t) \Bigr|_{0 }^ {\pi/2}$

$\displaystyle\implies B(x) = (\pi ) + \sin \pi = \pi$

$\boxed{ I = \pi = 3.14159}$

A smarter method using Analytic Geometry: A circle of radius 2 is

$x^2 + y^2 = 4 \implies y = \sqrt {4 -x^2}$