SciTech-Mathmatics-因式分解定理:UNIQUE FACTORIZATION THEOREM + Science Hackathon Prizes@Kaggle.com

SciTech-Mathmatics-UNIQUE FACTORIZATION THEOREM

因式分解趣谈:

一元 多项式方程：

解一元N次方程，其实就是对 一元N次方程“因式分解”的过程，
“数域”：

• Rational有理数域(+-, /), “数域”对 "+" 及 "" 及其"逆运算"的包容；
  XX - 1 = 0 => (X+1)(X-1)=0 => X=+1 或 X=-1
• Real实数域(+-, /, ^), “数集扩充的思想”; 将全体无理数“完备性”的纳入解方程;
  X^2 - 2 = 0 => (X+√2)(X-√2)=0 => X=+√2 或 X=-√2
• Complex复数域(+-, /, ^√), Gauss高斯提出的指数形式Complex 及 Radius模长 和 Angel幅角;
  X^2 + 1 = 0 => (X+i)(X-i) = 0 => X=+i 或 X=-i
  Complex 的 Radius模长 和 Angel幅角：
  对“乘法”运算时， 模长相乘，幅角相加(顺时针旋转),
  对“/除法”运算时，模长相除，幅角相减(逆时针旋转).
  重要的代数学理论：任何一个“复数域”上的方程式，复数域上一定有解。

多元 多项式方程组

解 K元N次方程组，大体也是对 K元N次方程组进行“因式分解”的过程：
例如: 解“二次型”(K元2次/齐次方程组)时用到的“配方法”或 高等代数的矩阵的“成对行列变换方法”就是很好示范；
统一建立模型为 K元N齐次方程组，使用“配方法”或 高等代数的矩阵的“成对行列变换方法”；

因式分解的理论:

https://brilliant.org/wiki/fundamental-theorem-of-arithmetic/

Elementary_Number_Theory:
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Clark)/01%3A_Chapters/1.11%3A_Unique_Factorization

https://public.csusm.edu/aitken_html/m422/Handout2.pdf

https://www.oxfordreference.com/display/10.1093/oi/authority.20110803110719647

https://math.stackexchange.com/questions/203383/proof-of-gcda-b-axby-bezouts-identity

UNIQUE FACTORIZATION THEOREM

In general, Every integer \(\large n > 1\) can be written UNIQUELY in the form:
\(\large \begin{array}{ccl} n &=& P_{1}^{a_1} \cdot P_{1}^{a_1} \cdot ... \cdot P_{s}^{a_s}\ \\ &=& \prod_{i=1}^{s} P_{i}^{a_i}\ \\ &where&\ P_1 < P_2 < ... < P_s \leq n\ ,\ \ P_i \ is\ PRIME\ ,\ \\ & &\ a_i \geq 1\ ,\ for\ all\ i\ \in [\ 1, s\ ]\ ,\ \ s \in\ [\ 1, n\ ] \cap Z^+\ . \\ \end{array}\)

Unique Factorization Theorem

https://brilliant.org/wiki/fundamental-theorem-of-arithmetic/
The fundamental theorem of arithmetic (FTA), also called the unique-prime-factorization theorem,
states that every integer greater than than 1:

• either is prime itself
• or is the product of a unique combination of prime numbers.

Music Hackathon: EMI Music Data Science Hackathon

https://www.kaggle.com/c/MusicHackathon
Music Data Science Hackathon Prizes (sponsored by EMI and EMC), - July 21st - 24 hours

• 1st Global Prize – £2,500
• 2nd Global Prize – £1,000
• 3rd Global Prize – £500
• London Venue Prize - £2,000
• Data Visualization Prize - £500 (sponsored by Adatis)

fastFM: A Library for Factorization Machines

https://jmlr.csail.mit.edu/papers/volume17/15-355/15-355.pdf

pyFM: Factorization Machines in Python

https://github.com/coreylynch/pyFM

polylearn: A library for factorization machines and polynomial networks

mainly for classification and regression in Python.
https://contrib.scikit-learn.org/polylearn/index.html

Maximum number of prime factors a number can have with exactly x factors

https://www.geeksforgeeks.org/maximum-number-of-prime-factors-a-number-can-have-with-exactly-x-factors/