The Spirit of Information Theory | 2017-06-28

“ISIT 2017 | David Tse |

5G Polar Codes Dr. Erdal Arikan

Dr. Erdal Arikan (1958 -) of Bilkent University, Turkey, recieved his PhD in MIT at the age of 27.

Huawei Award: The Father of Polar Codes (极化码)

Industry watchers believe that the selection of polar coding as the channel coding technique for control channels for 5G communications system may have put Huawei at the forefront in the 5G race (27 Gbps in 2016). 1 Gbps = 1000 Mbps

5G Basic Explained:

Part 1: The Math behind 5G Polar Codes

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Part 2:

Part 3:

Notations:

.. $\displaystyle F = \begin {bmatrix} 1 & 0\\ 1 & 1 \end{bmatrix}$ $A = \begin{pmatrix} a_{11} & a_{12}& \ldots & a_{1n}\\ a_{21} & a_{22} & \ldots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{n1}& a_{n2} &\ldots & a_{nn} \end{pmatrix}$ $B = \begin{pmatrix} b_{11} & b_{12}& \ldots & b_{1n}\\ b_{21} & b_{22} & \ldots & b_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ b_{n1}& b_{n2} &\ldots & b_{nn} \end{pmatrix}$ $\displaystyle F^{{\otimes}{n}} = F^{{\otimes}{(n-1) }} \otimes F = \begin {bmatrix} F^{{\otimes}{(n-1) }} & 0\\ F^{{\otimes}{(n-1) }} & F^{{\otimes}{(n-1) }} \end{bmatrix}$ $A \otimes B = \begin {pmatrix} a_{11}B & a_{12}B & \ldots & a_{1n}B\\ a_{21}B& a_{22}B & \ldots & a_{2n}B\\ \vdots & \vdots & \ddots & \vdots\\ a_{n1}B & a_{n2}B &\ldots & a_{nn}B \end{pmatrix}$

Singapore Math’s “CPA” Pedagogy Leads to World’s PISA Top # 1 Rank

【我们引以为傲的数学，竟然被新加坡孩子用这种画图法打败了】复制这条信息€13Rdg€5zXHMb€后打开👉今日头条极速版👈 CPA = Concrete – Picture – Abstract  Singapore Math = Chinese Math + English (UK/USA) Math

Chinese Math = Spoon-feeding 填鸭式|Computational focus | Applied |Memorization

English Math = Building-block 建构式|Practical focus |Less Theoretical

Comment:

Singapore Math inherits both the strengths as well as the weaknesses of Chinese Math & English Math:

Strengths: Applied, Practical, Building-block. (Up to secondary school math level, good for daily math usage).

Weaknesses : Computational focus, Less Theoretical. (Bad for abstract thinking in further study of advanced math).

Singapore Math is only good for kids from 7 to 15 years old (PISA age is 15). From upper secondary Math (at 16) to A-level or Baccalaureat(at 18) or 中国高一～高三, the French Math is better for laying foundation in abstract thinking, paving the way for advanced math in university & beyond – evidenced by the majority of Fields Medalists are French. (Read “How do the French Excel in Math” )

Python in Functional Programming Style

Functional Programming has the following key styles:

1) Lambda function:

2) Map, Filter, Reduce

Why the Proof of Fermat’s Last Theorem Doesn’t Need to Be Enhanced  … (Read on) from source :

https://www.quantamagazine.org/why-the-proof-of-fermats-last-theorem-doesnt-need-to-be-enhanced-via20190603/

Summary:

Andrew Wiles’ Proof of Fermat’s Last Theorem (FLT) by contradiction :

A. Assume FLT is true for all prime p (Why? sufficient to prove only for prime) such that: $a^p + b^p = c^p$

B. then a, b, c could be rearranged into an Elliptic Curve,

C. then leverage such Elliptic Curve into a Galois Represebtation.

D. then a Modular Form.

E. then leads to an impossible weight 2 level 2 Modular Form.

Hence,

¬E -> ¬D -> ¬C -> ¬B -> ¬A (proved)

1950s Taniyama-Shimura-Weil proved the link below:

B -> (via assume C) -> D

Andrew Wiles’ took 7 years to complete the whole proof in 1994 by proving the missing link C -> D.