Author Archives: rpflee

年年齡 10

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We don’t stop playing because we grow old; we grow old because we stop playing.

—

Gross well says that children are young because they play, and not vice versa; and he might have added, men grow old because they stop playing, and not conversely, for play is, at bottom, growth, and at the top of the intellectual scale it is the eternal type of research from sheer love of truth.

—

2012.09.15 Saturday ACHK

Godel’s completeness theorem

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Any proof of the Completeness Theorem consists always of two parts.

First we have show that all formulas that have a proof are tautologies. This implication is also called a Soundness Theorem, or soundness part of the Completeness Theorem.

The second implication says: if a formula is a tautology then it has a proof. This alone is often called a Completeness Theorem. In our case, we call it a completeness part of the Completeness Theorem.

— Cse371, Math371, LOGIC, Fall 2011

— Professor Anita Wasilewska

2012.09.14 Friday ACHK

diff 5

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這段改編自 2010 年 3 月 27 日的對話。

有一種電腦程式，叫做「diff」（差分），作用是比較兩個檔案，顯示它們的差別。例如，《維基百科》的文章不斷被人修改更新。如果你想知道《香港》條目，昨天和今天的版本有何不同，你只要「diff」了兩個版本就可以。亦即是話，「diff」程式會自動顯示兩個版本的差別，而毋須靠你自己「徒手」比較。

「diff」這個動詞概念，即使是對日常生活，也十分有用。例如，去年我和我物理碩士班的一位同學說：「我現在聽課時，再不會把教授講的所有東西，都記錄下來。無論是在心中還是紙上，做筆記時，我只會留意，我已知的和教授講的，有什麼差別。」換句話說，我 diff 了「自己所知」和「教授所講」。一方面，那節省了我大量心神時間。另一方面，那有助我把握重點，加強記憶。

— Me@2012.09.14

2012.09.14 Friday (c) All rights reserved by ACHK

Density matrix, 2

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Well, if you pick a particular state in the Hilbert space, it has a well-defined probability if it’s an eigenstate of the density matrix. This is an unusual operation that’s not usually talked about – because the density matrix isn’t an “observable” in the usual sense – like positions or momenta etc. But it’s still an operator on the Hilbert space. I will formally treat the density matrix rho as the “operator for the probability”.

— Density matrix and its classical counterpart

— Lubos Motl

2012.09.13 Thursday ACHK

Consistency

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There exist two definitions of consistency: semantical and syntactical.

Semantical definition uses the notion of a model and says:

a set is consistent if it has a model.

Syntactical definition uses the notion of provability and says:

a set is consistent if one can’t prove a contradiction from it.

— Cse371, Math371, LOGIC, Fall 2011

— Professor Anita Wasilewska

2012.09.12 Wednesday ACHK

Imaginary mass

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Complex mass and decay rate

The mass of an unstable particle is formally a complex number, with the real part being its mass in the usual sense, and the imaginary part being its decay rate in natural units.

When the imaginary part is large compared to the real part, the particle is usually thought of as a resonance more than a particle.

For a particle of mass , the particle can travel for time 1/M, but decays after time of order of . If > M then the particle usually decays before it completes its travel.

— Wikipedia on Particle decay

2012.09.11 Tuesday ACHK

無限年

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蘋果 與 Apple, 1.2 | Electric Field and Electric Potential, 1.2 | 物理語言 1.2

這段改編自 2010 年 6 月 9 日的對話。

你在讀數學和物理時遇到的困難，有時，其實不是「數學問題」或者「物理問題」，而是「語言問題」。

例如，如果我不跟你說，「electric field」（電場）和「electric potential」（電勢），其實是描述同一類物理現象的兩種不同語言，你可能一直會以為，它們是兩種不同的東西，繼而反覆思考它們的關係，庸人自擾一番。

所以，下次在 數學、物理 或 其他科 遇到疑惑時，你要知道，那很大機會是「語言問題」。只要把相關問題的重要字眼釐清，難題的神秘感往往會自然消失。

又例如，讀「微積分」時，「無限」好像十分驚嚇。其實，並不是真的有一樣東西，或者一個數字，叫做「無限」。它只是一個語言的技巧，說話的方法。每逢我們說「無限」時，即是沒有那樣東西。

例如，甲問乙：「你欠我的錢，什麼時候會歸還呢？」

乙答：「無限年之後。」

乙的意思，並不是真的有一個時間長度，叫做「無限年」。他等「無限年」之後，會把錢還給甲。乙的真正意思是，他不肯還錢。

再例如，「無人跑得快過我」，並不是指有一個人名叫「無人」，他跑得快過我。「無人」只是一種說話的方法。「無人跑得快過我」的真正意思是，「我是所有人之中，跑得最快的。」

— Me@2012.09.10

2012.09.10 Monday (c) All rights reserved by ACHK

香港科技, 2

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這段改編自 2010 年 3 月 27 日的對話。

美國 Silicon Valley（矽谷）的 intellectual power（才智實力），其實是建基於 科學、科技 和 科幻 的文化。而這類文化，是累積多年而得來的，而不是只靠大量金錢，在短短幾年間，就可以得到。

James C. Collins 在他的《Good to Great》中，用了 flywheel（飛輪）作比喻。香港見到矽谷的飛輪轉得很快，於是買了一個飛輪回來，企圖聘請很多「大力士」把它轉動至高速。殊不知，即使有很多「大力士」，飛輪都是極之難轉，加速得奇慢。別人的飛輪可以轉得那樣快，是因為別人的「大力士」，已為飛輪加速了幾十年。

— Me@2012.09.08

The Flywheel: The additive effect of many small initiatives; they act on each other like compound interest.

— Wikipedia on Good to Great

2012.09.08 Saturday (c) All rights reserved by ACHK

Quantum Mechanics

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In my current opinion, each of the three alternative dynamical equations

  •     equation for the density matrix
  •     Heisenberg equations for the operators
  •     Feynman’s path integral

is more pedagogical when it comes to the understanding of the actual relationship between quantum mechanics and classical physics.

— Density matrix and its classical counterpart

— Lubos Motl

2012.09.08 Saturday ACHK

蘋果 與 Apple, 2.3

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Electric Field and Electric Potential, 2.3 | 物理語言 2.3

這段改編自 2010 年 6 月 9 日的對話。

但是，表示 electric field（電場）時，如果真的在空間上的每一點，也畫一支箭的話，會十分麻煩。而且，那些箭會互相重疊，令你看不清楚每一支箭的長度。不知道箭的長度，就即是不知道空間上每一點，對應的 electric field 強度。

所以，物理學家把原本「箭的長度，代表乙所感受到的力的大小；箭的方向，代表力的方向」的計劃，修改了一點。現在，所有箭也是無限長，稱為 electric field lines 電力場線，或者「電力線」。箭的方向，仍然代表電荷乙所感受到的力的方向。而力的大小，則改為以 electric field lines 的密度來代表。

This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication.

— Me@2012.09.07

2012.09.07 Friday (c) All rights reserved by ACHK

Audiosurf

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Audiosurf is a puzzle/rhythm hybrid game created by Invisible Handlebar, a personal company created by Dylan Fitterer. Its track-like stages visually mimic the music the player chooses, while the player races across several lanes collecting colored blocks that appear in sync with the music.

— Wikipedia on Audiosurf

2012.09.06 Thursday ACHK

裝睡的人

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它要塑造的，不是鐵屋中沉睡的人，而是裝睡的人。

因為，沉睡的人，你總有辦法去喚醒。

但是，你永遠沒有辦法喚醒一個裝睡的人！

— 胡清心

2012.09.06 Thursday ACHK

香港科技

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這段改編自 2010 年 3 月 27 日的對話。

（安：你覺得香港有什麼主要的問題？）

很少人對 intellectual（智性）的東西有興趣。當然，其他地方的人都可能是那樣。這大概是整個地球的問題。

— Me@2012.09.06

2012.09.06 Thursday (c) All rights reserved by ACHK

Planck units, 2

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However, most Planck units are many orders of magnitude too large or too small to be of any practical use, so that Planck units as a system are really only relevant to theoretical physics. In fact, 1 Planck unit is often the largest or smallest value of a physical quantity that makes sense according to our current understanding.

For example:

  • A speed of 1 Planck length per Planck time is the speed of light in a vacuum, the maximum possible speed in special relativity;
  • Our understanding of the Big Bang begins with the Planck Epoch, when the universe was 1 Planck time old and 1 Planck length in diameter, and had a Planck temperature of 1. At that moment, quantum theory as presently understood becomes applicable. Understanding the universe when it was less than 1 Planck time old requires a theory of quantum gravity that would incorporate quantum effects into general relativity. Such a theory does not yet exist;   
  • At a Planck temperature of 1, all symmetries broken since the early Big Bang would be restored, and the four fundamental forces of contemporary physical theory would become one force.

— Wikipedia on Planck units

   
   
2012.09.05 Wednesday ACHK