Problem 14.4b1.4

Posted on May 24, 2018 by rpflee

Closed string degeneracies | A First Course in String Theory

What is the meaning of “With $a = 1, ..., 8$ and $\bar b = \bar 1, ..., \bar 8$ , …”?

— Me@2015.09.14 12:11 PM

p.315 “Explicitly, the eight states $| R_a \rangle, a = 1, 2, ..., 8$ , with an even number of creation operators are … ”

p.316 “The eight states $|R_{\bar{a}} \rangle, \bar a = \bar 1, \bar 2, ..., \bar 8$ , with an odd number of creation operators are … ”

— Me@2018-05-24 11:41:34 AM

Life, 3

Posted on May 23, 2018 by rpflee

生命 3

We exist in time because time is change.

Growing is part of the definition of life. Growing is a kind of change.

Also, without time/change, there would be no thinking and no thoughts.

— Me@2017-12-26 11:42 am

— Me@2018-05-23 10:05:03 PM

time ~ change

Time is logically necessary if change is necessary.

— Me@2018-02-04 09:07:48 PM

生命 2

Posted on May 22, 2018 by rpflee

life ~ changeable

time ~ change

— Me@2017-09-26 07:23:59 AM

多項選擇題 6

Posted on May 22, 2018 by rpflee

Multiple Choices 6

這段改編自 2010 年 8 月 24 日的對話。

有時，一題物理 MC（多項選擇題）會，同時有數學做法和物理做法。

那時，你就先用物理方法做一次，再用數學方法做多一次，以作驗算。

（問：哪有那麼多的時間？）

之前講過，那些做法，必須透過考試前，平日多加收集和練習而來；並不是在考試中途，才花額外時間發明。

— Me@2018-05-22 06:02:40 PM

Quantum Computing, 2

Posted on May 17, 2018 by rpflee

stcredzero 3 months ago

A note for the savvy: A quantum computer is not a magic bit-string that mysteriously flips to the correct answer. A n-qubit quantum computer is not like 2^n phantom computers running at the same time in some quantum superposition phantom-zone. That’s the popular misconception, but it’s effectively ignorant techno-woo.

Here’s what really happens. If you have a string of n-qubits, when you measure them, they might end up randomly in [one] of the 2^n possible configurations. However, if you apply some operations to your string of n-qubits using quantum gates, you can usefully bias their wave equations, such that the probabilities of certain configurations are much more likely to appear. (You can’t have too many of these operations, however, as that runs the risk of decoherence.) Hopefully, you can do this in such a way, that the biased configurations are the answer to a problem you want to solve.

So then, if you have a quantum computer in such a setup, you can run it a bunch of times, and if everything goes well after enough iterations, you will be able to notice a bias towards certain configurations of the string of bits. If you can do this often enough to get statistical significance, then you can be pretty confident you’ve found your answers.

— An Argument Against Quantum Computers

— Hacker News

2018.05.17 Thursday ACHK

Van der Waals equation 1.2

Posted on May 16, 2018 by rpflee

Whether $X_{\text{measured}}$ is bigger or smaller than $X_{\text{ideal}}$ ultimately depends on the assumptions and definitions used in the derivation of the ideal gas equation itself.

In the ideal gas equation derivation, the volume used in the equation refers to the volume that the gas molecules can move within. So

$V_{\text{ideal}} = V_{\text{available for a real gas' molecules to move within}}$

Then, when deriving the pressure, it is assumed that there are no intermolecular forces among gas molecules. So

$P_{\text{ideal}} = P_{\text{assuming no intermolecular forces}}$

These are the reasons that

$V_{\text{ideal}} < V_{\text{measured}}$

$P_{\text{ideal}} > P_{\text{measured}}$

—

$P_{\text{ideal}} V_{\text{ideal}} = nRT$

$\left(P_\text{measured} + a\left(\frac{n}{V}\right)^2\right) \left(V_\text{measured}-nb\right) = nRT$

— Me@2018-05-16 07:12:51 PM

~~~

… the thing to keep in mind is that the “pressure we use in the ideal gas law” is not the pressure of the gas itself. The pressure of the gas itself is too low: to relate that pressure to “pressure for the ideal gas law” we have to add a number. While the volume occupied by the real gas is too large – the “ideal volume” is less than that. – Floris Sep 30 ’16 at 17:34

— Physics Stackexchange

Tree rings, 2

Posted on May 16, 2018 by rpflee

This file is licensed under the Creative Commons Attribution 2.0 Generic license. Author: Lawrence Murray from Perth, Australia

Time-traveling to the past is like “making an outside ring more inside”, which is logically impossible.

— Me@2011.09.18

Me@2010

Richard Stallman

Posted on May 15, 2018 by rpflee

I’ve always lived cheaply. I live like a student, basically. And I like that, because it means that money is not telling me what to do. I can do what I think is important for me to do. It freed me to do what seemed worth doing. So make a real effort to avoid getting sucked into all the expensive lifestyle habits of typical Americans. Because if you do that, then people with the money will dictate what you do with your life. You won’t be able to do what’s really important to you.

— Richard Stallman

There is nothing wrong to be a student-having a lot of new learnings and new young friends, as long as you can earn enough money.

— Me@2011.08.20

On Keeping Your Soul, 3

Posted on May 15, 2018 by rpflee

這段改編自 2010 年 4 月 18 日的對話。

「學術研究的論文」和「網誌文章」的主要分別是，論文要花很多時間，才可寫成一篇，然後只有極少人去閱讀：

時間 ~ 1 至 2 年

潛在讀者人數 ~ 10 至 100 人

網誌文章則相反：

時間 ~ 1 至 2 星期

潛在讀者人數 ~ > 100 人

不過，曲高自然和寡：

如果一篇網誌文章的學術成份高的話，閱讀人數自然極少。

— Me@2018-05-15 05:00:47 PM

後記：

英雄所見略同，物理學家 Lubos Motl，有近乎相同的見解。

— Me@2018-05-15 05:00:55 PM

Ask users

Posted on May 14, 2018 by rpflee

gallerdude 3 months ago

There’s a good moral here. Everytime they had a question, they asked their users. Users don’t lie.

atYevP 3 months ago

Yev from Backblaze here -> That’s right! For better or worse they usually tell it like they see it, and that helps us inform decisions!

— How Backblaze Got Started (2017)

— Hacker News

2018.05.14 Monday ACHK

Problem 14.4b1.3

Posted on May 14, 2018 by rpflee

Closed string degeneracies | A First Course in String Theory

(b) State the values of $\alpha' M^2$ and give the separate degeneracies of bosons and fermions for the first five mass levels of the type IIA closed superstrings. Would the answer have the different for type IIB?

~~~

…

— This answer is my guess. —

Since for NS, the first 5 levels’ degeneracies are 8, 36, 128, 402, 1152, the degeneracies of (NS, NS) are $8^2, 36^2, 128^2, 402^2, 1152^2$ .

This is incorrect, for there are no (NS, NS) states. Instead, you should consider (NS+, NS+).

Since for NS+, the first 5 levels’ degeneracies are 8, 128, 1152, 7680, 42112, the degeneracies of (NS+, NS+) are $8^2, 128^2, 1152^2, 7680^2, 42112^2$ .

p.317 Consider the relationship of the degeneracy of R+ and that of R-:

How about the first 5 levels of R+?

The degeneracies are the same as those of R-.

p.317 Equation (14.54) “The appearance of an equal number of bosonic and fermionic states at every mass level is a signal of supersymmetry. This is, however, supersymmetry on the world-sheet.”

Equation (14.71):

$f_{R-}(x) = 8 + 128 x + 1152 x^{2} + 7680 x^{3} + 42112 x^{4} + ...$

p.321 “Indeed, the integer mass-squared levels in the NS generating function (14.67) have degeneracies that match those of (14.71) for the R- sector.”

— This answer is my guess. —

— Me@2018-05-14 02:51:55 PM

Van der Waals equation 1.1

Posted on May 13, 2018 by rpflee

Why do we add, and not subtract, the correction term for pressure in [Van der Waals] equation?

Since the pressure of real gases is lesser than the pressure exerted by (imaginary) ideal gases, shouldn’t we subtract some correction term to account for the decrease in pressure?

I mean, that’s what we have done for the volume correction: Subtracted a correction term from the volume of the container V since the total volume available for movement is reduced.

asked Sep 30 ’16 at 15:20
Ram Bharadwaj

— Physics Stackexchange

Ideal gas law:

$P_{\text{ideal}} V_{\text{ideal}} = nRT$

However, since in a real gas, there are attractions between molecules, so the measured value of pressure P is smaller than that in an ideal gas:

$P_{\text{measured}} = P_{\text{real}}$

$P_{\text{measured}} < P_{\text{ideal gas}}$

Also, since the gas molecules themselves occupy some space, the measured value of the volume V is bigger that the real gas really has:

$V_{\text{measured}} > V_{\text{real}}$

—

$P_{\text{ideal}} V_{\text{ideal}} = nRT$

If we substitute $P_{\text{measured}}$ onto the LHS, since $P_{\text{measured}} < P_{\text{ideal}}$ , the LHS will be smaller than the RHS:

$P_{\text{measured}} V_{\text{ideal}} < nRT$

So in order to maintain the equality, a correction term to the pressure must be added:

$\left(P_\text{measured} + a\left(\frac{n}{V}\right)^2\right) V_{\text{ideal}} = nRT$

—

$P_{\text{ideal}} V_{\text{ideal}} = nRT$

If we substitute $V_{\text{measured}}$ onto the LHS, since that volume is bigger that actual volume available for the gas molecules to move, the LHS will be bigger than the RHS:

$P_{\text{ideal}} V_{\text{measured}} > nRT$

So in order to maintain the equality, a correction term to the pressure must be subtracted:

$P_{\text{ideal}} \left(V_\text{measured}-nb\right) = nRT$

In other words,

$V_{\text{measured}} > V_{\text{real}}$

$V_{\text{ideal}} = V_{\text{real}}$

$V_{\text{measured}} > V_{\text{ideal}}$

— Me@2018-05-13 03:37:18 PM

Why? I still do not understand.

How come

$P_{\text{measured}} = P_{\text{real}}$

but

$V_{\text{measured}} \ne V_{\text{real}}$ ?

How come

$V_{\text{real}} = V_{\text{ideal}}$

but

$P_{\text{real}} \ne P_{\text{ideal}}$ ?

— Me@2018-05-13 03:22:54 PM

The above is wrong.

The “real volume” $V_{\text{real}}$ has 2 possible different meanings.

One is “the volume occupied by a real gas”. In other words, it is the volume of the gas container.

Another is “the volume available for a real gas’ molecules to move”.

To avoid confusion, we should define

$V_{\text{real}} \equiv V_{\text{measured}}$

$P_{\text{real}} \equiv P_{\text{measured}}$

Or even better, avoid the terms $P_{\text{real}}$ and $V_{\text{real}}$ altogether. Instead, just consider the relationship between $(P_{\text{ideal}}, P_{\text{measured}})$ and that between $(V_{\text{ideal}}, V_{\text{measured}})$ .

Whether $X_{\text{measured}}$ is bigger or smaller than $X_{\text{ideal}}$ ultimately depends on the assumptions and definitions used in the derivation of the ideal gas equation itself.

…

— Me@2018-05-13 04:15:34 PM

Bus Stop 3

Posted on May 12, 2018 by rpflee

If you are the smartest person in the room, then you are in the wrong room.

— Confucius

2018.05.12 Saturday ACHK

時空兌換率

Posted on May 11, 2018 by rpflee

這段改編自 2015 年的對話。

我的相對論教授說，所謂

$E = m c^2$ ，

在某些意思之下，沒有那麼特別，因為，你可以把它看成，貨幣的兌換。

$E = c^2 m$

能量 =（光速二次方） $\times$ 質量

$1 \text{USD} \approx 8 \times 1 \text{HKD}$

1 美元 $\approx 8 \times$ 1 港元

公式中的 $c^2$ （光速平方），角色其實正正就是，能量 E 和質量 m 之間的「貨幣兌換率」。

（而光速 c，則是時間和空間的兌換率。）

— Me@2018-05-11 09:10:00 PM

Sleep apnea

Posted on May 9, 2018 by rpflee

erentz 3 months ago

Slightly tangential to the study here but I’ve been going down the sleep apnea rabbit hole in recent months and I strongly encourage folks to investigate their sleep. A lot of people have sleep apnea and don’t realize it. You don’t have to snore to have it. A surprising statistic I found was 20-30% of people with ADHD have sleep apnea. A lot of people may be treating symptoms of sleep apnea like ADHD and high blood pressure with medications while ignoring the root cause. My experience with this has been that doctors are surprisingly ignorant. They’ll happy prescribe you medications for anxiety, ADHD, blood pressure for years, and never think to ask you about your sleep. Do some of your own investigations or ask about it if you have any suspicions.

copperx 3 months ago

I’m going to add myself as a data point. After suffering years of feeling sleepy regardless of how much I slept and frequently unmotivated and withdrawn, I was diagnosed with severe sleep apnea. A few years ago I was given Dexedrine, Ritalin, Adderall, and I even tried self-medicating with modafinil to ameliorate the symptoms of what doctors thought was ADHD with varying levels of success; but the drug-free treatment of sleep apnea with a BiPAP machine got rid of all of these issues; in addition, I feel about ten years younger.

— Sleep and Mortality: A Population-Based 22-Year Follow-Up Study (2007)

— Hacker News

2018.05.09 Wednesday ACHK

Problem 14.4b1.2

Posted on May 8, 2018 by rpflee

Closed string degeneracies | A First Course in String Theory

~~~

…

NS+ equations of (14.38):

$\alpha'M^2=0,$	$~~N^\perp = \frac{1}{2}:$	$~~~~b_{-1/2}^I \|NS \rangle \otimes \|p^+, \vec p_T \rangle,$
$\alpha'M^2=1,$	$~~N^\perp = \frac{3}{2}:$	$~~~~\{ \alpha_{-1}^I b_{\frac{-1}{2}}^J, b_{\frac{-3}{2}}^I, b_{\frac{-1}{2}}^I b_{\frac{-1}{2}}^J b_{\frac{-1}{2}}^K \} \|NS \rangle \otimes \|p^+, \vec p_T \rangle,$

$\alpha'M^2=2,$	$~~N^\perp = \frac{5}{2}:$	$~~~~\{\alpha_{-2}^I b_{\frac{-1}{2}}^J, \alpha_{-1}^I \alpha_{-1}^J b_{\frac{-1}{2}}^K, \alpha_{-1}^I b_{\frac{-3}{2}}^J, \alpha_{-1}^I b_{\frac{-1}{2}}^J b_{\frac{-1}{2}}^K b_{\frac{-1}{2}}^M, ...\}$
		$\{ ..., b_{\frac{-5}{2}}^I, b_{\frac{-3}{2}}^I b_{\frac{-1}{2}}^J b_{\frac{-1}{2}}^K, b_{\frac{-1}{2}}^I b_{\frac{-1}{2}}^J b_{\frac{-1}{2}}^K b_{\frac{-1}{2}}^M b_{\frac{-1}{2}}^N \} \|NS \rangle \otimes \|p^+, \vec p_T \rangle,$
…	…	…

For $N^\perp = \frac{5}{2}$ , the number of states is

$8^2 + \left[ \frac{(8)(7)}{2!} + 8 \right] (8) + 8^2$
$+ 8 + \frac{(8)(7)(6)}{3!} + 8 + (8) \left[ \frac{(8)(7)}{2!} \right] + \frac{(8)(7)(6)(5)(4)}{5!}$
$= 1152$

Since $\alpha' M^2 = N^\perp - \frac{1}{2}$ , when $N^\perp = \frac{5}{2}$ , $\alpha' M^2 = 2$ .

Equation (14.67):

$f_{NS} (x) = \frac{1}{\sqrt{x}} + 8 + 36 \sqrt{x} + 128 x + 402 x \sqrt{x} + 1152 x^2 + ...$

Equation (14.66):

$f_{NS} (x) = \frac{1}{\sqrt{x}} \prod_{n=1}^\infty \left( \frac{1+x^{n-\frac{1}{2}}}{1-x^n} \right)^8$

p.321

If we take $f_{NS} (x)$ in (14.66) and change the sign inside each factor in the numerator

Equation (14.72):

$\frac{1}{\sqrt{x}} \prod_{n=1}^\infty \left( \frac{1-x^{n-\frac{1}{2}}}{1-x^n} \right)^8$

the _only_ effect is changing the sign of each term in the generating function whose states arise with an odd number of fermions.

$\prod_{n=1}^\infty \left( \frac{1}{1-x^n} \right)^8$ is the boson contribution.

$\prod_{n=1}^\infty \left( 1+x^{n-\frac{1}{2}} \right)^8$ is the fermion contribution.

Turning Equation (14.67) into

$f_{NS?} (x)$
$= - \frac{1}{\sqrt{x}} + 8 - 36 \sqrt{x} + 128 x - 402 x \sqrt{x} + 1152 x^2 - ...$

is equivalent to turning all $\sqrt{x}$ into $- \sqrt{x}$ :

$f_{NS?} (x)$
$= \frac{1}{\sqrt{x}} \prod_{n=1}^\infty \left( \frac{1-x^{n-\frac{1}{2}}}{1-x^n} \right)^8$

Me@2015.08.29 12:49 PM: Somehow, $\sqrt{x}$ represents “contribution from fermions”.

Me@2015.08.29 12:50 PM: If you still cannot understand, try replace all $\sqrt{x}$ with $y$ .

$f_{NS+} (x) = \frac{1}{2} \left( f_{NS} - f_{NS?} \right)$

$f_{NS} (x)$
$= \frac{1}{\sqrt{x}} + 8 + 36 \sqrt{x} + 128 x + 402 x \sqrt{x} + 1152 x^2 + ...$

$f_{NS?} (x)$
$= - \frac{1}{\sqrt{x}} + 8 - 36 \sqrt{x} + 128 x - 402 x \sqrt{x} + 1152 x^2 - ...$

$f_{NS+} (x)$
$= \frac{1}{2} \left( f_{NS} - f_{NS?} \right)$
$= \frac{1}{\sqrt{x}} + 36 \sqrt{x} + 402 x \sqrt{x} + ...$

It is _not_ correct. Just consider it as $\left(\sqrt{x} \to -\sqrt{x} \right)$ is not correct, since the beginning factor $\frac{1}{\sqrt{x}}$ is not considered yet.

Instead, we should present in the following way:

$f_{NS} (x)$
$= \frac{1}{\sqrt{x}} \prod_{n=1}^\infty \left( \frac{1+x^{n-\frac{1}{2}}}{1-x^n} \right)^8$
$= \frac{1}{\sqrt{x}} g_{NS}(x)$
$= \frac{1}{\sqrt{x}} + 8 + 36 \sqrt{x} + 128 x + 402 x \sqrt{x} + 1152 x^2 + ...$

$g (\sqrt{x})$
$= \prod_{n=1}^\infty \left( \frac{1+x^{n-\frac{1}{2}}}{1-x^n} \right)^8$
$= 1 + 8 \, \sqrt{x} + 36 \, x + 128 \, x^{\frac{3}{2}} + 402 \, x^{2} + 1152 \, x^{\frac{5}{2}} + 3064 \, x^{3} + ...$

$g (-\sqrt{x})$
$= \prod_{n=1}^\infty \left( \frac{1-x^{n-\frac{1}{2}}}{1-x^n} \right)^8$
$= 1 -8 \, \sqrt{x} + 36 \, x -128 \, x^{\frac{3}{2}} + 402 \, x^{2} -1152 \, x^{\frac{5}{2}} + 3064 \, x^{3} + ...$

$g (\sqrt{x}) - g (-\sqrt{x})$
$= \prod_{n=1}^\infty \left( \frac{1+x^{n-\frac{1}{2}}}{1-x^n} \right)^8 - \prod_{n=1}^\infty \left( \frac{1-x^{n-\frac{1}{2}}}{1-x^n} \right)^8$
$= 16 \, \sqrt{x} + 256 \, x^{\frac{3}{2}} + 2304 \, x^{\frac{5}{2}} + 15360 \, x^{\frac{7}{2}} + ...$

$f_{NS+}(x) = \frac{1}{2 \sqrt{x}} \left[ g (\sqrt{x}) - g (-\sqrt{x}) \right]$
$= \frac{1}{2 \sqrt{x}} \left[ \prod_{n=1}^\infty \left( \frac{1+x^{n-\frac{1}{2}}}{1-x^n} \right)^8 - \prod_{n=1}^\infty \left( \frac{1-x^{n-\frac{1}{2}}}{1-x^n} \right)^8 \right]$
$= \frac{1}{2 \sqrt{x}} \left[ 16 \, \sqrt{x} + 256 \, x^{\frac{3}{2}} + 2304 \, x^{\frac{5}{2}} + 15360 \, x^{\frac{7}{2}} + ... \right]$
$= 8 + 128 \, x + 1152 \, x^{2} + 7680 \, x^{3} + 42112 \, x^{4} + ...$

— Me@2018-05-08 08:50:32 PM

Music | Painting

Posted on May 7, 2018 by rpflee

Software is independent of space (music).

Hardware is independent of time (painting).

— Me@2011.09.12

旅程 4

Posted on May 6, 2018 by rpflee

Long Traveller 4

destination ~ the summation of all of the journey

— Me@2011.08.20

技術細節 3

Posted on May 6, 2018 by rpflee

親歷其境 3

這段改編自 2010 年 4 月 18 日的對話。

一切文學，余愛以血書者。

— 尼采

剛才講到，我想去升學那間大學的物理系，其獨特優秀之處。

世界上，只有極少數大學，提供超弦理論的課程。而且，那個碩士課程，不是純修課式。它為期兩年：第一年是上課，第二年是研究。

對於我現在這個年紀，單純知識的吸收（簡稱「學習」），沒有大意義。對我有大意義的是，研究和製造。

或者說，即使一個人對學術有極大興趣，十分喜歡學習，都不應做「永恆學生」。「永恆學生」在這裡，作負面意思用。「永恆學生」的特徵是，知識多而能力差，簡稱「高分低能」。

留意，所謂「知識多而能力差」，其實就即是，知識流於表面，未詳細到可以應用。例如，即使你對中國象棋中的每一隻，行走的方法，非常熟悉，那並不代表，你「懂」怎樣下棋。

要真正下到棋，你要靠不斷嘗試，長期的實戰。那樣，你才可以獲得下一個層次的知識；令自己對於象棋，知得足夠詳細，詳細到足以有效實行。

「

有時，知易行難，是因為知得不夠深入；

有時，行而無效，是因為行得不夠徹底。

」

永久單純地，吸收新知識，而沒有下一個層次的發展，是沒有意義的。或者說，知識上，只有輸入而沒有輸出，是病態。

所謂「下一個層次的發展」，可以是：

1. 應用

隨著越來越多次的應用，你於該個領域的知識，再不是依書直說，而是真才實學。

2. 傳播

如果你要傳播知識，必須靠說話或文章。

無論用兩者的哪一個，你必須透過自己的「語言」。換句話說，你必須為舊知識，創造一個新的包裝；即使，有時，你是不自覺地那樣做，未必意識到自己正在從是創作。

還有，有時，你亦會無意中，有知識上的新發現—在為舊知識創造新包裝時，竟然引發了新知識。

3. 研究

透過舊知識，發掘新知識。

— Me@2018-05-05 10:06:15 AM

Know the subject—At the Dale Carnegie public speaking class, which I highly recommend—they say anyone can make a good speech if he or she has _earned the right_ to speak on the subject in question. How do you earn the right? By living through the subject or by doing extensive research on it—which is arguably another form of living through it. Same principle applies to how-to writing. You cannot do high school student research. That is, find 21 facts and write an essay that consists solely of those 21 facts. Rather you need the proverbial iceberg of unused facts under the “tip” that is your book. The good news is virtually everyone has lived through something that fits that criterion. And, with a year or so, we can all research something that interest us to the point where we can write about it. The key to bad writing is assigned topics—the standard of high school and college teachers.

— John T. Reed

The best software

Posted on May 1, 2018 by rpflee

somethingsimple 3 months ago

> Software is the currency that we pay to solve problems, which is our actual goal. We should endeavor to build as little software as possible to solve our problems.

ScottBurson 3 months ago

My point today is that, if we wish to count lines of code, we should not regard them as “lines produced” but as “lines spent”: the current conventional wisdom is so foolish as to book that count on the wrong side of the ledger. — E. W. Dijkstra

That’s why the best choice of software is often no software …

— Coding Horror

— by Jeff Atwood

— Write dumb code

— Hacker News

2018.05.01 Tuesday ACHK

Physics Town

continues

Monthly Archives: May 2018

Problem 14.4b1.4

Life, 3

生命 2

多項選擇題 6

Quantum Computing, 2

Van der Waals equation 1.2

Tree rings, 2

Richard Stallman

On Keeping Your Soul, 3

Ask users

Problem 14.4b1.3

Van der Waals equation 1.1

Bus Stop 3

時空兌換率

Sleep apnea

Problem 14.4b1.2

Music | Painting

旅程 4

技術細節 3

The best software