財政自由 1.1

(問:根據你的講法,好像大部分情況下,都不應該讀研究院似的。)

在理想的情況下,你可能應該讀研究院。

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(問:那樣,你心目中的理想情況是什麼?)

假設你已經有財政自由,你就有可能,適合讀研究院;…

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(問:那就即是話,要在還年青時,就賺到一生夠用的金錢?
為什麼要選擇,那麼難的目標?)

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(問:但是,財政自由,又可以如何實現呢?)

首先,要有一個概念,就是:

長遠而言,為生不可靠「售賣時間」,而要靠「創造價值」。

例如,你是一位音樂演奏家。如果你的收入,主要來自去現場演出的話,那就為之,以「售賣時間」為生。

「售賣時間」的致命傷是,收入不可倍增放大。那就是所謂的「一分耕耘,一分收穫」。

(問:「一分耕耘,一分收穫」有什麼問題?)

連基本正常的生活,也可能推持不到。

例如,衣食住行是必須的。即使只講「住」,如果你的收入來源,只有薪金的話,你也要花十至三十年(或以上)的時間,才可以累積到有足夠的金錢,去支付買房子所需的首期。

— Me@2019-08-21 07:40:35 PM

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2019.08.23 Friday (c) All rights reserved by ACHK

Physical laws are low-energy approximations to reality, 1.6

d_2019_01_31__23_13_36_PM_a

QM GR

too many particles

when you cool down the system, you see something that your equations cannot predict

only by experiment, you discover that you can go to there

the system state goes from one local minimum to another

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Since each theory is valid only when near a particular energy local minimum, we cannot use it to predict other local minima (other physical realities). That’s why we need to keep doing experiments that are designed for stress-testing existing theories. For example, one experiment can put a system in extreme high pressure. Another experiment can put a system in extreme high temperature.

Once a theory breaks down in one of those experiments, we see phenomena that we have never seen before. From there, we construct new theories to explain those phenomena.

d_2019_01_31__23_13_36_PM_b

Once a theory breaks down in one of those experiments, we see phenomena that we have never seen before. In this sense, experimental physics is much richer.

Computational physics goes further in only one direction. Only experiments let you try randomly.

— Me@2019-08-18 07:51:05 PM

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2019.08.18 Sunday (c) All rights reserved by ACHK

Literal numbers

All primitive mathematical procedures are extended to be generic over
symbolic arguments. When given symbolic arguments, these procedures
construct a symbolic representation of the required answer. There are
primitive literal numbers. We can make a literal number that is
represented as an expression by the symbol “a” as follows:

(literal-number 'a)        ==>  (*number* (expression a))

The literal number is an object that has the type of a number, but its
representation as an expression is the symbol “a”.

(type (literal-number 'a))          ==>  *number*

(expression (literal-number 'a))    ==>  a

— SCMUTILS Reference Manual

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2019.08.17 Saturday ACHK

Multiple dimensions of time

What would be the implications of multiple dimensions of time?

That means the (past) history itself can change, as commonly seen in time travel stories.

But wouldn’t that be the case with one dimension also?

In reality, there is only one dimension of time, meaning that the state of a system keeps changing, forming the timeline. But the timeline itself cannot be changed once formed. In other words, (past) history cannot be changed.

— Me@2019-08-11 04:07:48 PM

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2019.08.11 Sunday (c) All rights reserved by ACHK

(反對)開夜車 4.2

平均而言,你仍必須要有,充足的睡眠。亦即是話,某一天睡少了,就必須於在當個星期,還回「睡債」。

例如,如果你的充足睡眠是,每天七小時,而你在某一天只睡了六小時的話,你就有義務,在當個星期的另一天,睡多一小時。一般而言,「另一天」是指週末,或者其他假日。

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另外,有時,只要跳出框框,破格思考,你會發現,或者,只要你的時間表稍改一點,就根本毋須「開夜車」。

例如,我在大學一年級時,發覺日間的課堂,加課外活動,異常充實。回到家時,就已經累到,根本集中不到精神研習。

我當時靈機一觸想到,與其沒有精神地研習,倒不如每晚九時半就睡,凌晨三或四時才起床。那樣,七時多出門前,我就會有,三至四個小時的專心時空。

— Me@2019-07-30 11:11:42 PM

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2019.08.03 Saturday (c) All rights reserved by ACHK

Quick Calculation 15.1.2

A First Course in String Theory

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Recall that a group is a set which is closed under an associative multiplication; it contains an identity element, and each element has a multiplicative inverse. Verify that \displaystyle{U(1)} and \displaystyle{U(N)}, as described above, are groups.

~~~

Definition

A group is a set, G, together with an operation \displaystyle{\bullet} (called the group law of G) that combines any two elements a and b to form another element, denoted \displaystyle{a \bullet b} or \displaystyle{ab}. To qualify as a group, the set and operation, \displaystyle{(G, \bullet)}, must satisfy four requirements known as the group axioms:

Closure

For all a, b in G, the result of the operation, \displaystyle{a \bullet b}, is also in G.

Associativity

For all a, b and c in G, \displaystyle{(a \bullet b) \bullet c = a \bullet (b \bullet c)}.

Identity element

There exists an element e in G such that, for every element a in G, the equation \displaystyle{e \bullet a = a \bullet e = a} holds. Such an element is unique, and thus one speaks of the identity element.

Inverse element

For each a in G, there exists an element b in G, commonly denoted \displaystyle{a^{-1}} (or \displaystyle{-a}, if the operation is denoted “+”), such that \displaystyle{a \bullet b = b \bullet a = e}, where e is the identity element.

— Wikipedia on Group (mathematics)

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The axioms for a group are short and natural… Yet somehow hidden behind these axioms is the monster simple group, a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists.

— Richard Borcherds in Mathematicians: An Outer View of the Inner World

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2019.07.28 Sunday ACHK

Alfred Tarski, 3

The undefinability theorem shows that this encoding cannot be done for semantic concepts such as truth. It shows that no sufficiently rich interpreted language can represent its own semantics. A corollary is that any metalanguage capable of expressing the semantics of some object language must have expressive power exceeding that of the object language. The metalanguage includes primitive notions, axioms, and rules absent from the object language, so that there are theorems provable in the metalanguage not provable in the object language.

— Wikipedia on Tarski’s undefinability theorem

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Tarski’s 1969 “Truth and proof” considered both Gödel’s incompleteness theorems and Tarski’s undefinability theorem, and mulled over their consequences for the axiomatic method in mathematics.

— Wikipedia on Alfred Tarski

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2019.07.20 Saturday ACHK

PhD, 3.7.2

碩士 4.7.2 | On Keeping Your Soul, 2.2.7.2

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(問:根據你的講法,好像大部分情況下,都不應該讀研究院似的。)

在理想的情況下,你可能應該讀研究院。

(問:那樣,你心目中的理想情況是什麼?)

假設你已經有財政自由,你就有可能,適合讀研究院;因為,以前講有關讀研究院的一堆大問題,將會細很多,例如:

  1. 如果你是自資,就即是不拿學校的資助。那樣,你就不是僱員。研究以外的工作,例如做助教等,可以一概不理。

  2. 同理,你的博士導師再不會是,你工作上的上司。反而,你是消費者,他是你的僱員。

(問:怎樣為之「有財政自由」呢?)

即是你當時的積蓄,已足夠你一生人的使用。

(問:那就是即是「退了休」?)

不太一樣。

一來,「退休」通常是指,年紀大時才發生。追求財政自由的人,通常不會計劃,在年老時才財政自由;因為,追求財政自由的主要目的是,不用再每天上班,從而,有足夠的時間,去追求自己的理想。

二來,大有部分人「退休」時的積蓄,其實不夠餘生的使用。亦即是話,他們的財政,其實未有自由。

(問:那就即是話,要在還年青時,就賺到一生夠用的金錢?

為什麼要選擇,那麼難的目標?)

假設,你的理想是要,做一個物理學家。如果沒有財政自由,即使你做了物理學家,你也做不到物理學家。

(問:那麼玄?什麼意思?)

如果要做物理學家,通常的方法是,做物理教授。但是,做了教授後,你會發覺,教授的主要工作,其實是為其研究團隊,爭取研究撥款。

那樣,你只會剩下極少時間,給自己研究物理。

— Me@2019-07-06 10:57:22 PM

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2019.07.10 Wednesday (c) All rights reserved by ACHK

Ex 1.7 Properties of $\delta$

Let \displaystyle{F} be a path-independent function and \displaystyle{g} be a path-dependent function; then

\displaystyle{\delta_\eta h[q] = \left( DF \circ g[q] \right) \delta_\eta g[q]~~~~~\text{with}~~~~~h[q] = F \circ g[q].~~~~~(1.26)}

— 1.5.1 Varying a path

— Structure and Interpretation of Classical Mechanics

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Prove that

\displaystyle{\delta_\eta F \circ g[q] = \left( DF \circ g[q] \right) \delta_\eta g[q]}

~~~

\displaystyle{RHS = \lim_{\Delta t \to 0} \left( \frac{F \circ g[q](t+\Delta t) - F \circ g[q](t)}{\Delta t} \right) \lim_{\epsilon \to 0} \left( \frac{g[q+\epsilon \eta]-g[q]}{\epsilon} \right)}

\displaystyle{ \begin{aligned} LHS &= \delta_\eta F \circ g[q] \\   &=  \lim_{\epsilon \to 0} \left( \frac{F \circ g[q+\epsilon \eta]-F \circ g[q]}{\epsilon} \right) \\    &=  \lim_{\epsilon \to 0} \left( \frac{F \left[ g[q+\epsilon \eta] \right] - F \left[ g[q] \right]}{\epsilon} \right) \\   \end{aligned}}

Since \displaystyle{F} is path-independent,

\displaystyle{ \begin{aligned} LHS   &= \lim_{\epsilon \to 0} \left(  \frac{F \left( g[q+\epsilon \eta ] \right) - F \left( g[q] \right)}{\epsilon} \right) \\   \end{aligned}}

Let \displaystyle{ g[q+\epsilon \eta] = g + \Delta g}.

\displaystyle{ \begin{aligned} LHS   &= \lim_{\epsilon \to 0} \left(  \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\epsilon} \right) \\   &= \lim_{\epsilon \to 0} \left(  \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]}\frac{\Delta g[q]}{\epsilon} \right) \\   \end{aligned}}

When \displaystyle{ \epsilon \to 0}, \displaystyle{ \Delta g \to 0 }.

\displaystyle{ \begin{aligned} LHS   &= \lim_{\substack{\epsilon \to 0 \\ \Delta g \to 0}} \left(  \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]}\frac{\Delta g[q]}{\epsilon} \right) \\   &= \lim_{\Delta g \to 0} \left(  \frac{F \left( g[q] + \Delta g[q]] \right) - F \left( g[q] \right)}{\Delta g[q]} \lim_{\epsilon \to 0} \frac{g[q + \epsilon \eta] - g[q]}{\epsilon} \right) \\   &= DF \left( g[q] \right) \delta_\eta g[q] \\   &= RHS \\  \end{aligned}}

— Me@2019-06-24 10:55:28 PM

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2019.06.25 Tuesday (c) All rights reserved by ACHK

From classical to quantum

From this viewpoint, the move from a classical to a quantum mechanical system is not a move from a comutative to a non-commutative algebra \displaystyle{\mathcal{A}} of a real-valued observables, but, instead, a move from a commutative algebra to a partial commutative algebra of observables.

Of course, every non-commutative algebra determines an underlying partial commutative algebra and also its diagram of commutative subalgebras.

That fact that assuming the structure of a non-commutative algebra is the wrong assumption has already been observed in the literature (see, for example, [19]),

but it is often replaced by another wrong assumption, namely that of assuming the structure of a Jordan algebra.

These differing assumptions on the structure of \displaystyle{\mathcal A} affect the size of its automorphisum group and, hence, of the allowable symmetries of the system (the weaker the assumed structure on \displaystyle{\mathcal A}, the larger is its automorphism group).

— The Mathematical Foundations of Quantum Mechanics

— David A. Edwards

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2019.06.18 Tuesday ACHK

(反對)開夜車 4.1

本文章並(!)不(!)可作為醫學建議。如需醫學意見,請諮詢專業人士。

(問:但是,在現今社會,無論是上班,或是讀書,完全不「開夜車」,又好像不切實際。)

其實,主要是講讀書時代。如果在工作時代,你的職位需要,你時常「開夜車」的話,你根本就應該另謀高就。

試問,世間上,有什麼工作,竟然值得你冒生命危險,去時常「開夜車」呢?

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言歸正傳,讀書時代,如果時間管理得宜,需要「開夜車」的情況,其實是很少。

(問:那樣說有意思嗎?我正正是問你,在時間管理失宜,需要「開夜車」時,該如何自處?)

一定「開夜車」的話,你至少要做到以下幾點,去保障自己的安全:

  • 只可以間中,不可以經常。

  • 日間中途要有小睡。

  • 平均而言,你仍必須要有,充足的睡眠。亦即是話,某一天睡少了,必須於在當個星期,還回「睡債」。

    • 例如,如果你的充足睡眠是,每天七小時,而你在某一天只睡了六小時的話,你就有義務,在當個星期的另一天,睡多一小時。

.

另外,有時,只要跳出框框,破格思考,或者,只要你的時間表稍改一點,就根本毋須「開夜車」。

本文章並(!)不(!)可作為醫學建議。如需醫學意見,請諮詢專業人士。

— Me@2019-06-06 08:23:56 PM

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2019.06.08 Saturday (c) All rights reserved by ACHK

Quick Calculation 15.1

A First Course in String Theory

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Recall that a group is a set which is closed under an associative multiplication; it contains an identity element, and each element has a multiplicative inverse. Verify that \displaystyle{U(1)} and \displaystyle{U(N)}, as described above, are groups.

~~~

What is \displaystyle{U(1)}?

— Me@2019-05-24 11:25:41 PM

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The set of all \displaystyle{1 \times 1} unitary matrices clearly coincides with the circle group; the unitary condition is equivalent to the condition that its element have absolute value 1. Therefore, the circle group is canonically isomorphic to \displaystyle{U(1)}, the first unitary group.

— Wikipedia on Circle group

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In mathematics, a complex square matrix \displaystyle{U} is unitary if its conjugate transpose \displaystyle{U^*} is also its inverse—that is, if

\displaystyle{U^{*}U=UU^{*}=I,}

where \displaystyle{I} is the identity matrix.

In physics, especially in quantum mechanics, the Hermitian conjugate of a matrix is denoted by a dagger (\displaystyle{\dagger}) and the equation above becomes

\displaystyle{U^{\dagger }U=UU^{\dagger }=I.}

The real analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.

— Wikipedia on Unitary matrix

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2019.05.25 Saturday ACHK