3.1 Lorentz covariance for motion in electromagnetic fields, 1

A First Course in String Theory

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The Lorentz force equation (3.5) can be written relativistically as

\displaystyle{\frac{d p_\mu}{ds} = \frac{q}{c} F_{\mu \nu} \frac{d x^\nu}{ds}},

where \displaystyle{p_{\mu}} is the four-momentum.

(a) Check explicitly that this equation reproduces (3.5) when \displaystyle{\mu} is a spatial index.

(b) What does (1) gives when \displaystyle{\mu = 0}? Does it make sense?

~~~

Eq. (3.5):

\displaystyle{\frac{d \vec p}{dt} = q \left( \vec E + \frac{\vec v}{c} \times \vec B \right)}

Eq. (2.20):

\displaystyle{ds \equiv \sqrt{ds^2}}    if    \displaystyle{ds^2 > 0}

Eq. (2.21):

\displaystyle{-ds^2 = \eta_{\mu \nu} dx^\mu dx^\nu}

The spacetime interval \displaystyle{ds^2} is Lorentz invariant. If \displaystyle{ds^2 > 0}, we have Eq. (2.27) and (2.28):

\displaystyle{\begin{aligned}     ds &= c dt_p \\     ds &= c dt \sqrt{1 - \beta^2}     \end{aligned}}

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\displaystyle{\frac{d p_\mu}{ds} \left( \frac{ds}{dt} \right) = \frac{q}{c} F_{\mu \nu} \frac{d x^\nu}{ds}} \left( \frac{ds}{dt} \right)

\displaystyle{    \begin{aligned}     \frac{d p_1}{dt}     &= \frac{q}{c} F_{1 \nu} \frac{d x^\nu}{dt} \\    &= \frac{q}{c} \left( F_{1 0} \frac{d x^0}{dt} + F_{1 1} \frac{d x^1}{dt} + F_{1 2} \frac{d x^2}{dt} + F_{1 3} \frac{d x^3}{dt} \right) \\    \frac{d p_x}{dt} &= \frac{q}{c} \left( E_x c \frac{d t}{dt} + (0) \frac{d x^1}{dt} + B_z \frac{d x^2}{dt} - B_y \frac{d x^3}{dt} \right) \\    &= q \left( E_x + \frac{1}{c} \left( \vec v \times \vec B \right)_x \right) \\    \end{aligned}}

\displaystyle{    \begin{aligned}     \frac{d p_0}{dt}     &= \frac{q}{c} \left( F_{0 0} \frac{d x^0}{dt} + F_{0 1} \frac{d x^1}{dt} + F_{0 2} \frac{d x^2}{dt} + F_{0 3} \frac{d x^3}{dt} \right) \\    &= \frac{q}{c} \left( 0 \frac{d x^0}{dt} - E_x \frac{d x^1}{dt} - E_y \frac{d x^2}{dt} - E_z \frac{d x^3}{dt} \right) \\    \frac{d }{dt} \left( \frac{-E}{c} \right)&= - \frac{q}{c} \left( \vec v \cdot \vec   E \right) \\    \frac{d E}{dt}  &= \vec v \cdot \vec F_E \\      \end{aligned}}

— Me@2022-08-04 04:17:59 PM

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2022.08.06 Saturday (c) All rights reserved by ACHK

Bravado

Anyone who expects to create, be it as scientist or artist, scholar or writer, needs self-confidence, even bravado. How else can one dare to imagine understanding what no one else has understood, discovering what no one else has discovered? Where does this confidence come from? Fortunately, every young person is blessed with some of it. It is part of human character. What of the girl or boy who reads about Newton and Maxwell and Bohr and Einstein and says, “I want to build on what they have built; I want to add to the sum of knowledge about the most basic laws of nature”?

– Geons, Black Holes & Quantum Foam, p. 84

– 2000
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2022.08.03 Wednesday (c) All rights reserved by ACHK

CSCI3420


~~~

砌摩打

~~~

Design:
1: make car moving
2: make car stop itself
3: follow the track
4: change direction
5: tuning

~~~

CSCI3420 Project Phase 2

Deadline: 23:59

~~~

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2022.08.03 Wednesday (c) All rights reserved by ACHK

排列組合 1.1

nCr, 0

這段改編自 2010 年 7 月 27 日的對話。

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\displaystyle{n!}

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1.1  意思:

\displaystyle{n} 個人 \displaystyle{n} 個座位的話,有多少種坐法?

1.2.1  算式:

\displaystyle{n! = (n)(n-1)(n-2) \cdots (3)(2)(1)}

1.2.2  由來:

第一個位,有 \displaystyle{n} 個可能的人選;第一個位被坐後,第二個位只有 \displaystyle{(n-1)} 個,可能的人選;如此類推,直到最後一個位,被餘下的唯一個人,選了為此。

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1.3 

留意,\displaystyle{n} 是多少,就有多少項。

\displaystyle{n! = (n)(n-1)(n-2) \cdots (3)(2)(1)}

例如,\displaystyle{5!},就有 5 項相乘;

\displaystyle{\begin{aligned}     5! &= (5)(4)(3)(2)(1) \\ \\     \end{aligned}}

\displaystyle{3!},就有 3 項;等等。

\displaystyle{\begin{aligned}     3! &= (3)(2)(1) \\    \end{aligned}}

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1.4.0  零的階乘,\displaystyle{0!},還未有定義,因為,算式 \displaystyle{n! = (n)(n-1) \cdots (2)(1)} 中的 \displaystyle{n},只可以是正整體,不可以零。

階乘零,\displaystyle{0!},需要額外定義,因為會常用到。那樣,\displaystyle{0!} 應該定義為,什麼數值呢?

1.4.1  既然 \displaystyle{n!}意思是「\displaystyle{n} 個人 \displaystyle{n} 個座位,有多少種坐法」,那樣,你就可以視,\displaystyle{0!} 的意思是「\displaystyle{0} 個人 \displaystyle{0} 個座位,有多少種坐法」;那明顯是一,因為,那個情況之下,只有一個「坐法」,就是「沒有人又沒有位」這個唯一的可能性。

\displaystyle{0! = 1}

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1.4.2  另外一種設計定義的想法是,由 \displaystyle{n!}算式取靈感。

既然 \displaystyle{n} 是多少,\displaystyle{n!} 就有多少項相乘,那樣,零的階乘,\displaystyle{0!},理應只有零項相乘,\displaystyle{~a^0~}

\displaystyle{0! = a^0}

但是,「零項相乘」的數值,又應該是什麼呢?

「零項相乘」即是「乘了等如沒有乘」,所以是一,因為任何數乘了一,效果都等於沒有乘。

\displaystyle{a^0 = 1}

(「零項相乘」的正式學名是,「空積」或「零項積」。)

\displaystyle{0! = 1}

— Me@2022-08-02 02:41:43 PM

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2022.08.02 Tuesday (c) All rights reserved by ACHK

Mega Man Zero 3

Euler problem 1

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(proclaim '(optimize speed))

(reduce #'+ '(1 2 3 4))

; 10

(loop :for n :below 10 :collect n)

; (0 1 2 3 4 5 6 7 8 9)

(describe :below)

(defun range (max &key (min 0) (step 1))
   (loop :for n :from min :below max :by step
      collect n))
      
(- (+ (* 3 (reduce #'+ (range 334 :min 1 :step 1)))
      (* 5 (reduce #'+ (range 200 :min 1 :step 1))))
   (* 15 (reduce #'+ (range 67 :min 1 :step 1))))
   
; 233168

(defun sum-1-n (n)
  (/ (* n (+ n 1)) 2))
  
(- (+ (* 3 (sum-1-n 333))
      (* 5 (sum-1-n 199)))
   (* 15 (sum-1-n 66)))
   
; 233168

— Me@2022-08-01 03:29:01 PM

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2022.08.01 Monday (c) All rights reserved by ACHK

Jupyter Notebook

SICMUtils, 3

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The goal of this post to setup Jupyter Notebook.

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1. Read and follow the exact steps of my post titled “SICMUtils“.

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2. To install the Jupyter Notebook software in Ubuntu, use this command:

sudo apt-get install sagemath-jupyter 

3. Try to open the SageMath program.

4. It will open a Jupyter notebook page.

5. Click the “New” button at the top right corner and then select “SageMath“.

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6. Type

1 + 1

onto the input line.

7. Hit the keys shift-enter to get the output

2

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8. Input

Integrate(x^3, x)

9. Hit shift-enter:

NameError: name 'Integrate' is not defined

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10. Input

from sage.symbolic.integration.integral import *

indefinite_integral(x^3, x)

11. Hit shift-enter:

1/4*x^4

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12. Input

%display latex

13. Hit shift-enter.

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14. Input

indefinite_integral(x^3, x)

15. Hit shift-enter:

\displaystyle{\frac{1}{4} \, x^{4}}

— Me@2022-07-30 12:18:50 PM

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2022.08.01 Monday (c) All rights reserved by ACHK

Quantum as potential, 2

Only measurement results (aka physical phenomena) form the physical reality.

Quantum Mechanics is a theory of measurement results.

Quantum Mechanics is a theory of reality.

Quantum Mechanics is not a theory of unobservables (undefined-observables).

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Quantum mechanics is a story of reality, not a story of story.

— Me@2022-07-27 10:38:32 AM

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2022.07.29 Friday (c) All rights reserved by ACHK

女敵人 2

這段改編自 2021 年 12 月 15 日的對話。

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男女友人,如果長期(一年)時常(每週)見面的話,不會是純友誼。

在這個上文下理中,男女之間不會有「純友誼」,除非外貌驚奇。

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不會有「純友誼」,但可不可以有「純仇恨」?

— Me@2022-07-29 02:37:02 PM

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2022.07.29 Friday (c) All rights reserved by ACHK

Org-babel-clojure

SICMUtils, 2

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The goal of this post to setup the Emacs editor for Clojure programming.

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1. Read and follow the exact steps of the last post.

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2. Open the .emacs file. Go to the end of the file. Add the following code:

(require 'org)
(require 'ob-clojure)

(setq org-babel-clojure-backend 'cider)
(require 'cider)

(set-register ?c '(file . "~/my-stuff/my-stuff.org"))

(setq org-confirm-babel-evaluate nil)

(setq org-src-tab-acts-natively t)

3. Close Emacs.

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4. Go to the directory “~/my-stuff/” and then create a file named “my-stuff.org“.

5. Use Emacs to open the file.

6. Within the file, add the following code:

#+BEGIN_SRC emacs-lisp

  (+ 1 1)

#+END_SRC

7. Place the text cursor in the code block (between the line #+BEGIN_SRC and the line #+END_SRC).

8. Hit the Emacs command

C-c C-c

9. You will get the evaluation result:

#+RESULTS:
: 2

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10. Hit the Emacs command:

M-x cider-jack-in

11. Within the file “my-stuff.org“, add the code:

#+BEGIN_SRC clojure :results value		  

  (require '[sicmutils.env :as env])

#+END_SRC

12. Place the text cursor in the code block.

13. Hit the Emacs command

C-c C-c

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14. Add the code:

#+BEGIN_SRC clojure :results value		  

  (env/bootstrap-repl!) 

#+END_SRC

15. Place the text cursor in the code block and then hit the Emacs command

C-c C-c

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16. Add the code:


#+BEGIN_SRC clojure :results replace drawer
 
(->TeX (simplify ((D cube) 'x)))
 
#+END_SRC

17. Place the text cursor and then hit

C-c C-c

It will give you the \LaTeX code

#+RESULTS:
:RESULTS:
"3\\,{x}^{2}"
:END:

— Me@2022-07-27 11:34:28 PM

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2022.07.28 Thursday (c) All rights reserved by ACHK

SICMUtils

A Clojure(script) implementation of the scmutils system for math and physics investigations in the Clojure and Clojurescript languages.

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1. To install Clojure in Ubuntu, just this command is enough:

sudo apt-get install elpa-cider

Although the Clojure version you get is probably not the most updated one, that is not important, because you can specify which version you want in the config file of each project.

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2. Then use this command to generate a new project named my-stuff:

lein new app my-stuff

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3. Use Emacs to open the file:

~/my-stuff/project.clj

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4. Replace the existing :dependencies line with this one

  :dependencies [[org.clojure/clojure "1.11.1"]
                 [sicmutils "0.22.0"]]

And make sure that both clojure and sicmutils have the most updated version numbers.

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5. In Emacs, type the command

M-x cider-jack-in

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6. In the clojure window (cider-repl), type

(clojure-version)

with enter at the end.

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7. Type

(require '[sicmutils.env :as env])

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8. Type

(env/bootstrap-repl!) 

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9. Code

((D cube) 'x)

will result

(+ (* x x) (* x (+ x x)))

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10. Type the Emacs command

M-p

to access the last input. Then modify it into

(simplify ((D cube) 'x))

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It will result

(* 3 (expt x 2))

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11. Code

(->TeX (simplify ((D cube) 'x)))

will give the LaTeX code

3\\,{x}^{2}

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12. You can exit by the Emacs command

<C-c C-q> 

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For the time being, SICMUtils is not suitable for the book SICM (Structure and Interpretation of Classical Mechanics). In other words, SICMUtils cannot replace the scmutils library yet, because:

a. You would have to do the translation manually, from the scmutils code in the book to SICMUtils.

b. Although it can generate LaTeX source code, it does NOT do the LaTeX rendering.

c. It cannot plot graphs.

However, SICMUtils has one advantage over scmutils. It can generate LaTeX source of an expression, but scmutils cannot. So I am planning to use both scmutils and SICMUtils.

Also, I will learn how to use SICMUtils with other Clojure libraries and the Jupyter Notebook. That would get LaTeX rendering and graph plotting running.

— Me@2022-07-26 11:03:51 AM

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2022.07.26 Tuesday (c) All rights reserved by ACHK

Quantum as potential

Realist view is wrong.

Before measurement, there are quantum potentials only.

quantum ~ potential

Note that it is NOT the “quantum potential” in the Bohm interpretation.

— Me@2016-08-21 06:13:49 PM

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A wave function encodes the probabilities of different potential measurement results of a physical experiment. It is not a physical wave.

Quantum superposition is NOT a superposition of realities.

Physics should consider only measurement results and their probabilities. Only measurement results are realities.

No measurement result, aka physical phenomenon, is in a superposition.

For example, in the double-slit experiment, the measurement results are (the locations of) the dots on the final screen. Every dot location is not in a superposition.

— Me@2022-07-25 06:43:05 PM

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2022.07.25 Monday (c) All rights reserved by ACHK

Presentation 基本原理 1.2.2

反不相關推薦 3

這段改編自 2010 年 4 月 24 日的對話。

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(安:雖然我不是從事教學工作,但是,在公司解釋東西給上司時,其實很多時也需要使用到,所謂的「教學技巧」。

你有沒有一些推介呢?)

.

我在初入行時,教學頗為混亂。我以為在每一課中,教得越多東西越好。

在教了約兩年後,大約 2006/7 年,閱到數學家 John Baez 的幾點教學建議。對我最重要的一點是:

Keep the Sheep Moving in the Same Direction – a lecture should have a clear and simple plot. Avoid anything that distracts from this. Don’t make too many points. Don’t be afraid to repeat yourself.

— How to Teach Stuff

不是講得越多越好,而是相反——講得越少越好

Perfection is achieved, not when there is nothing more to add, but when there is nothing left to take away.

— Antoine de Saint-Exupéry

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(假設每一節課,都是一小時以內。)

每一節課應該,只有一個重點。在該課中,你要用不同的句子、字眼、例子,來重複釐清和闡述,那個重點。講課時,一切的細節,都要圍繞著,那個重點來運行。

Every lecture should make only one main point.

The German philosopher G. W. F. Hegel wrote that any philosopher who uses the word “and” too often cannot be a good philosopher. I think he was right, at least insofar as lecturing goes. Every lecture should state one main point and repeat it over and over, like a theme with variations.

— Advice for the Young Scientist

— John Baez

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又例如,見工面試時,你重點必須是,

1. 你將如何為公司賺錢?

2. 為何要選你,而不是其他應徵者?

你的一切才能,都要圍繞著,這個重點來介紹。換而言之,千萬要記住,要的是「賺錢介紹」,不是「自我介紹」。沒有人想理會,你的「自我」。

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(安:那是兩個重點,而不是一個。)

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可合體為:

在「為本公司賺錢」這方面,你如何優勝過,其他應徵者?

— Me@2022-07-24 04:41:17 PM

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2022.07.25 Monday (c) All rights reserved by ACHK

Common Lisp Reloaded

; sudo apt-get install sbcl

; sudo apt-get install slime

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(setq inferior-lisp-program "sbcl")

(defun load-slime-with-correct-buffer-position ()

  (save-excursion (slime))
     
 
  (delete-other-windows) 
) 

(defun prelude-start-slime ()
  (unless (slime-connected-p)
    (load-slime-with-correct-buffer-position)))

(add-hook 'slime-mode-hook 'prelude-start-slime)

(set-register ?f '(file . "/path_to/lisp_file.lisp"))

— Me@2022-07-23 05:20:32 PM

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2022.07.23 Saturday (c) All rights reserved by ACHK

Cyclic coordinate

A generalized coordinate component that does not appear explicitly in the Lagrangian is called a cyclic coordinate. The generalized momentum component conjugate to any cyclic coordinate is a constant of the motion.

— 1.8 Conserved Quantities

— Structure and Interpretation of Classical Mechanics

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This is a special case of Noether’s theorem. Such coordinates are called “cyclic” or “ignorable”.

— Wikipedia on Lagrangian mechanics

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If only the cyclic coordinate \displaystyle{q(t)} varies with time (if it doesn’t, \displaystyle{q} is superfluous), the Lagrangian, or the essential physical situation, doesn’t vary. Hence the initial value of \displaystyle{q} doesn’t determine the path, which is only possible if the path is closed.

— edited Jul 28, 2014 at 15:55

— ACuriousMind

— answered Jul 28, 2014 at 15:50

— Pieter Kockx

— Why are they called “cyclic” coordinates?

— Physics Stack Exchange

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2022.07.22 Friday ACHK

Lectures On Computation

Feynman’s philosophy of learning and discovery comes through strongly in these lectures. He constantly points out the benefits of play around with concepts and working out solutions to problems on your own before looking at the back for the answers. As Feynman says in the lectures: “If you keep proving stuff that others have done, getting confidence, increasing the complexities of your solutions — for fun of it — then one day you’ll turn around and discovered that nobody actually did that one! And that’s the way, to become a computer scientist.” Imagine that you are explaining your ideas to your former smart but ignorant, self, at the beginning of your studies!

— Feynman[ Lectures on ]Computation

— 30[.]06[.]2003

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2022.07.21 Thursday ACHK

ENGG 2011

~~~

ENGG 2011 - Advanced Engineering Maths - Assignment 1

~~~

C++: function(...)
            Many argument[s]

~~~

C++: mid-term test (31/10/2011)

~~~

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2022.07.21 Thursday (c) All rights reserved by ACHK

機會再生論 1.2

本來沒有「社會」,有的是「個人」。要「社會」供養你,其實就是要,「其他人」供養你。

「自力更生」的好處是,你的才能會越來越強。「他力更生」的壞處是,不一定有;即使有,你的財能會遇弱越弱。

「求之於己」的好處是,自己會越來越強,自己成自己的主人。「依賴他人」的壞處是,他人會越來越強,自己成了他人的奴隸。

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免費的東西是最昂貴,因為,它把代價隱藏了,對你溫水煮蛙。

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已有財富有盡,創造財富無窮。賺錢,就是接收,來自未來的金錢。

過去有限,未來無限。

— Me@2022-07-14 03:27:39 PM

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2022.07.19 Tuesday (c) All rights reserved by ACHK