原因非理由

Posted on May 21, 2023 by rpflee

原因 ~ 先決條件

理由 ~ 充分條件

— Me@2016-03-31 08:35:43 PM

負數無大細

Posted on May 20, 2023 by rpflee

92-CE-PHY II – 4 前傳 2

VirtualBox Start

Posted on May 19, 2023 by rpflee

In KDE, when a Virtualbox virtual machine is turned on, sometimes, both the guest OS and the host OS would capture the Super key at the same time. In order to avoid that, use the following steps:

1. Right-click the VirtualBox title bar

2. Choose:

More Actions
$\to$ Special Window Settings
$\to$ Ignore Global Shortcuts (in Appearance and Fixes)
$\to$ Force $\to$ Yes

— Me@2023-05-17 07:52:55 PM

3.4 Electric fields and potentials of point charges, 2

Posted on May 17, 2023 by rpflee

A First Course in String Theory

(b) Show that with $d$ spatial dimensions, the potential $\Phi$ due to a point charge $q$ is given by

$\displaystyle{\Phi(r) = \frac{\Gamma \left( \frac{d}{2} - 1 \right)}{4 \pi^{d/2}} \frac{q}{r^{d-2}}}$

~~~

Eq. (3.74):

$\displaystyle{E(r) = \frac{\Gamma\left( \frac{d}{2} \right)}{2 \pi^{\frac{d}{2}}} \frac{q}{r^{d-1}}}$

$\displaystyle{\begin{aligned} E(r) &= -\frac{d\Phi(r)}{dr} \\ \\ \int_r^{\infty} d\Phi &= - \frac{\Gamma\left( \frac{d}{2} \right)}{2 \pi^{\frac{d}{2}}} q \int_r^{\infty} r^{-d+1} dr \\ \\ \Phi(\infty) - \Phi(r) &= \frac{\Gamma\left( \frac{d}{2} \right)}{4 \pi^{\frac{d}{2}}} \frac{q}{\frac{d}{2}-1} \left[\frac{1}{r^{d-2}} \right]_r^{\infty} \\ \\ \Gamma (z) &= \frac{\Gamma (z+1)}{z} \\ \\ \Phi(r) &= \frac{\Gamma\left( \frac{d}{2} - 1\right)}{4 \pi^{\frac{d}{2}}} \frac{q}{r^{d-2}} \\ \\ \end{aligned} }$

— Me@2023-05-17 09:11:07 AM

Random variable

Posted on May 16, 2023 by rpflee

Random variable $X$ represents a single-valued result of a random event. Its value is unknown to us, not because of our ignorance, but because of its non-existence. The value exists only after the happening of that random event.

Symbol $x$ represents a particular value of $X$ . It is an existing value that can be substituted to $X$ . We use symbol $x$ instead of a number because we have not yet known what that particular number is.

— Me@2016-04-08 05:24:45 PM

X ~ random variable

It is a variable due to the fact that the “identical” random process can result differently at different times.

x ~ a value of X

Since it is a particular value of X, it is not a variable. However, it may seem to be a variable because it may still be unknown to us.

Symbol $P(X)$ is meaningless because inside, it must be a statement (representing an event). Symbol $X$ is a random variable, not a statement.

Instead, “ $X=x$ ” is a statement. So expression $P(X=x)$ is meaningful, such as

$P(X=x) = {\begin{cases}{\frac {1}{2}},&x=0,\\{\frac {1}{2}},&x=1,\\0,&x\notin \{0,1\} \end{cases}}$

From another point of view, $X$ is a noun phrase, such as “my monthly salary”, not a number. Symbol $x$ is a number, although maybe not known yet. That’s why whatever the formula, it contains no $X$ ‘s, but $x$ ‘s. For example,

$\cdots = {\begin{cases}{\frac {1}{2}},&x=0,\\{\frac {1}{2}},&x=1,\\0,&x\notin \{0,1\} \end{cases}}$

— Me@2016-05-04 06:32:24 PM

NotesOrganization

Posted on May 14, 2023 by rpflee

"A real person has two reasons for doing anything... 
a good reason and the real reason."

… 

One today worths two tomorrows.

Do one thing well. Do exactly what a customer wants, 
nothing more, nothing less. 

You can't solve all the problems. You can't satisfy 
all the people.

Constructive laziness is one of the cardinal virtues 
of a master programmer.

"such a final theory must be concerned not with fields, 
not even with spacetime, but rather with information 
exchange among physical processes."

… 

04[.]08[.]2003

A good scientist is a person with original ideas. 
A good engineer is a person who makes a design 
that works with as few original ideas as possible. 
There are no prima donnas in engineering. 
-- Freeman Dyson

"There are two kinds of researchers: 
those that have implemented something and 
those that have not. The latter will tell you that 
there are 142 ways of doing things and that 
there isn't consensus on which is best.  
The former will simply tell you that 141 of them 
don't work."

ResearchNotesOrganization

2023.05.14 Sunday ACHK

電腦革命

Posted on May 11, 2023 by rpflee

這段改編自 2021 年 12 月 15 日的對話。

雖然我也偏好用，全新的電腦，但是我原本覺得，不能強求。有電腦用已經難得。並不是每個人，都有機會用到電腦。預算不足時，我也不會否定，買二手電腦來用。

有一次閱讀網民的對答，提點了我，令我的價值觀大改。

如果上一手時已經有病毒，你買了後，仍然會有。

還有，電腦或其零件，其實只有新舊之分。如果是舊的話，你不會知道，它是二手、三手，還是更多。

另外，如果是大型公司，淘汰出來的舊電腦的話，已有的勞損就會特別嚴重，因為，公司電腦會近乎，全天候、無間斷地運行著。

如果更換電腦，並不輕易的話，我就寧願暫時不買；待儲多一點錢時，買一部全新的電腦。

— Me@2023-03-10 11:00:16 AM

Euler problem 12.2

Posted on May 4, 2023 by rpflee

primes :: [Integer]
primes = 2 : filter (null . tail . primeFactors) [3, 5 ..]

primeFactors :: Integer -> [Integer]
primeFactors n = factor n primes
  where
    factor n (p : ps)
      | p * p > n = [n]
      | n `mod` p == 0 = p : factor (n `div` p) (p : ps)
      | otherwise = factor n ps

groupFactors :: [Integer] -> [[Integer]]
groupFactors = gf []
  where
    gf acc lst
      | null lst = reverse acc
      | null acc = gf [[p,1]] ps
      --
      | p == head (head acc) =
        gf ([p, head (tail (head acc)) + 1]:tail acc) ps
      --  
      | otherwise = gf ([p,1]:acc) ps
      where
        p = head lst
        ps = tail lst
        
nDiv :: Integer -> Integer
nDiv n = product (map ((1+) . head . tail)
                  (groupFactors (primeFactors n)))

fm :: Integer -> Integer -> Integer
fm m n
  | triDiv n > m = n*(n+1) `div` 2
  | otherwise = fm m (n+1)
  where
    triDiv n
      | even n = nDiv (n `div` 2)*nDiv (n+1)
      | otherwise = nDiv n*nDiv ((n+1) `div` 2)

λ> :set +s
λ> fm 500 1
76576500
(0.20 secs, 199,313,728 bytes)
λ>

— Me@2023-05-04 09:51:19 AM

Ex 2.2 Stereographic Projection

Posted on April 25, 2023 by rpflee

Functional Differential Geometry

The points on the plane can also be specified with polar coordinates $\displaystyle{(\rho, \theta)}$ and the points on the sphere are specified both by Riemann coordinates and the traditional colatitude and longitude $\displaystyle{(\phi, \lambda)}$ .

(show-expression
 ((compose
   (chart S2-spherical)
   (point S2-Riemann)
   (chart R2-rect)
   (point R2-polar))
  (up 'rho 'theta)))

~~~

1. The code

 
(up 'rho 'theta)

represents the polar coordinates of a point.

2. The function

 
(point R2-polar)

generates an abstract point from a point in R2-polar coordinates.

3. The function

 
(chart R2-rect)

gives the rect coordinates given an abstract point on the plane R2.

(show-expression
 ((compose
   (chart R2-rect)
   (point R2-polar))
  (up 'rho 'theta)))

The procedure (point S2-Riemann) gives the point on the sphere given rectangular coordinates on the plane.

In other words, the function

 
(point S2-Riemann)

generates an abstract point-on-the-sphere (S2) from a point-on-the-plane (R2) in rect coordinates. In other words,

 
S2-Riemann

means

 
S2-rect

Perform an analogous computation to get the polar coordinates of the point on the plane corresponding to a point on the sphere given by its colatitude and longitude.

(show-expression
 ((compose
   (chart R2-polar)
   (point R2-rect)
   (chart S2-Riemann)
   (point S2-spherical))
  (up 'phi 'lambda)))

— Me@2023-04-22 10:42:50 PM

Split-brain, 2.1

Posted on April 20, 2023 by rpflee

y-c-o-m-b 8 months ago | next [–]

I have schizotypal personality disorder. For me it feels like a collective, kind of like the borg. I say “me” or “I” when communicating externally, but all thoughts are actually communicated as “us”. All entities speak with the same voice (what most people think of as their inside voice), but each one speaks differently, with a different pace/tone, and has its own personality, thoughts, and desires. There are a handful of dominant ones that sit in the captain’s chair so to speak, and they all have a say in how we behave externally.

For example one of us is a pacifist, another one loves to socialize and party, one is cautious and anxious, the other very confident, and there’s one that has a severe thirst for violence and blood-lust (this one we work hard to keep in check). I’ve literally been in fights where immediately after knocking my opponent down, I ask if they’re ok and then help them back up and let them go. Then there’s hundreds of transient entities that are usually clones of personalities observed elsewhere (movie characters, celebrities, or other influential people). These transient types will actually adopt the mannerisms, voice, and even accent of the personalities and display them outwardly! Makes for very weird interactions with family and friends lol.

I’ve yet to see anyone on reddit or elsewhere with this description of the disorder. The closest thing is DID or maybe even borderline personality disorder (which is on the same schizo spectrum), but there’s no disassociation with what I have, we’re all fully aware of what the other is thinking.

— Split Brain Psychology

— Hacker News

2023.04.20 Thursday ACHK

Heat

Posted on April 18, 2023 by rpflee

When you can’t make them see the light, make them feel the heat.

— Ronald Reagan

2023.04.18 Tuesday ACHK

權力來源 1.4

Posted on April 17, 2023 by rpflee

這段改編自 2010 年 4 月 24 日的對話。

勢

~ environmental default

— Me@2021-11-16 07:59:12 PM

勢

~ potential

~ 未來事件s對現在的影響

— Me@2023-03-12 09:14:50 AM

potential energy

~ energy of potential motion

~ energy of future motion

— Me@2023-04-16 12:08:08 PM

權力

~ 勢力

大概而言，「權」即是「勢」。但是，兩詞給人的感覺不同。根本的原因是，它們的用法，不大相同。

「權力」著重個人，例如：「甲擁有很大的權力。」「勢」則著重外在。甲身處環境的人事心理結構，形成了一個「勢」。如果當時的那個「勢」對甲有利，就簡稱為：「甲擁有很大的勢力。」

— Me@2023-04-17 12:56:11 AM

MacBook Pro A1502

Posted on April 11, 2023 by rpflee

Euler problem 12.1.2

(defun good-reverse (lst)
  (labels ((rev (lst acc)
             (if (null lst)
                 acc
                 (rev
                  (cdr lst)
                  (cons (car lst) acc)))))
    (rev lst nil)))

(defparameter *primes* '(2 3 5))

(defun primeFactors (n)
  (factor-iter n *primes* '()))

(defun range (max &key (min 0) (step 1))
  (loop :for n :from min :below max :by step
        collect n))

(defun primep (x)
  (NULL (cdr (primeFactors x))))

(defun factor-iter (n p-list acc-list)
  (let* ((p (car p-list))
         (ps (cdr p-list)))
    (cond ((> (* p p) n)
           (good-reverse (cons n
                               acc-list)))
          ((eql (mod n p) 0)
           (factor-iter (floor n p)
                        p-list
                        (cons p acc-list)))
          ((NULL ps)
           (let* ((op
                    *primes*)
                  (num-extend
                    (range (1+
                            (ceiling (sqrt n)))
                           :min (+ p 2)
                           :step 2))
                  (primes-extend
                    (remove-if-not
                     #'primep
                     num-extend)))
             (if (NULL primes-extend)
                 (cons n acc-list)                
                 (progn
                   (setf *primes*
                         (append op primes-extend))
                   (factor-iter n
                                primes-extend
                                acc-list)))))
          ('t
           (factor-iter n ps acc-list)))))

(defmacro prime-filter (n)
  `(remove-if-not #'primep
                  (cons 2
                        (range (1+ ,n)
                               :min 3
                               :step 2))))

(time (length (prime-filter 20000000)))

;; Evaluation took:
;; 13.056 seconds of real time
 
;; 1270607

— Me@2023-03-28 11:51:24 AM

Ex 1.28 The energy function

Posted on April 7, 2023 by rpflee

Structure and Interpretation of Classical Mechanics

An analogous result holds when the $f_\alpha$ ‘s depend explicitly on time.

…

c. Show, using Euler’s theorem, that the energy function is $\mathcal{E} = A + B$ .

~~~

$\displaystyle{ \begin{aligned} f(t x_1, t x_2, ...) &= t^n f( x_1, x_2 , ...), \\ \end{aligned}}$

then

$\displaystyle{ \begin{aligned} \sum_{i} x_i \frac{\partial f}{\partial x_i} &= n f(\mathbf{x}) \\ \end{aligned}}$

— Euler’s Homogeneous Function Theorem

$\displaystyle{\begin{aligned} L + D_t F &= A - B \\ \\ D_t F &= - \frac{1}{2} \sum_\alpha m_\alpha \left \{ g_\alpha(t,q) + [\partial_0 f_\alpha (t,q)]^2 + 2 \partial_0 f_\alpha (t,q) \partial_1 f_\alpha (t,q) v \right \} \\ \\ A &= \frac{1}{2} \sum_\alpha m_\alpha [\partial_1 f_\alpha (t,q) ]v^2 \\ - B &= - V(t, q) - \frac{1}{2} \sum_\alpha m_\alpha g_\alpha(t,q) \\ \end{aligned}}$

This answer is not totally correct, since the generalized velocity, $v$ , should be a vector.

— Me@2022-11-01 08:58:52 AM

Eq. (1.133):

$\displaystyle{\mathcal{P}_i = (\partial_2 L)_i}$

Eq. (1.144):

$\displaystyle{\mathcal{E} = \mathcal{P} \dot Q - L}$ ,

where $\displaystyle{\mathcal{P}}$ is the momentum state function.

$\displaystyle{\begin{aligned} \mathcal{E} &= \mathcal{P} \dot Q - A + B \\ &= \left( \partial_2 L \right) \dot Q - A + B \\ &= \left( \partial_2 (A - B) \right) \dot Q - A + B \\ \end{aligned}}$

Since $B$ has no velocity dependence,

$\displaystyle{\begin{aligned} \mathcal{E} &= \left( \partial_2 A \right) \dot Q - A + B \\ &= \begin{bmatrix} (\partial_2 A)_1 & (\partial_2 A)_2 & ... \end{bmatrix} \begin{bmatrix} \dot Q_1 \\ \dot Q_2 \\ \vdots \end{bmatrix} - A + B \\ \\ \end{aligned}}$

Since $A$ is a homogeneous function of the generalized velocities of degree 2, by Euler’s Homogeneous Function Theorem,

$\displaystyle{ \begin{aligned} \sum_{i} v_i \frac{\partial A}{\partial v_i} &= 2 A(v) \\ \end{aligned}}$

— Me@2023-04-06 12:49:49 PM

無限思維

Posted on April 2, 2023 by rpflee

Thinking in terms of infinity

「無限」不是數字，不能用來比較大小。

使用「無限」，即是不再權衡輕重。

— Me@2016-05-24 12:12:33 PM

無限思維 = 無思維

— Me@2023-04-02 03:38:07 PM

Infinity is not a number. So it cannot be used for comparing.

As long as you use “infinity”, you stop weighing the pros and cons.

In other words, you stop reasoning.

— Me@2023-04-02 01:51:06 PM

“Optimization without constraints” is not optimization.

— Me@2023-04-02 03:41:19 PM

History 5

Posted on April 1, 2023 by rpflee

bambax on June 17, 2018

Not just music history, but history in general.

History told forwards is superstition: trying to show that what came after was inevitable, when in fact it was one future among an infinite number of different possible futures.

History told backwards makes sense: look for the seeds of post events, in the past.

However the human mind loves nothing more than causation; we see cause and effect everywhere even where it’s not, we like “stories”, we don’t understand, don’t believe in, and outright reject chance.

— We Should Teach Music History Backwards

— Hacker News

2023.04.01 Saturday ACHK

運動即速度；速度即所見

Posted on March 30, 2023 by rpflee

92-CE-PHY II – 4 前傳

— Me@2023-03-30 10:10:29 AM

Euler problem 12.1.1

Posted on March 27, 2023 by rpflee

(defmacro sq (x)
  `(* ,x ,x))

(defmacro last-item (lst)
  `(car (last ,lst)))

(defun good-reverse (lst)
  (labels ((rev (lst acc)
             (if (null lst)
                 acc
                 (rev
                  (cdr lst)
                  (cons (car lst) acc)))))
    (rev lst nil)))

(defun prime-sieve-a-list (input-lst)
  (labels ((sieve-iter (go-lst acc-list)
             (if (not go-lst) 
                 acc-list        
                 (if (> (sq (car go-lst))
                        (last-item go-lst))

                     (append (good-reverse acc-list)
                             go-lst)
                     
                     (sieve-iter
                      (remove-if #'(lambda (x)
                                     (=
                                      (mod x
                                           (car go-lst))
                                      0))
                                 (cdr go-lst))
                      (cons (car go-lst)
                            acc-list))))))

    (sieve-iter input-lst '())))

(defun range (max &key (min 0) (step 1))
  (loop :for n :from min :below max :by step
        collect n))

(defmacro prime-sieve (n)
  `(prime-sieve-a-list (cons 2
                             (range (1+ ,n)
                                    :min 3
                                    :step 2))))


(time (length (prime-sieve 20000000)))

;; Evaluation took:
;; 19.907 seconds of real time

What is the value of the first triangle number to have over five hundred divisors?

(defun factor-iter (n p-list acc-list)
  (if (NULL p-list)
      acc-list         
      (let* ((p (car p-list))                                   
             (ps (cdr p-list)))  
        (cond ((> (* p p) n)           
               (good-reverse (cons n
                                   acc-list)))
              ((eql (mod n p) 0)
               (factor-iter (floor n p)
                            p-list
                            (cons p acc-list)))   
              ('t
               (factor-iter n ps acc-list))))))

(defparameter *pm* 2000000)

(defparameter *psi* (prime-sieve *pm*))

(defun factor (n)
  (if (> n (expt *pm* 2))
      
      (let ((m (floor (sqrt n))))
        (factor-iter n (prime-sieve m) '()))
      
      (factor-iter n *psi* '())))

(defun group-factors (lst)
  (labels ((gf (acc lst)
             (if (NULL lst)
                 (good-reverse acc)
                 (let* ((p (car lst))
                        (ps (cdr lst))
                        (lp1 (list p 1)))
                   (if (NULL acc)                      
                       (gf (list lp1) ps)                      
                       (if (eql p (caar acc))
                           (gf (cons
                                (list p
                                      (+ 1
                                         (cadar acc)))
                                (cdr acc))
                               ps)                  
                           (gf (cons lp1 acc) ps)))))))    
    (gf '() lst)))

(defmacro sum (lst)
  `(reduce #'+ ,lst))

(defmacro product (lst)
  `(reduce #'* ,lst))

(defun nDiv (n)
  (product (mapcar #'(lambda (x) (1+ (cadr x)))
                   (group-factors (factor n)))))

(defun fm (m n)
  (labels ((tri-div (n)
             (if (evenp n)
                 (* (nDiv (/ n 2)) (nDiv (1+ n)))
                 (* (nDiv n) (nDiv (/ (1+ n) 2))))))    

    (if (> (tri-div n) m)
        (/  (* n (1+ n)) 2) 
        (fm m (+ 1 n)))))

;

(time (fm 500 1))

;; Evaluation took:
;; 0.007 seconds of real time

;; 76576500

— Me@2023-03-25 07:51:18 PM

3.4 Electric fields and potentials of point charges

Posted on March 18, 2023 by rpflee

A First Course in String Theory

(a) Show that for time-independent fields, the Maxwell equation $\displaystyle{T_{0ij}=0}$ implies that $\displaystyle{\partial_i E_j - \partial_j E_i = 0}$ . Explain why this condition is satisfied by the ansatz $\displaystyle{\vec E = - \nabla \Phi}$ .

~~~

Eq. (3.23):

$\displaystyle{ \begin{aligned} T_{\mu \lambda \nu} &= \partial_\lambda F_{\mu \nu} + \partial_\mu F_{\nu \lambda} + \partial_\nu F_{\lambda \mu} \\ \end{aligned}}$

…

$\displaystyle{ \begin{aligned} &\vec E \\ &= - \nabla \Phi \\ &= - \left( \partial_x, \partial_y, \partial_z \right) \Phi \\ \end{aligned}}$

$\displaystyle{ \begin{aligned} &\partial_i E_j - \partial_j E_i \\ &= \partial_i \partial_j \Phi - \partial_j \partial_i \Phi \\ \end{aligned}}$

— Me@2023-03-18 11:08:24 AM

Debugging quantum concepts

Posted on March 17, 2023 by rpflee

Reality is a superposition of eigenstates. Even if we use the pilot-wave formalism, in which a particle has definite position or momentum, the pilot wave itself is in a superposition.

— Me@2012-04-16 2:27:20 PM

Physics reality is NOT a superposition of eigenstates. If physics reality was a superposition of eigenstates, there would have never been any interference patterns.

For an experimental setup, what is in a superposition is the quantum state, which is a tool for deducing probabilities of different potential measurement results.

“A quantum state is a superposition of eigenstates” just means nothing more than that we need to use individual probabilities of the eigenstates to calculate the probabilities.

A quantum state, which is represented by a wave function, is logical, mathematical, conceptual, and linguistic in nature. A quantum state is NOT physical. A quantum state is NOT reality. A quantum state is NOT directly corresponding to a physical reality (aka observable events, measurement results, etc.)

A quantum state is NOT even corresponding to a probability directly. (If a quantum state was a probability, there would have never been the phenomenon of interference.) Instead, a quantum state is corresponding to a probability amplitude, which is used for calculating probabilities.

— Me@2023-03-16 09:57:07 AM

Physics Town

continues