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

Physics Town

continues

Monthly Archives: April 2023