Privacy Statement

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Last Updated: November 14, 2025

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Apps Covered:

1. Keyboard Calendar
2. Keyboard Date Stamp Light
3. Keyboard Date Stamp Pro

Introduction: Our Android apps are designed to enhance user experience by storing settings locally on the device without transmitting them over the internet. We prioritize user privacy and ensure that no personal data is collected or shared.

Data Collection: We want to assure our users that our apps do not collect any personal information, such as names, email addresses, or location data. The apps solely store user preferences and settings locally on the device.

Permissions Note: In particular, Keyboard Calendar requires access to the device’s calendar (via READ_CALENDAR and WRITE_CALENDAR permissions) to enable adding events while typing; this access is local-only and does not involve data transmission.

Keyboard Date Stamp Light and Pro function as input methods and may use standard Android keyboard permissions (e.g., BIND_INPUT_METHOD). They require no special permissions beyond basic app functionality.

Data Security: We take the privacy and security of our users’ data seriously. Since our apps do not transmit any data over the internet, there is minimal risk of data breaches or unauthorized access to personal information.

App Settings Backup: As a default Android setting, the apps allow backup of local app data via the device’s backup service (e.g., to Google Drive). This occurs through Android’s default auto-backup feature, which is enabled by default but configurable in device settings (e.g., Settings > System > Backup). The apps do not send any data over the internet by themselves, and backups are handled entirely by the Android system.

User Control: We believe in empowering our users to have full control over their settings. Users can easily modify or delete their preferences within the apps at any time.

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2024.01.20 Saturday ACHK

眾害取最輕

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眾害取其輕 | The least of all evils, 13 | 友情演義 2 | 相聚零刻 3 | 無拍之拖 3 | 原來是你 4

這段改編自 2023 年 6 月 20 日的對話。

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第一，我們萬不可在，自己的生存並未受威脅時，為了換取現實利益，而犧牲道德原則。

第二，在我們的生活勉強可過時，萬不可因要得到較佳報酬，而犧牲他人。

第三，當我們因生活困難，而被迫不得不放棄，若干作人的原則時，我們必須儘可能作「道德的抗戰」，把道德的領土放棄的愈少愈好；而且要存心待機「收復道德的失地」。

— 人生的意義

— 殷海光

— 加了格式和標點

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有時，你會被迫做，一些「必要之惡」。但是，那並不代表，你沒有任何責任，去盡快盡量減輕，那「必要之惡」。

例如，結婚前叫做「拍拖時期」。而拍拖前，則是「曖昧時期」。

那是一段模稄兩可，事後既可以標籤為「友情」，亦可能標籤成「愛情」的時期。如果成功發展成情侶的話，那就是愛情的開端時期；如果不成功的話，那就不屬於什麼特別時期，因為那段時空，只是日常一般，朋友間的交往。

維持「曖昧時期」，不是健康的關係，因為，如果一邊覺得只是友情，而另一邊卻覺得不是的話，即引起極大誤會，所以我視「曖昧時期」為「害」。

但那是「必要之惡」，因為人不會未卜先知；你總不能，亦總不會，在從來未相處暸解過，就喜歡別人，向她示愛。

…

— Me@2023-10-12 07:55:14 PM

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2024.01.18 Thursday (c) All rights reserved by ACHK

7z to encrypt folder with AES 256

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Sequential speed, 2 | Restore, 2

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7za a -mmt4 -tzip -p -mem=AES256 /target_folder/the_zip_file.zip /source_folder/

a

Add files to archive

-mmt[N]

set number of CPU threads

-t{Type}

Set type of archive

-p{Password}

set Password

-m{Parameters}

set compression Method

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How to check the encryption algorithm of a 7z/zip file?

7z l -slt /target_folder/file.zip

l

List contents of archive

-slt

show technical information for l (List) command

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To unzip:

7za x /source_folder/the_zip_file.7z -o/target_folder/

x

eXtract files with full paths

-o{Directory}

set Output directory

— Me@2023-11-26 12:17:46 PM

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2024.01.14 Sunday (c) All rights reserved by ACHK

3.6 Analytic continuation for gamma function, 4

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A First Course in String Theory

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…

Explain why the above right-hand side is well defined for \displaystyle{ \Re (z) > - N - 1 }.

…

~~

…

\displaystyle{ \begin{aligned} \Gamma (z) &= \int_{0}^{1}t^{z-1}e^{-t}dt + \int_{1}^{\infty}t^{z-1}e^{-t}dt \\\\ \end{aligned} \\ }

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Prove that \displaystyle{\int_{1}^{\infty}t^{z-1}e^{-t}dt} converges.

…

Prove that \displaystyle{\int_{0}^{1}t^{z-1}e^{-t}dt} converges.

…

Under this condition,

\displaystyle{ \begin{aligned} &\int_{0}^{1}t^{z-1}e^{-t}dt \end{aligned} }
\displaystyle{ = \int_{0}^{1} t^{z-1} \left( \sum_{n=0}^N \frac{(-t)^n}{n!} +\sum_{n=N+1}^\infty \frac{(-t)^n}{n!} \right) dt, }
\displaystyle{ = \int_{0}^{1} t^{z-1} \sum_{n=0}^N \frac{(-t)^n}{n!} dt + \int_{0}^{1} t^{z-1} \sum_{n=N+1}^\infty \frac{(-t)^n}{n!} dt}

\displaystyle{ = \sum_{n=0}^N \frac{(-1)^n}{n!} \frac{1}{z+n} + \int_{0}^{1} t^{z-1} \sum_{n=N+1}^\infty \frac{(-t)^n}{n!} dt}

So for \displaystyle{\Re (z) > 0},

\displaystyle{ \begin{aligned} &\Gamma (z) \\ &= \int_{0}^{\infty}t^{z-1}e^{-t}dt \\ &= \int_{0}^{1}t^{z-1}e^{-t}dt + \int_{1}^{\infty}t^{z-1}e^{-t}dt \\ &= \sum_{n=0}^N \frac{(-1)^n}{n!} \frac{1}{z+n} + \int_{0}^{1} t^{z-1} \sum_{n=N+1}^\infty \frac{(-t)^n}{n!} dt + \int_{1}^{\infty}t^{z-1}e^{-t}dt \\ \end{aligned} \\ }

If we define this as the analytic continuation of Gamma function:

\displaystyle{ \begin{aligned} \Gamma_c (z) \end{aligned} \\ }

\displaystyle{ \begin{aligned} = \sum_{n=0}^N \frac{(-1)^n}{n!} \frac{1}{z+n} + \int_{0}^{1} t^{z-1} \sum_{n=N+1}^\infty \frac{(-t)^n}{n!} dt + \int_{1}^{\infty}t^{z-1}e^{-t}dt \\ \end{aligned} \\ },

its convergence depends on the convergence of the middle term:

\displaystyle{ \begin{aligned} \int_{0}^{1} t^{z-1} \sum_{n=N+1}^\infty \frac{(-t)^n}{n!} dt \\ \end{aligned} \\ }

By the argument above, a necessary condition for this term to converge is

\displaystyle{ \Re (z) + n_{\text{min}} > 0 }

So

\displaystyle{ \Re (z) + N + 1 > 0 }

\displaystyle{ \Re (z) > - N - 1 }

However, besides the integral convergence, we have to consider also the summation convergence.

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To check whether the order of integration and summation can be interchanged, we rewrite the term as

\displaystyle{ \begin{aligned} &\int_{0}^{1} t^{z-1} \sum_{n=N+1}^\infty \frac{(-t)^n}{n!} dt \\ &= \int_{0}^{1} t^{z-1} \left( e^{-t} - \sum_{n=0}^N \frac{(-t)^n}{n!} \right) dt \\ \end{aligned} \\ }

Since both \displaystyle{e^{-t}} and \displaystyle{\sum_{n=0}^N \frac{(-t)^n}{n!}} are finite, \displaystyle{ \begin{aligned} \sum_{n=N+1}^\infty \frac{(-t)^n}{n!} dt \end{aligned} \\ } is also finite.

…

— Me@2023-09-07 07:38:17 PM

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2024.01.10 Wednesday (c) All rights reserved by ACHK

What is absolute is the change of shape

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@dialectphilosophy, 1.3.4

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…

The key point is that the observer within the car can see the separation changes among some objects, including the car itself, within the car.

Besides seeing the separation changes, the observer can also feel the acceleration directly within his body. The feeling of force is also due to the separation changes, but among points of his body.

How come a free falling frame is equivalent to an inertia frame? In other words, if acceleration is really absolute, how come a free falling observer cannot feel the acceleration?

Short answer: Acceleration is absolute in general, but not in all cases. What is absolute are the particles’ positions relative to each other in a physical system.

Long answer:

For two free falling objects, if they start to fall at the same time, they have equal initial velocities. Then, according to the equation

s = ut + \frac{1}{2}at^2,

the displacements of them are always the same. So their separation is always constant.

For the two objects, their displacements are

\begin{aligned} s_1 &= u_1 t_1 + \frac{1}{2} a_1 t_1^2 \\ s_2 &= u_2 t_2 + \frac{1}{2} a_2 t_2^2 \\ \end{aligned}

If u_1 = u_2, t_1 = t_2, and a_1 = a_2, then s_1 = s_2. As a result, their separation remains unchanged.

Note that the separation would remain unchanged only if

1. the acceleration is uniform (in space) so that a_1 = a_2 always; and

2. both initial velocities have an identical value, i.e. u_1 = u_2.

So what really is absolute is not acceleration, but the separation changes among points seen by the observer within the physical system. Let us label “the separations among points” as “the spatial configuration”, or as an even simpler term: “the shape of the physical system”.

— Me@2023-12-06 11:06:23 AM

— Me@2023-12-24 05:58:54 PM

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2024.01.10 Wednesday (c) All rights reserved by ACHK

上山尋寶, 3

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It’s the time that you spent on your rose that makes your rose so important.

— The Little Prince

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In economics and business decision-making, a sunk cost (also known as retrospective cost) is a cost that has already been incurred and cannot be recovered. Sunk costs are contrasted with prospective costs, which are future costs that may be avoided if action is taken. In other words, a sunk cost is a sum paid in the past that is no longer relevant to decisions about the future.

Even though economists argue that sunk costs are no longer relevant to future rational decision-making, people in everyday life often take previous expenditures in situations, such as repairing a car or house, into their future decisions regarding those properties.

— Wikipedia on Sunk cost

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2024.01.06 Saturday ACHK

她有騙我嗎？

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我們不會無緣無故地，憑空，知道世界。

我們只能透過（直接或間接的）觀察世界（，加上推理），去認識世界。

我們只能透過現象，去瞭解本體。

我們不會憑空知道「全部事實」，那是「全知」。

我們只能透過，收集事實的部分，去了解事實（的全部）。

— Me@2023-11-14 10:59:22 AM

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2024.01.02 Tuesday (c) All rights reserved by ACHK

Euler problem 18.2

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tri = [[75],
       [95,64],
       [17,47,82],
       [18,35,87,10],
       [20,04,82,47,65],
       [19,01,23,75,03,34],
       [88,02,77,73,07,63,67],
       [99,65,04,28,06,16,70,92],
       [41,41,26,56,83,40,80,70,33],
       [41,48,72,33,47,32,37,16,94,29],
       [53,71,44,65,25,43,91,52,97,51,14],
       [70,11,33,28,77,73,17,78,39,68,17,57],
       [91,71,52,38,17,14,91,43,58,50,27,29,48],
       [63,66,04,68,89,53,67,30,73,16,69,87,40,31],
       [04,62,98,27,23,09,70,98,73,93,38,53,60,04,23]]
  
pathAddItem x (y:ys) (z:zs)
  = (x+m):x:ms
  where 
    (m:ms) | y > z = y:ys
           | otherwise = z:zs
           
listApplyPathAddItem xs ys
  = zipWith3 f xs ys $ tail ys 
  where
    f = pathAddItem
   
e18 = foldr g acc (init tri) 
  where
    toPair x = [x,x]
    g = listApplyPathAddItem
    acc = map toPair $ last tri

— Me@2023-12-28 11:25:22 AM

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2023.12.28 Thursday (c) All rights reserved by ACHK

3 Vector Fields and One-Form Fields, 1.2

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p. 21

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5.

\displaystyle{(Df(x)) b(x) \ne (Df(x)) \Delta x}

Instead, \displaystyle{(Df(x)) b(x)} is a generalization of \displaystyle{(Df(x)) \Delta x}.

6.

However, how to calculate \displaystyle{(Df(x)) b(x)}?

By this:

\displaystyle{f(x + \Delta x) \approx f(x) + (Df(x)) \Delta x}

Then:

\displaystyle{(Df(x)) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}}

So:

\displaystyle{(Df(x)) b(x) = \left( \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} \right) b(x)}

7.

b is written as subscript to capture the meaning of “with respect to b“. The original directional derivative uses the same convention:

So the spatial rate of change of \displaystyle{f} along the direction of the vector \displaystyle{\mathbf{v}} is

\displaystyle{\begin{aligned} D_{\mathbf{v}}(f) &= \frac{\left(\delta f\right)_{\mathbf{v}}}{|\mathbf{v}|} \\ &= \frac{1}{|\mathbf{v}|} \left( \frac{\partial f}{\partial x} \delta x + \frac{\partial f}{\partial x} \delta y \right) \\ &= \frac{\partial f}{\partial x} \frac{\delta x}{\sqrt{(\delta x)^2 + (\delta y)^2}} + \frac{\partial f}{\partial x} \frac{\delta y}{\sqrt{(\delta x)^2 + (\delta y)^2}} \\ &= \left(\nabla f\right) \cdot \frac{\mathbf{v}}{|\mathbf{v}|} \\ &= \left(\nabla f\right) \cdot \hat{\mathbf{v}} \\ \end{aligned}}

\displaystyle{D_{\mathbf{v}}(f)} is called directional derivative.

— Me@2016-02-06 09:49:22 PM

— Me@2023-12-27 01:37:32 PM

— Functional Differential Geometry

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2023.12.27 Wednesday (c) All rights reserved by ACHK

@dialectphilosophy, 1.3.3

Posted on by

…

That is what “acceleration is absolute” means. The observer can notice different phenomena, compared with the no-acceleration case, even without seeing outside the car window.

The additional meaning of “acceleration is absolute” is that we can deduce the acceleration value by measuring v_{2c}(0), v_{2c}(t), and t. And, unlike velocity values, this acceleration value is identical for any two observers related by a Galilean transformation. (They are called inertial observers.)

.

To see a more direct consequence of accelerating the car, we can consider an even simpler case where v_{2c}=0. Then we split the case into two, one with acceleration and one without.

In the no acceleration case, the particle 2 keeps co-moving with the car at the same velocity. In this case, the distance between particle 2 and a point of the car is always the same, no matter what the car’s velocity is. So it is impossible to deduce that velocity by observing only the distance between any two points within the car.

The car velocity value observed is different for different observers, depending on each observer’s velocity. That is what “velocity is relative” means.

However, if the car accelerates, although for an observer co-moving with the car, what he sees is not the car accelerating, but the particle 2 accelerates in the opposite direction; he can measure the acceleration based on the separations (between points within the car), the separation changes, and rate of separation changes; and he will get an acceleration value of the same magnitude.

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The key point is that the observer within the car can see the separation changes among some objects, including the car itself, within the car.

…

— Me@2023-12-06 11:06:23 AM

— Me@2023-12-24 05:58:54 PM

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2023.12.26 Tuesday (c) All rights reserved by ACHK

NDE

Posted on by

22 07 2003

“Do not concentrate too much on being useful to others, but concentrate more on being useful to yourself.”
 

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2023.12.24 Sunday ACHK

宇宙的琴弦

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這段改編自 2010 年 4 月 24 日的對話。

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中學時代讀男校，少與女性相處，所以印象中，男仕是怪獸，女仕是仙子。多年後發現，那不一定。

比較內心而言，有一部分男仕，其實優雅過一部分女仕。

（你又怎會知道，他人的內心優不優美呢？）

觀察她的言行舉止就可以。例如，如果她有網誌的話，你可以閱讀之，看看她通常寫什麼題材。

— Me@2023-12-22 04:42:28 PM

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2023.12.23 Saturday (c) All rights reserved by ACHK

Euler problem 18.1

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(ql:quickload "str")

;; (defun file-get-contents (filename)
;;   (with-open-file (stream filename)
;;     (let ((contents (make-string
;;                      (file-length stream))))
;;       (read-sequence contents stream)
;;       contents)))

(defun file-get-lines (filename)
  (with-open-file (stream filename)
    (loop :for line = (read-line stream nil)
          :while line
          :collect line)))

;; (defun string-to-list (the-string)
;;   (loop :for char :across the-string
;;         :collect char))

;; (defun char-to-integer-list (char-list)
;;   (mapcar #'digit-char-p char-list))

(defun path-add-item (x yys zzs)
  (let* ((y (car yys))
         (ys (cdr yys))
         (z (car zzs))
         (zs (cdr zzs))
         (m (if (> y z) y z))
         (ms (if (> y z) ys zs)))
    (cons (+ x m) (cons x ms))))

(defmacro zipWith3 (f xs ys zs)
  `(mapcar ,f ,xs ,ys ,zs))

(defun path-add-item-zip (acc xs)
  (zipWith3 #'path-add-item xs acc (cdr acc)))

(defun foldl (f acc xs)
  (if (not xs)
      acc
      (foldl f (funcall f acc (car xs)) (cdr xs))))

(defun string-split (x)
  (str:split #\space x))

(defun map-parse-integer (xs)
  (mapcar #'parse-integer xs))

(defmacro map-string-split (xs)
  `(mapcar #'string-split ,xs))

(defmacro head (xs)
  `(car ,xs))

(defmacro tail (xs)
  `(cdr ,xs))

(defun pair-double (x)
  (list x x))

(defun e18 ()
  (let* ((string-lines (file-get-lines #P"n.txt"))
         (string-lists (map-string-split string-lines))
         (number-tower (mapcar #'map-parse-integer
                               string-lists))
         (r-tower (reverse number-tower))
         (acc (mapcar #'pair-double (head r-tower))))
    
    (foldl #'path-add-item-zip acc (tail r-tower))))


CL-USER> (e18)
((1074 75 64 82 87 82 75 73 28 83 32 91 78 58 73 93))
CL-USER>

— Me@2023-12-20 07:32:00 PM

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2023.12.21 Thursday (c) All rights reserved by ACHK

Taylor expansion of f around x

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Series can also be formed by processes such as exponentiation of an
operator or a square matrix. For example, if f is any function of one
argument, and if x and dx are numerical expressions, then this
expression denotes the Taylor expansion of f around x.

(((exp (* dx D)) f) x)
= (+ (f x) (* dx ((D f) x)) (* 1/2 (expt dx 2) (((expt D 2) f) x)) ...)

— refman

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(((exp (* 'dx D)) (literal-function 'f)) 'x)
#| (*series* (0 . 0) (*number* (expression (f x)) (literal-function #[apply-hook 40]) (type-expression Real)) . #[promise 41 (unevaluated)]) |#

(show-expression
 (series:print (((exp (* 'dx D))
                 (literal-function 'f)) 'x)))

(f x)
(* dx ((D f) x))
(* 1/2 (expt dx 2) (((expt D 2) f) x))
(* 1/6 (expt dx 3) (((expt D 3) f) x))
(* 1/24 (expt dx 4) (((expt D 4) f) x))
(* 1/120 (expt dx 5) (((expt D 5) f) x))
(* 1/720 (expt dx 6) (((expt D 6) f) x))
(* 1/5040 (expt dx 7) (((expt D 7) f) x))
(* 1/40320 (expt dx 8) (((expt D 8) f) x))
(* 1/362880 (expt dx 9) (((expt D 9) f) x))
(* 1/3628800 (expt dx 10) (((expt D 10) f) x))
(* 1/39916800 (expt dx 11) (((expt D 11) f) x))
(* 1/479001600 (expt dx 12) (((expt D 12) f) x))
(* 1/6227020800 (expt dx 13) (((expt D 13) f) x))
(* 1/87178291200 (expt dx 14) (((expt D 14) f) x))
;Aborting!: maximum recursion depth exceeded

— Me@2023-12-20 11:30:50 AM

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2023.12.20 Wednesday (c) All rights reserved by ACHK

@dialectphilosophy, 1.3.2

Posted on by

…

He can deduce the relative velocity v_2 - v_1 by the separations x_2(t) - x_1(t) and x_2(0) - x_1(0). However, he still cannot deduce v_2 nor v_1 unless he is able to look outside the car window. Thus, he cannot deduce the car speed v just by observing the positions and velocities of the objects inside the car.

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For simplicity, assume object 1 is actually a point of the car itself. So v_1 is actually the speed of the car, v. Then the calculation

\displaystyle{ \begin{aligned} &x_2(t) - x_1(t) \\ &= \Bigl( x_2(0) + v_2 t \Bigr) - \Bigl(x_1(0) + v_1 t \Bigr) \\ &= \Bigl( x_2(0) - x_1(0) \Bigr) + \Bigl( v_2 - v_1 \Bigr) t \\ \end{aligned}}

becomes

\displaystyle{ \begin{aligned} &x_2(t) - x(t) \\ &= \Bigl( x_2(0) - x(0) \Bigr) + \Bigl( v_2 - v \Bigr) t \\ \end{aligned}}

where x is a point of the car.

In this case, x_{2c} = x_2(t) - x(t) becomes the position of object 2 relative to car; and v_{2c} = v_2 - v becomes the velocity of object 2 relative to the car. The equation can be simplified to

\displaystyle{ \begin{aligned} x_{2c}(t) &= x_{2c}(0) + v_{2c} t \\ \end{aligned}}

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If the car has acceleration, the story is totally different. In short, for the observer inside the car, the path of each particle is not a straight line anymore. In long, the previous calculation becomes

\displaystyle{ \begin{aligned} &x_2(t) - x(t) \\ &= \Bigl( x_2(0) + v_2 t - 1/2 a t^2 \Bigr) - \Bigl(x(0) + v t - 1/2 a t^2 \Bigr) \\ &= \Bigl( x_2(0) - x(0) \Bigr) + \Bigl( v_2 - v \Bigr) t \\ \end{aligned}}

where a is the acceleration of the car. Here, we assume that v_1, v_2, and a are all pointing in the same direction.

Although the result is the same as before:

\displaystyle{ \begin{aligned} x_{2c}(t) &= x_{2c}(0) + v_{2c} t, \\ \end{aligned}}

the velocity v_{2c} is no longer a constant; it would keep decreasing.

In the no acceleration case, even if the particle velocity and the car velocity are not in parallel, the observer will see a straight path. However, in the accelerated case where the acceleration and velocity directions are not in parallel, the path of the particle will no longer be a straight line.

That is what “acceleration is absolute” means. The observer can notice different phenomena, compared with the no-acceleration case, even without seeing outside the car window.

The additional meaning of “acceleration is absolute” is that we deduce the acceleration value by measuring v_{2c}(0), v_{2c}(t), and t. And, unlike velocity values, this acceleration value is identical for any two observers related by a Galilean transformation. (They are called inertial observers.)

— Me@2023-12-06 11:06:23 AM

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2023.12.16 Saturday (c) All rights reserved by ACHK

神必成魔

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Some of the biggest cases of mistaken identity are among intellectuals who have trouble remembering that they are not God.

— Thomas Sowell

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1. 魔來自神；

2. 神必成魔，

因為神自以為，高人一等。

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間中做好事，永不做好人。

— Me@2023-12-02 12:37:01 AM

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贓官可恨，人人知之，清官尤可恨人多不知，蓋贓官自知有病，不敢公然為非；清官則自以為不要錢，何所不可，剛愎自用，小則殺人，大則誤國，吾人親目所見，不知凡幾矣。

唉！天下大事壞於奸臣手上的十之三四，壞於不通世故的君子手上的倒有十分之六七！

— 老殘遊記

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壞人是壞

好人也壞

因為時刻干涉他人的人生

沒有邊界感，非禮也

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好人比壞人更恐怖

壞人害怕報復

好人無所顧忌

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壞人企圖，謀財而害命

好人無知，殺人於無形

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壞人你會有所提防

好人你會不知不覺

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壞人可以武力對付

好人只能積極逃避

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邪惡可以激烈抵抗

愚蠢只能無可奈何

— Me@2023-12-02 12:37:01 AM

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魔鬼，也是由天使變成的。

不企圖做天使，就不會變成魔鬼。

— Me@2016-08-06 03:02:36 PM

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You either “die” a hero, or live long enough to see yourself become the villain.

— Harvey Dent

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Don’t try to be a god.

Remember, reaching the maximum also means going to fall.

— Me@2023-12-02 11:40:09 PM

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2023.12.13 Wednesday (c) All rights reserved by ACHK

Eval the buffer

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(defun common-lisp-eval (str)
  (unless (slime-current-connection)
    (let ((wnd (current-window-configuration)))
      (slime)
      (while (not (and (slime-current-connection)
                       (get-buffer-window
                        (slime-output-buffer))))
        (sit-for 0.2))
      (set-window-configuration wnd)))
  (let (deactivate-mark)
    (slime-eval `(swank:eval-and-grab-output ,str))))

(defun file-eval ()
  (interactive)
  (message
   (car (common-lisp-eval
         (concat "(format t \"~{~a~%~}\" (list "
                 (string-make-unibyte
                  (buffer-string)) "))")))))

(defun region-eval (from to)
  (interactive "r")
  (message
   (car (common-lisp-eval
         (concat "(format t \"~{~a~%~}\" (list "
                 (string-make-unibyte
                  (buffer-substring
                   from to)) "))")))))

(global-set-key (kbd "C-F") 'file-eval)
(global-set-key (kbd "C-R") 'region-eval)

— Me@2023-12-10 09:59:57 AM

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2023.12.11 Monday (c) All rights reserved by ACHK

3.6 Analytic continuation for gamma function, 3

Posted on by

A First Course in String Theory

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…

Explain why the above right-hand side is well defined for \displaystyle{ \Re (z) > - N - 1 }.

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~~

…

\displaystyle{ \begin{aligned} \Gamma (z) &= \int_{0}^{1}t^{z-1}e^{-t}dt + \int_{1}^{\infty}t^{z-1}e^{-t}dt \\\\ \end{aligned} \\ }

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Prove that \displaystyle{\int_{1}^{\infty}t^{z-1}e^{-t}dt} converges.

…

Prove that \displaystyle{\int_{0}^{1}t^{z-1}e^{-t}dt} converges.

\displaystyle{ \begin{aligned} \Re (z) &> 0 \\ \Gamma (z) &= \int_{0}^{\infty}t^{z-1}e^{-t}dt \\ &= \int_{0}^{1}t^{z-1}e^{-t}dt + \int_{1}^{\infty}t^{z-1}e^{-t}dt \\\\ \end{aligned} \\ }

\displaystyle{ \begin{aligned} &\int_{0}^{1}t^{z-1}e^{-t}dt \\ &= \int_{0}^{1} t^{z-1} \sum_{n=0}^\infty \frac{(-t)^n}{n!} dt \\ &= \int_{0}^{1} t^{z-1} \left( \sum_{n=0}^N \frac{(-t)^n}{n!} +\sum_{n=N+1}^\infty \frac{(-t)^n}{n!} \right) dt \\ \end{aligned} \\ }

\displaystyle{ \begin{aligned} &= \int_{0}^{1} t^{z-1} \left( \sum_{n=0}^N \frac{(-t)^n}{n!} + e^{-t} - \sum_{n=0}^N \frac{(-t)^n}{n!} \right) dt \\ \end{aligned} \\ }

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\displaystyle{ \begin{aligned} &\int_{0}^{1} t^{z-1} \frac{(-t)^n}{n!} dt \\ &= \frac{(-1)^n}{n!} \int_{0}^{1} t^{z+n-1} dt \\ &= \frac{(-1)^n}{n!} \frac{1}{z+n} \left[ 1 - 0^{z+n} \right] \\ \end{aligned} \\ }

Therefore, for the integral to converge, \displaystyle{ \Re (z) + n > 0 } is required. So for the overall integral \displaystyle{ \int_{0}^{1} t^{z-1} \sum_{n=0}^\infty \frac{(-t)^n}{n!} dt} to converge, a necessary condition is

\displaystyle{ \Re (z) + n_{\text{min}} > 0 }

Therefore,

\displaystyle{ \Re (z) > 0 }

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Under this condition,

\displaystyle{ \begin{aligned} &\int_{0}^{1}t^{z-1}e^{-t}dt \end{aligned} }
\displaystyle{ = \int_{0}^{1} t^{z-1} \left( \sum_{n=0}^N \frac{(-t)^n}{n!} +\sum_{n=N+1}^\infty \frac{(-t)^n}{n!} \right) dt, }

\displaystyle{ = \sum_{n=0}^N \frac{(-1)^n}{n!} \frac{1}{z+n} + \int_{0}^{1} t^{z-1} \sum_{n=N+1}^\infty \frac{(-t)^n}{n!} dt}

So for \displaystyle{\Re (z) > 0},

…

— Me@2023-09-07 07:38:17 PM

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2023.12.08 Friday (c) All rights reserved by ACHK

@dialectphilosophy, 1.3.1

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…

As long as object 1 and object 2 have the same velocity-relative-to-the-ground as that of the car, \displaystyle{v}, i.e.

\displaystyle{v_1=v_2=v},

no matter what the value of \displaystyle{v} is, the distance between object 1 and object 2 is always the same. In other words, you cannot deduce the value of the \displaystyle{v} by observing the separation changes between any two objects/points within the car.

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Even in another case where \displaystyle{v_2 \ne v_1},

the separation between the 2 objects is

\displaystyle{ \begin{aligned} &x_2(t) - x_1(t) \\ &= \Bigl( x_2(0) + v_2 t \Bigr) - \Bigl(x_1(0) + v_1 t \Bigr) \\ &= \Bigl( x_2(0) - x_1(0) \Bigr) + \Bigl( v_2 - v_1 \Bigr) t \\ \end{aligned}}

In other words, the separation \displaystyle{(x_2(t) - x_1(t))} depends only on the initial separation \displaystyle{(x_2(0) - x_1(0))} and the velocity of object 2 relative to the object 1, \displaystyle{(v_2 - v_1)}.

Let this relative velocity be \displaystyle{v_{21}}:

\displaystyle{ \begin{aligned} v_{21} &= v_2 - v_1 \\ \end{aligned}}

Although the value of either v_1 or v_2 depends on the observer’s own velocity, v_{21} does not. In other words, although either v_1 or v_2 would have a different value under Galilean transformation, v_{21} would stay the same.

\displaystyle{ \begin{aligned} v_{21}' &= v_2' - v_1' \\ &= (v_2 + V) - (v_1 + V) \\ &= v_2 - v_1 \\ &= v_{21} \\ \end{aligned}}

The observer can directly see are positions x_1(0), x_2(0), x_1(t), and x_2(t). With them and the time measured t, he can deduce the value of v_{21}.

He can deduce the relative velocity v_2 - v_1 by the separations x_2(t) - x_1(t) and x_2(0) - x_1(0). However, he still cannot deduce v_2 nor v_1 unless he is able to look outside the car window. Thus, he cannot deduce the car speed v just by observing the positions and velocities of the objects inside the car.

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For simplicity, assume object 1 is actually a point of the car itself. So v_1 is actually the speed of the car, v. Then the calculation

\displaystyle{ \begin{aligned} &x_2(t) - x_1(t) \\ &= \Bigl( x_2(0) + v_2 t \Bigr) - \Bigl(x_1(0) + v_1 t \Bigr) \\ &= \Bigl( x_2(0) - x_1(0) \Bigr) + \Bigl( v_2 - v_1 \Bigr) t \\ \end{aligned}}

becomes

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— Me@2023-12-06 11:06:23 AM

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2023.12.06 Wednesday (c) All rights reserved by ACHK