能知過去未來, 2

Posted on June 28, 2024 by rpflee

我能知過去未來，偏偏不掌握現在。

— Me@2011.09.02

— Me@2024-06-28

無拍之拖 3.2

Posted on June 27, 2024 by rpflee

眾害取其輕 | The least of all evils, 13.2 | 友情演義 2.2 | 相聚零刻 3.2 | 原來是你 4.2

這段改編自 2023 年 6 月 20 日的對話。

…

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。

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」。亦即是話，「曖昧時期」一但有結果，就應立即終止。

「結果」就是，愛情或友情。「終止」就是，要麼進化成情侶，要麼退化成朋友。

— Me@2024-06-26 04:05:21 PM

Batman: Return of the Joker

Posted on June 26, 2024 by rpflee

Euler problem 20.1

Find the sum of the digits in the number $\displaystyle{100!}$ .

(reduce #'+
        (map 'list #'digit-char-p
             (write-to-string
              (reduce #'* (loop for i
                                from 1 to 100
                                collect i)))))

CL-USER> 
648

— Me@2024-06-25 08:26:22 PM

1.9 Abstraction of Path Functions, 2

Posted on June 15, 2024 by rpflee

Structure and Interpretation of Classical Mechanics

Let $\displaystyle{\bar \Gamma}$ be a function such that

$\displaystyle{\begin{aligned} f &= \bar \Gamma (\bar f) \\ \end{aligned}}$

So, to get a value of $\displaystyle{\begin{aligned} f \end{aligned}}$ , we input a local tuple:

$\displaystyle{\begin{aligned} f (t, q(t), v(t), \cdots, q^{(n)}(t)) &= f (t, q, v, \cdots, q^{(n)})(t) \\ &= f(\Gamma[q])(t) \\ &= f \circ \Gamma[q](t) \\ &= \bar f [q](t) \\ \bar \Gamma (\bar f) (t, q(t), v(t), \cdots, q^{(n)}(t)) &= \bar f [q](t) \\ \end{aligned}}$

(define ((Gamma-bar f-bar) path-q-local-tuple)
  (let* ((tqva path-q-local-tuple)
         (t (time tqva))         
         (O-tqva (osculating-path tqva)))   
    ((f-bar O-tqva) t)))

The procedure osculating-path takes a number of local components and returns a path with these components; it is implemented as a power series.

In scmutils, you can use the pp function to get the definition of a function. For example, the code

(pp osculating-path)

results

(define ((osculating-path state0) t)
  (let ((t0 (time state0))
        (q0 (coordinate state0))
        (k (vector-length state0)))
    (let ((dt (- t t0)))
      (let loop ((n 2) (sum q0) (dt^n/n! dt))
        (if (fix:= n k)
            sum
            (loop (+ n 1)
                  (+ sum
                     (* (vector-ref state0 n)
                        dt^n/n!))
                  (/ (* dt^n/n! dt) n)))))))

$\displaystyle{\begin{aligned} q(t) &= q_0 + v_0 (t-t_0) + \frac{1}{2} a_0 (t-t_0)^2 + ... +\frac{1}{n!} q^{(n)}_0 (t-t_0)^n \\ &= q_0 + v_0 (dt) + \frac{1}{2} a_0 (dt)^2 + ... +\frac{1}{n!} q^{(n)}_0 (dt)^n \\ \end{aligned}}$

Note that for the function in the program, $\displaystyle{\bar \Gamma (\bar f)}$ , the input is actually not $\displaystyle{(t, q, v, \cdots, q^{(n)})}$ but $\displaystyle{(t, q(t), v(t), \cdots, q^{(n)}(t))}$ .

$\displaystyle{\begin{aligned} \bar \Gamma (\bar f) (t, q(t), v(t), \cdots) &= \bar f[O(t, q, v, \cdots)](t) \\ \end{aligned}}$

(define (F->C F)
  (define (f-bar q-prime)
    (define q
      (compose F (Gamma q-prime)))
    (Gamma q))
  (Gamma-bar f-bar))

(show-expression 
 ((F->C p->r)
  (->local 't
           (up 'r 'theta)
           (up 'rdot 'thetadot))))

— Me@2023-12-19 08:16:40 PM

Philosophical Investigations, 2

Posted on June 12, 2024 by rpflee

According to Wittgenstein, philosophical problems arise when language is forced from its proper home into a metaphysical environment, where all the familiar and necessary landmarks and contextual clues are removed. He describes this metaphysical environment as like being on frictionless ice: where the conditions are apparently perfect for a philosophically and logically perfect language, all philosophical problems can be solved without the muddying effects of everyday contexts; but where, precisely because of the lack of friction, language can in fact do no work at all. Wittgenstein argues that philosophers must leave the frictionless ice and return to the “rough ground” of ordinary language in use. Much of the Investigations consists of examples of how the first false steps can be avoided, so that philosophical problems are dissolved, rather than solved: “The clarity we are aiming at is indeed complete clarity. But this simply means that the philosophical problems should completely disappear.”

— Wikipedia on Ludwig Wittgenstein

2024.06.12 Wednesday ACHK

天使與傻瓜 3

Posted on June 10, 2024 by rpflee

自己的世界，以自己為中心，並不是「自我中心」；

要求別人的世界，也以你自己為中心，才是「自我中心」。

而「他我中心」，則是自己的世界，以他人為中心。

不要自我中心；

不要他我中心，

因其兩者皆錯。

— Me@2023-09-13 09:33:19 PM

所有第一次

Posted on June 6, 2024 by rpflee

這段改編自 2021 年 12 月 15 日的對話。

《我的第一次》這是我給學生訂的作文題目，批閱作文的時候，忽然憶起自己重要的第一次。我第一次真正愛上國文課，是因為來了一位姓盛的老師。我第一次暗戀，對象是一位同班同學，她叫陳文靖，就坐在我前面。每天上課時，我總會間歇嗅到，她髮鬢的爽身粉香味。我第一次感到自己重要，是在你出生的那天。嬰兒床上的你，臉上滿是皺紋，口中嗚嗚地發出，小貓般微弱哭叫。我抱起你，有點不知所措，但是一種奇妙感覺，同時洶湧而來，世界變得溫柔，我也從此變得重要。一晃眼，小貓不再是小貓，早已出發尋找，他自己的第一次去了，在一個靜靜的角落裡，尚有一個全新的『第一次』在等著他。

–《男人四十》

拍拖，不要只為互相陪伴。拍拖，應以結婚有目的。肯定不是你未來太太的人，應立刻中止當時（或潛在）的戀愛關係。

與女子第一次看星空；第一次看煙花；第一次討論人生大計；第一次討論廣義相對論；第一次討論羅馬帝園衰亡史；等等，都應全部留給，唯一的真正未來太太，盡量。

— Me@2024-06-04 12:11:17 PM

NixOS config

Posted on February 29, 2024 by rpflee

Zsh, 4

# Edit this configuration file to define what should be installed on your system.  Help is available in the configuration.nix(5) man page and in the NixOS manual (accessible by running ‘nixos-help’).

{ config, pkgs, ... }:

{
  imports =
    [ # Include the results of the hardware scan.
      ./hardware-configuration.nix
    ];

  # Bootloader.
  boot.loader.grub.enable = true;
  boot.loader.grub.device = "/dev/sda";
  boot.loader.grub.useOSProber = true;

  networking.hostName = "hi"; # Define your hostname.
  # networking.wireless.enable = true;  # Enables wireless support via wpa_supplicant.
  networking.wireless.enable = false;
  hardware.bluetooth.enable = false;

  # Configure network proxy if necessary
  # networking.proxy.default = "http://user:password@proxy:port/";
  # networking.proxy.noProxy = "127.0.0.1,localhost,internal.domain";

  # Enable networking
  networking.networkmanager.enable = true;

  # Set your time zone.
  time.timeZone = "Australia/Perth";

  # Select internationalisation properties.
  i18n.defaultLocale = "en_US.UTF-8";

  i18n.extraLocaleSettings = {
    LC_ADDRESS = "en_GB.UTF-8";
    LC_IDENTIFICATION = "en_GB.UTF-8";
    LC_MEASUREMENT = "en_GB.UTF-8";
    LC_MONETARY = "en_GB.UTF-8";
    LC_NAME = "en_GB.UTF-8";
    LC_NUMERIC = "en_GB.UTF-8";
    LC_PAPER = "en_GB.UTF-8";
    LC_TELEPHONE = "en_GB.UTF-8";
    LC_TIME = "en_GB.UTF-8";
  };

  # Enable the X11 windowing system.
  services.xserver.enable = true;

  # Enable the KDE Plasma Desktop Environment.
  services.xserver.displayManager.sddm.enable = true;
  services.xserver.desktopManager.plasma5.enable = true;

  # Configure keymap in X11
  services.xserver = {
    layout = "us";
    xkbVariant = "";
  };

  # Enable CUPS to print documents.
  services.printing.enable = true;

  # Enable sound with pipewire.
  sound.enable = true;
  hardware.pulseaudio.enable = false;
  security.rtkit.enable = true;
  services.pipewire = {
    enable = true;
    alsa.enable = true;
    alsa.support32Bit = true;
    pulse.enable = true;
    # If you want to use JACK applications, uncomment this
    #jack.enable = true;

    # use the example session manager (no others are packaged yet so this is enabled by default, no need to redefine it in your config for now)
    #media-session.enable = true;
  };

  # Enable touchpad support (enabled default in most desktopManager).
  # services.xserver.libinput.enable = true;

  # Define a user account. Don't forget to set a password with ‘passwd’.
  users.users.hi = {
    isNormalUser = true;
    description = "hi";
    extraGroups = [ "networkmanager" "wheel" "vboxsf" "vboxusers"];
    shell = pkgs.zsh;
    packages = with pkgs; [
      firefox
      kate
      thunderbird
      chromium
    ];
  };

  # Allow unfree packages
  nixpkgs.config.allowUnfree = true;

  # virtualisation.virtualbox.guest.enable = true;
  # virtualisation.virtualbox.guest.x11 = true;
  # virtualisation.virtualbox.host.enable = true;
  # virtualisation.virtualbox.host.enableExtensionPack = true;

  # List packages installed in system profile. To search, run:
  # $ nix search wget
  environment.systemPackages = with pkgs; [
  #  vim # Do not forget to add an editor to edit configuration.nix! The Nano editor is also installed by default.
  #  wget
    zsh
    zsh-autosuggestions
    zsh-syntax-highlighting
    zsh-powerlevel10k
    meslo-lgs-nf

    oh-my-zsh
    corefonts
    fira
    powerline-fonts
    ubuntu_font_family
    unifont

    p7zip
    unrar
    unzip

    util-linux
    meld

    home-manager

    git
    git-cola
    github-desktop

    emacs29

    autokey
    virtualbox
    librewolf
    palemoon-bin
    ventoy-full

    flatpak
    steam
    libsForQt5.falkon
    libsForQt5.korganizer
    krusader

    psensor
    playonlinux
    protonvpn-gui

    # vmware-workstation

    quickemu
    quickgui

    neofetch
  ];

  # Some programs need SUID wrappers, can be configured further or are   # started in user sessions.
  # programs.mtr.enable = true;
  # programs.gnupg.agent = {
  #   enable = true;
  #   enableSSHSupport = true;
  # };

  programs.zsh = {
    enable = true;
    enableCompletion = true;
    autosuggestions.enable = true;
    interactiveShellInit = ''
      source /run/current-system/sw/share/zsh-syntax-highlighting/zsh-syntax-highlighting.zsh
      source /run/current-system/sw/share/zsh-autosuggestions/zsh-autosuggestions.zsh
      source /run/current-system/sw/share/zsh-powerlevel10k/powerlevel10k.zsh-theme
    '';
  };

  fonts = {
    fontDir.enable = true;
    packages = with pkgs; [
      corefonts
      fira
      powerline-fonts
      ubuntu_font_family
      unifont
    ];
  };

  # List services that you want to enable:

   systemd.services.rfkill-block-all = {
     path = [ pkgs.networkmanager ];
     script = ''
       nmcli radio wifi off
     '';
     wantedBy = [ "multi-user.target" ];
   };

  # Enable the OpenSSH daemon.
  # services.openssh.enable = true;

  # Open ports in the firewall.
  # networking.firewall.allowedTCPPorts = [ ... ];
  # networking.firewall.allowedUDPPorts = [ ... ];
  # Or disable the firewall altogether.
  # networking.firewall.enable = false;

  networking.firewall.enable = true;
  powerManagement.enable = true;
  services.thermald.enable = true;

  # This value determines the NixOS release from which the default settings for stateful data, like file locations and database versions on your system were taken. It‘s perfectly fine and recommended to leave this value at the release version of the first install of this system. Before changing this value read the documentation for this option (e.g. man configuration.nix or on ... ).
  system.stateVersion = "23.11"; # Did you read the comment?
}

— Me@2024-01-23 09:55:24 AM

3.6 Analytic continuation for gamma function, 5

Posted on February 21, 2024 by rpflee

A First Course in String Theory

…

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

…

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

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.

…

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.

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

$\displaystyle{ \begin{aligned} &= \sum_{n=N+1}^\infty \frac{(-1)^n}{n!} \frac{1}{z+n} \left[ 1 - 0^{z+n} \right] \\ \end{aligned} \\ }$

This term converges as long as $\displaystyle{ \Re (z) > - N - 1 }$ .

So this condition is not only necessary but also sufficient for the convergence of the analytic continuation of Gamma function,

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

— Me@2023-09-07 07:38:17 PM

@dialectphilosophy, 1.6

Posted on February 20, 2024 by rpflee

Comments on Dialect’s Newton vs. Mach: The Bucket Experiment

1.1 In dialectphilosophy, the author claimed that acceleration is not really absolute, because measuring its value within a physical system actually requires your prior knowledge about the world and the initial calibration of the accelerometer.

While the point is correct, it is irrelevant here, because it is not what the original “acceleration is absolute” refers to.

In other words, the statement “acceleration is absolute” is with respect to Galilean transformation. It is not with respect to every kind of transformation. Confusing these two meanings is a major bug of @dialectphilosophy.

1.2 Actually, calibration is a process that lets you define what “acceleration is zero” means in terms of physical phenomenon. In other words, you decide under what condition that you should set the accelerometer reading to zero.

1.3 Note that it is always the case that you have to define the value of a physical quantity in terms of a state of the measuring device. That is exactly what “calibration” means.

1.4 More fundamentally, it is just the normal process of defining new words. We define new words either in terms of other words or in terms of physical phenomena.

2. Even though the value of acceleration, and thus also the answer to “whether the acceleration is zero”, is relative to the accelerometer calibration, the answer to “whether the acceleration is increasing, decreasing, or constant” is not.

— Me@2023-08-07 05:56:31 AM

Akin’s Laws

Posted on February 18, 2024 by rpflee

tlb on May 4, 2018 | next [–]

Akin’s rules for spacecraft design [0] include: “Any exploration program which “just happens” to include a new launch vehicle is, de facto, a launch vehicle program.”

By analogy, any software project that includes writing a database is, de facto, a database project.

Strom on May 5, 2018 | parent | prev | next [–]

Another common version of this is: video game project turning into a video game engine project, usually not even reaching the actual content phase.

k__ on May 5, 2018 | root | parent | next [–]

But weren’t such projects rather common?

faitswulff on May 4, 2018 | parent | next [–]

The title is misleading – it’s actually about how and why they did end up writing their own db. From the article:

There’s an old adage in software engineering — “never write your own database”. So why did the OneNote team go ahead and write one as part of upgrading OneNote’s local cache implementation?

— Never Write Your Own Database (2017)

— Hacker News

2024.02.18 Sunday ACHK

宇宙的琴弦 2

Posted on February 15, 2024 by rpflee

這段改編自 2010 年 4 月 24 日的對話。

自己的世界，以自己為中心，並不是「自我中心」；

要求別人的世界，也以你自己為中心，才是「自我中心」。

（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。那會反映她的一些性格特質，例如，她是否自我中心。

有很多網誌（的作者），都是自我中心的。我都不知道，閱讀來有什麼意思。

（你是指，她會講當天做過什麼事和吃過什麼東西等等？）

無錯。那些內容對讀者，完全沒有任何形式的得益。

當然，如果她寫自己的生活，寫得十分有趣，則另計；我則視之為對我有用，我仍然會有動機去欣賞。人的主要娛樂之一，就是經歷各種，不同的人生。

網誌（作者）的另一個極端，則是「完全不自我中心」。那亦不是好事，因為，那會把網誌變成，完全沒有個人魅力，沒有人性，沉悶非常，有如字典工具書。這個極端，我簡稱為「他我中心」。

「自我中心」和「他我中心」都是錯的。

— Me@2024-02-14 11:15:43 AM

I like to think that the important part of my blog is the content, not me. After all, users don’t care about me. It’s about what you, the reader, get out of this blog. I struggled with this for a while until I realized what I was missing. Blogs, for better or worse, are as much about the writer as they are any other topic. Personality is the essential ingredient that makes blogs so interesting, so compelling, so.. human. To avoid writing about yourself is just as much of a mistake as writing about yourself in every post. So Five Things isn’t off topic at all. It’s very much on topic. Behind every fascinating blog is a fascinating person.

— Five Things You Didn’t Know About Me (and my office)

— Coding Horror

— by Jeff Atwood

— 15 Jan 2007

Euler problem 19.2

Posted on February 11, 2024 by rpflee

How many Sundays fell on the first of the month during the twentieth century (1 Jan 1901 to 31 Dec 2000)?

isLeap :: Int -> Bool
isLeap year
  = (mod year 4 == 0
     && mod year 100 /= 0)
  || mod year 400 == 0

daysInMonth :: Int -> Int -> Int
daysInMonth year month
  | month `elem` [1, 3, 5, 7, 8, 10, 12] = 31
  | month `elem` [4, 6, 9, 11] = 30
  | month == 2 = if isLeap year
                 then 29
                 else 28

nextMonth :: Int -> Int -> Int -> Int
nextMonth dayOfWeek y m 
  = (dayOfWeek + daysInMonth y m) `mod` 7

countSundays :: Int -> Int -> Int -> Int -> Int
countSundays startYear startMonth dayOfWeek sundays
  | y > 2000  = sundays
  | m > 12    = countSundays (y+1) 1 dayOfWeek sundays
  | otherwise = countSundays y (m+1) nextDayOfWeek sun
  where
    y = startYear
    m = startMonth
    nextDayOfWeek = nextMonth dayOfWeek y m 
    sun | dayOfWeek == 0 = sundays + 1
        | otherwise      = sundays

e19 :: Int
e19 = countSundays 1901 1 2 0
-- The third argument is dayOfWeek.
-- Its value is 2 here  
-- because 1 Jan 1901 was a Tuesday.

λ> e19
171
λ>

— Me@2024-02-10 12:00:39 PM

3 Vector Fields and One-Form Fields, 2.2

Posted on February 8, 2024 by rpflee

Functional Differential Geometry

p. 21

…

2.4

Let

$\displaystyle{ \begin{aligned} D_b(f)(x) &= ((Df)\bigg|_x b \bigg|_x \\ \end{aligned} }$ ,

where $\displaystyle{ b(x) }$ is a coordinate function.

p. 12

A coordinate function $\displaystyle{\chi}$ maps points in a coordinate patch of a manifold to a coordinate tuple:

$\displaystyle{x = \chi(\mathbf{m})}$ ,

where $\displaystyle{x}$ may have a convenient tuple structure.

2.5

$\displaystyle{ \begin{aligned} \textbf{v}(\text{f})(\textbf{m}) &\ne D_b(f)(x) \\ \end{aligned} }$

Instead, $\displaystyle{ \textbf{v}(\text{f})(\textbf{m})}$ is a further generalization of $\displaystyle{ D_b(f)(x)}$ .

p. 25

The vector field $\displaystyle{ \textbf{v} }$ has a coordinate representation $\displaystyle{ v}$ :

$\displaystyle{ \begin{aligned} \textbf{v}(\text{f})(\textbf{m}) &= D( \textbf{f} \circ \chi^{-1})(\chi(\mathbf{m})) b(\chi(\mathbf{m})) \\ &= Df(x) b(x) \\ &= v(f)(x), \end{aligned} }$

with the definitions $\displaystyle{ f = \mathbf{f} \circ \chi^{-1} }$ and $\displaystyle{ x = \chi (\mathbf{m}) }$ .

So, actually,

$\displaystyle{ \begin{aligned} \textbf{v}(\text{f})(\textbf{m}) &= D_b(f)(x) \\ &= ((D(f)) \bigg|_x) b \bigg|_x \\ &= (\nabla f)\bigg|_x \cdot \vec b\bigg|_x \\ \end{aligned} }$

2.6

While $\displaystyle{ x }$ is the tuple of coordinate components of a point, $\displaystyle{ \textbf{m} }$ is the abstract point itself.

3.1

… ; they measure how quickly the coordinate functions change in the direction of the vector field, scaled by the magnitude of the vector field: …

$\displaystyle{ \begin{aligned} b^i_{\chi, \mathbf{v}} &= \mathbf{v}(\chi^i) \circ \chi^{-1} \\ \end{aligned} }$

$\displaystyle{ \chi }$ inputs an abstract point and outputs its coordinates.

The first factor $\displaystyle{ \mathbf{v}(\chi^i) }$ is just the meaning of the definition of $\displaystyle{ b^i_{\chi, \mathbf{v}} }$ . The second factor is needed because the input of $\displaystyle{ b^i_{\chi, \mathbf{v}} }$ is $\displaystyle{ x \\ }$ , not $\displaystyle{ \mathbf{m} \\ }$ .

$\displaystyle{ \begin{aligned} b^i_{\chi, \mathbf{v}} (x) &= \mathbf{v}(\chi^i) (\mathbf{m}) \\ &= \mathbf{v}(\chi^i) ( \chi^{-1} (\chi (\mathbf{m}))) \\ &= \mathbf{v}(\chi^i) \circ \chi^{-1} (x) \\ \end{aligned} }$

In other words, $\displaystyle{ \mathbf{v}(\chi^i) (\mathbf{m}) \\ }$ is just the definition of $\displaystyle{ b^i_{\chi, \mathbf{v}} (x) }$ .

3.2

… $\displaystyle{ \textbf{v}(\text{f})(\textbf{m})}$ is the direction derivative of the function $\displaystyle{\text{f}}$ at the point $\displaystyle{ \textbf{m} }$ .

Note that it is not the ordinary directional derivative.

3.2.1

Instead, the ordinary directional derivative is

$\displaystyle{(Df(x)) \Delta x}$

$\displaystyle{\begin{aligned} D_{\mathbf{v}}(f) &= \frac{\left(\delta f\right)_{\mathbf{v}}}{|\mathbf{v}|} &= \left(\nabla f\right) \cdot \hat{\mathbf{v}} \\ \end{aligned}}$

3.2.2

The generalization of directional derivative is replacing $\displaystyle{\Delta x}$ , a vector independent of $\displaystyle{x}$ , with $\displaystyle{b(x)}$ , a vector function of $\displaystyle{x}$ .

— Me@2024-02-03 04:45:17 PM

@dialectphilosophy, 1.5

Posted on February 3, 2024 by rpflee

1. Velocity is by definition relative because displacement is by definition relative.

$\displaystyle{ \begin{aligned} v &= \frac{s}{\Delta t} \\ s &= x_2 - x_1 \\ \end{aligned} }$

2. Even for either coordinate ( $x_1$ or $x_2$ ), its value is relative because it is defined with respect to an origin chosen by you.

3.1 Furthermore, even for the origin itself, it is relative in a sense. When you choose a point as the origin of the coordinate system, you have to choose a static point. However, “whether a point is static or not” is subjective for different observers.

3.2 In other words, to be an origin, it has to be the same physical location. However, whether the physical location is the “same” could be up to debate.

— Me@2024-02-03 01:43:32 PM

1017

Posted on February 2, 2024 by rpflee

Professor Lee: SHB1017
Proposal: timetable

宇宙無外面

Posted on February 1, 2024 by rpflee

有限而無邊

— Me@2024-02-01 09:05:46 AM

Euler problem 19.1

Posted on January 30, 2024 by rpflee

How many Sundays fell on the first of the month during the twentieth century (1 Jan 1901 to 31 Dec 2000)?

(defun is-leap-year (year)
  (or (and (zerop (mod year 4))
           (not (zerop (mod year 100))))
      (zerop (mod year 400))))

(defun days-in-month (month year)
  (case month
    ((1 3 5 7 8 10 12) 31)
    ((4 6 9 11) 30)
    (2 (if (is-leap-year year)
           29
           28))))

(defun euler-19 ()
  (let ((day-of-week 2) ; 1 Jan 1901 was a Tuesday
        (sundays 0))
    (loop :for year :from 1901 :to 2000 :do
      (loop :for month :from 1 :to 12 :do
        (when (zerop day-of-week) ; 0 represents Sunday
          (incf sundays))
        (setf day-of-week
              (mod
               (+ day-of-week
                  (days-in-month month year))
               7))))
    sundays))

(print (euler-19))

CL-USER> (euler-19)
171
CL-USER>

— Me@2024-01-30 01:46:11 PM

1.9 Abstraction of Path Functions

Posted on January 29, 2024 by rpflee

Structure and Interpretation of Classical Mechanics

Given a function $\displaystyle{f}$ of a local tuple, a corresponding path-dependent function $\displaystyle{\bar f[q]}$ is

$\displaystyle{\bar f[q] = f \circ \Gamma[q]}$ .

So while the input of $\displaystyle{f}$ is a tuple $\displaystyle{(t, q, v, \cdots)}$ , the input of $\displaystyle{\bar f}$ is an abstract path $\displaystyle{q}$ .

$\displaystyle{\begin{aligned} \Gamma [q] &= (t, q, v, \cdots) \\ \bar f &= f \circ \Gamma \\ \end{aligned}}$

Given $\displaystyle{\bar f}$ we can reconstitute $\displaystyle{f}$ by taking the argument of $\displaystyle{f}$ , which is a finite initial segment of a local tuple, constructing a path that has this local description, and finding the value of $\displaystyle{\bar f}$ for this path.

1. The goal is to use $\displaystyle{\bar f}$ to reconstitute $\displaystyle{f}$ .

2. The argument of $\displaystyle{f}$ is

$\displaystyle{\begin{aligned} \Gamma [q] &= (t, q, v, \cdots, q^{(n)}) \\ &= (t, q^{(0)}, q^{(1)}, \cdots, q^{(n)}) \\ \end{aligned}}$

3. Assume that we have the value of the initial position of the path, $\displaystyle{q_0 = q(t=0)}$ . Then the path $\displaystyle{q(t)}$ can be constructed by

$\displaystyle{q(t) = q_0 + v_0 t + \frac{1}{2} a_0 t^2 + ... +\frac{1}{n!} q^{(n)}_0 t^n}$

Note that while $\displaystyle{q}$ represents a path, $\displaystyle{q(t)}$ represents the coordinates of the particle location on the path at time $\displaystyle{t}$ .

Knowing the value of $\displaystyle{q(t)}$ for every moment $\displaystyle{t}$ is equivalent to knowing the path $\displaystyle{q}$ as a whole.

— Me@2023-12-19 08:16:40 PM

@dialectphilosophy, 1.4

Posted on January 24, 2024 by rpflee

Velocity is relative in the following sense:

Subjective value of an objective velocity changes under Galilean transformation. Different inertial observers would see different velocity values.

Acceleration is absolute in the following sense:

Subjective value of an objective accelerative remains unchanged under Galilean transformation. Different inertial observers would see identical acceleration values.

The proof:

$\displaystyle{a' = \frac{dv'}{dt} = \frac{d}{dt} (v+v_0) = a}$