diff options
Diffstat (limited to 'source')
43 files changed, 320 insertions, 565 deletions
diff --git a/source/_includes/image.html b/source/_includes/image.html index a9bc3fd..5b4f35f 100644 --- a/source/_includes/image.html +++ b/source/_includes/image.html @@ -12,22 +12,29 @@ {% comment %} If 'name' ends in "full", half-size image must exist {% endcomment %} {% assign name_full = name %} {% if suffix == "full" %} - {% assign name_half = prefix | append: "half" %} + {% assign name_half = prefix | append: "half" %} {% else %} - {% assign name_half = name_full %} + {% assign name_half = name_full %} {% endif %} {% comment %} Insert the image, linking to the full-size version {% endcomment %} +{% if include.class %} + {% assign add_class = 'class="' | append: include.class | append: '"' %} +{% endif %} +{% if include.style or include.width %} + {% assign set_width = "width:" | append: include.width %} + {% assign set_style = include.style | default: set_width %} + {% assign add_style = 'style="' | append: set_style | append: '"' %} +{% endif %} <p> <a href="{{ name_full }}.{{ format }}"> - {% assign set_width = "width:" | append: include.width %} - <picture markdown="0"> + <picture> <source srcset="{{ name_half }}.avif" type="image/avif"> {% if name_full == name_half %} - <img src="{{ name_half }}.{{ format }}" style="{{ include.style | default: set_width }}" alt="{{ include.alt }}" title="{{ include.alt }}"> + <img src="{{ name_half }}.{{ format }}" {{ add_class }} {{ add_style }} alt="{{ include.alt }}" title="{{ include.alt }}"> {% else %} <source srcset="{{ name_half }}.webp" type="image/webp"> - <img src="{{ name_half }}.jpg" style="{{ include.style | default: set_width }}" alt="{{ include.alt }}" title="{{ include.alt }}"> + <img src="{{ name_half }}.jpg" {{ add_class }} {{ add_style }} alt="{{ include.alt }}" title="{{ include.alt }}"> {% endif %} </picture> </a> diff --git a/source/_includes/preamble.html b/source/_includes/preamble.html index 312f1cf..c94741b 100644 --- a/source/_includes/preamble.html +++ b/source/_includes/preamble.html @@ -1,9 +1,9 @@ <meta charset="utf-8"> {% if page.layout == "blog" or page.syntax %} -<link rel="stylesheet" href="/infra/css/syntax.css?v=20221008"> +<link rel="stylesheet" href="/infra/css/syntax.css?v=20221027"> {% endif %} {% if page.layout == "concept" or page.maths %} <link rel="stylesheet" href="/infra/css/katex.min.css?v=20221008"> {% endif %} -<link rel="stylesheet" href="/infra/css/main.css?v=20221001"> +<link rel="stylesheet" href="/infra/css/main.css?v=20221027"> <script data-goatcounter="https://prefetch.goatcounter.com/count" async src="https://gc.zgo.at/count.js"></script> diff --git a/source/_includes/proof/end.html b/source/_includes/proof/end.html new file mode 100644 index 0000000..29533a4 --- /dev/null +++ b/source/_includes/proof/end.html @@ -0,0 +1,3 @@ + +</div> +</div> diff --git a/source/_includes/proof/start.html b/source/_includes/proof/start.html new file mode 100644 index 0000000..780b159 --- /dev/null +++ b/source/_includes/proof/start.html @@ -0,0 +1,6 @@ +<div class="proof"> +<input type="checkbox" class="proof" id="{{ include.id }}"/> +<label class="proof" for="{{ include.id }}">{{ include.label | default: "Proof" }}</label> +<div class="proof-hidden" markdown="1"> +<div class="proof-starts"></div> +<label class="proof" for="{{ include.id }}">{{ include.label | default: "Proof" }}.</label> diff --git a/source/blog/2022/email-server-revisited/index.md b/source/blog/2022/email-server-revisited/index.md index 4519bae..eb3fc18 100644 --- a/source/blog/2022/email-server-revisited/index.md +++ b/source/blog/2022/email-server-revisited/index.md @@ -181,9 +181,7 @@ One day, I tried to send an email to an Outlook-based account, and OpenSMTPD reported it had been unable to make the delivery, because Microsoft had thrown an error: -<a href="microsoft-bounce.png"> -<img src="microsoft-bounce.png" class="darkinv" style="width:100%"> -</a> +{% include image.html file="microsoft-bounce.png" width="100%" class="darkinv" alt="Bounce message due to error from Microsoft" %} To their credit, they seem to be offering a way out. This approach is reasonable: preventively ban high-risk IP ranges, diff --git a/source/blog/2022/email-server-revisited/microsoft-bounce.avif b/source/blog/2022/email-server-revisited/microsoft-bounce.avif Binary files differnew file mode 100644 index 0000000..bc1c4f5 --- /dev/null +++ b/source/blog/2022/email-server-revisited/microsoft-bounce.avif diff --git a/source/index.md b/source/index.md index 680b577..7847d5f 100644 --- a/source/index.md +++ b/source/index.md @@ -4,7 +4,14 @@ date: 2021-02-22 layout: "default" --- -<img src="/infra/image/logo256.png" class="darkinv" style="border: 2px solid #121212;"> +{% comment %} Not using includes/image.html to avoid link {% endcomment %} +<p> +<picture> +<source srcset="/infra/image/logo-yu4qu3.avif" type="image/avif"> +<img src="/infra/image/logo-yu4qu3.png" class="darkinv" style="border: 2px solid #121212" +title="預取 (yùqǔ): 'prefetch'; literally 'take in advance'" alt="預取 (yùqǔ): 'prefetch'; literally 'take in advance'"> +</picture> +</p> Welcome to my website. @@ -15,5 +22,5 @@ where I explain STEM concepts I've learned over the years. This website is made by me using [Hugo](https://gohugo.io), and served to you by [nginx](https://nginx.org/). -I intend to keep it free of advertising and trackers, +I intend to keep it free of advertising, and to maintain my A+ score for [TLS quality](https://www.ssllabs.com/ssltest/analyze.html?d=prefetch.eu). diff --git a/source/infra/css/main.css b/source/infra/css/main.css index 5127873..3f64520 100644 --- a/source/infra/css/main.css +++ b/source/infra/css/main.css @@ -11,10 +11,21 @@ --f: #ededed; --a: #ffff00; } - .darkinv {filter: invert(100%)} + .darkinv {filter: invert(100%);} } /* Basic elements */ +/* See also https://github.com/necolas/normalize.css/blob/master/normalize.css */ +html { + line-height: 1.5; + -webkit-text-size-adjust: none; + text-size-adjust: none; +} +hr { + box-sizing: content-box; + height: 0; + overflow: visible; +} body { background: #ededed; background: var(--b); @@ -26,43 +37,42 @@ body { padding: 1rem 0; font-family: sans-serif; - line-height: 1.5; } -h1,h2,h3 {text-align: center;} +h1, h2, h3 {text-align: center;} h1 {font-size: 2.0rem;} h2 {font-size: 1.6rem;} h3 {font-size: 1.2rem;} a { + background-color: transparent; text-decoration: none; color: #0000ff; color: var(--a); } img { + border-style: none; display: block; max-width: 100%; height: auto; margin: auto; } +table {margin: auto;} +td {padding: 0 1.5rem;} +/*code, pre { + font-family: monospace, monospace; + font-size: 1em; +}*/ +b, strong {font-weight: bolder;} /* Header and footer */ .nav { height: 3rem; font-size: 2.4rem; - padding: 0.25rem 0; -} -.navl { - float: left; - text-align: left; } -.navr { - float: right; - text-align: right; -} -.footl { +.navl, .footl { float: left; text-align: left; } -.footr { +.navr, .footr { float: right; text-align: right; } @@ -82,18 +92,19 @@ img { color: var(--a); } -/* Collapsible boxes */ -label { +/* IE-compatible collapsible proofs */ +div.proof { + display: block; + border: dotted; + padding: 0.25rem 0.5rem; +} +label.proof { color: #0000ff; color: var(--a); cursor: pointer; } -.accordion { - display: block; - padding: 0.3rem 0.5rem; - border-style: dotted; -} -input {display: none;} -input + label + .hidden {display: none;} -input:checked + label {display: none;} -input:checked + label + .hidden {display: block;} +.proof-hidden {display: none;} +input.proof {display: none;} +input.proof:checked + label {display: none;} +input.proof:checked + label + .proof-hidden {display: block;} +.proof-starts + p {margin-top: 0;} diff --git a/source/infra/css/syntax.css b/source/infra/css/syntax.css index 8772351..50191d2 100644 --- a/source/infra/css/syntax.css +++ b/source/infra/css/syntax.css @@ -1,3 +1,19 @@ +.highlighter-rouge { + border: 1px solid; + border-color: #999988; + background-color: #f8f8f8; +} +@media only screen and (prefers-color-scheme: dark) { +.highlighter-rouge { + background-color: #0d1117; + border-color: #8b949e; +} +} +div.highlight { overflow: auto; } +pre.highlight { margin: 0.5em; } + + + /* From https://github.com/brazacz/rouge-themes/blob/main/css/github.css */ .highlight { background-color: #f8f8f8; } .highlight .bp { color: #999999; } diff --git a/source/infra/image/logo-cgit.png b/source/infra/image/logo-cgit.png Binary files differnew file mode 100644 index 0000000..9b7b783 --- /dev/null +++ b/source/infra/image/logo-cgit.png diff --git a/source/infra/image/logo-yu4qu3.avif b/source/infra/image/logo-yu4qu3.avif Binary files differnew file mode 100644 index 0000000..b0be209 --- /dev/null +++ b/source/infra/image/logo-yu4qu3.avif diff --git a/source/infra/image/logo256.png b/source/infra/image/logo-yu4qu3.png Binary files differindex d6b9961..d6b9961 100644 --- a/source/infra/image/logo256.png +++ b/source/infra/image/logo-yu4qu3.png diff --git a/source/infra/image/logo96x64.gif b/source/infra/image/logo96x64.gif Binary files differdeleted file mode 100644 index 869890b..0000000 --- a/source/infra/image/logo96x64.gif +++ /dev/null diff --git a/source/know/concept/binomial-distribution/index.md b/source/know/concept/binomial-distribution/index.md index 1193a93..dc75221 100644 --- a/source/know/concept/binomial-distribution/index.md +++ b/source/know/concept/binomial-distribution/index.md @@ -44,11 +44,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-mean"/> -<label for="proof-mean">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-mean">Proof.</label> + +{% include proof/start.html id="proof-mean" -%} The trick is to treat $$p$$ and $$q$$ as independent until the last moment: $$\begin{aligned} @@ -62,8 +59,8 @@ $$\begin{aligned} \end{aligned}$$ Inserting $$q = 1 - p$$ then gives the desired result. -</div> -</div> +{% include proof/end.html id="proof-mean" %} + Meanwhile, we find the following variance $$\sigma^2$$, with $$\sigma$$ being the standard deviation: @@ -74,12 +71,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-var"/> -<label for="proof-var">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-var">Proof.</label> -We use the same trick to calculate $$\overline{n^2}$$ + +{% include proof/start.html id="proof-var" -%} (the mean squared number of successes): $$\begin{aligned} @@ -106,8 +99,8 @@ $$\begin{aligned} \end{aligned}$$ By inserting $$q = 1 - p$$, we arrive at the desired expression. -</div> -</div> +{% include proof/end.html id="proof-var" %} + As $$N \to \infty$$, the binomial distribution turns into the continuous normal distribution, @@ -119,11 +112,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-normal"/> -<label for="proof-normal">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-normal">Proof.</label> + +{% include proof/start.html id="proof-normal" -%} We take the Taylor expansion of $$\ln\!\big(P_N(n)\big)$$ around the mean $$\mu = Np$$: @@ -211,8 +201,7 @@ $$\begin{aligned} \end{aligned}$$ Taking $$\exp$$ of this expression then yields a normalized Gaussian distribution. -</div> -</div> +{% include proof/end.html id="proof-normal" %} ## References diff --git a/source/know/concept/boltzmann-equation/index.md b/source/know/concept/boltzmann-equation/index.md index 9ed2fd2..d2631b2 100644 --- a/source/know/concept/boltzmann-equation/index.md +++ b/source/know/concept/boltzmann-equation/index.md @@ -145,11 +145,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-moment0"/> -<label for="proof-moment0">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-moment0">Proof.</label> + +{% include proof/start.html id="proof-moment0" -%} We insert $$Q = m$$ into our prototype, and since $$m$$ is constant, the rest is trivial: @@ -159,9 +156,8 @@ $$\begin{aligned} \\ &= \pdv{\rho}{t} + \nabla \cdot \big(\rho \Expval{\vb{v}}\big) - 0 \end{aligned}$$ +{% include proof/end.html id="proof-moment0" %} -</div> -</div> If we instead choose the momentum $$Q = m \vb{v}$$, we find that the **first moment** of the BTE describes conservation of momentum, @@ -174,11 +170,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-moment1"/> -<label for="proof-moment1">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-moment1">Proof.</label> + +{% include proof/start.html id="proof-moment1" -%} We insert $$Q = m \vb{v}$$ into our prototype and recognize $$\rho$$ wherever possible: $$\begin{aligned} @@ -220,9 +213,8 @@ $$\begin{aligned} 0 &= \pdv{}{t}\big(\rho \vb{V}\big) + \nabla \cdot \big(\rho \vb{V} \vb{V} + \hat{P}\big) - n \vb{F} \end{aligned}$$ +{% include proof/end.html id="proof-moment1" %} -</div> -</div> Finally, if we choose the kinetic energy $$Q = m |\vb{v}|^2 / 2$$, we find that the **second moment** gives conservation of energy, @@ -237,11 +229,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-moment2"/> -<label for="proof-moment2">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-moment2">Proof.</label> + +{% include proof/start.html id="proof-moment2" -%} We insert $$Q = m |\vb{v}|^2 / 2$$ into our prototype and recognize $$\rho$$ wherever possible: $$\begin{aligned} @@ -349,9 +338,7 @@ $$\begin{aligned} \end{bmatrix} = \sum_{i=1}^{3} \sum_{j=1}^{3} \pdv{P_{ij}}{x_j} V_i \end{aligned}$$ - -</div> -</div> +{% include proof/end.html id="proof-moment2" %} diff --git a/source/know/concept/convolution-theorem/index.md b/source/know/concept/convolution-theorem/index.md index 742c8ff..510417a 100644 --- a/source/know/concept/convolution-theorem/index.md +++ b/source/know/concept/convolution-theorem/index.md @@ -12,6 +12,8 @@ is equal to a product in the frequency domain. This is especially useful for computation, replacing an $$\mathcal{O}(n^2)$$ convolution with an $$\mathcal{O}(n \log(n))$$ transform and product. + + ## Fourier transform The convolution theorem is usually expressed as follows, where @@ -27,11 +29,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-fourier"/> -<label for="proof-fourier">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-fourier">Proof.</label> + +{% include proof/start.html id="proof-fourier" -%} We expand the right-hand side of the theorem and rearrange the integrals: @@ -57,8 +56,8 @@ $$\begin{aligned} &= B \int_{-\infty}^\infty \tilde{g}(k') \: \tilde{f}(k - k') \dd{k'} = B \cdot (\tilde{f} * \tilde{g})(k) \end{aligned}$$ -</div> -</div> +{% include proof/end.html id="proof-fourier" %} + ## Laplace transform @@ -79,11 +78,8 @@ $$\begin{aligned} \boxed{\hat{\mathcal{L}}\{(f * g)(t)\} = \tilde{f}(s) \: \tilde{g}(s)} \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-laplace"/> -<label for="proof-laplace">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-laplace">Proof.</label> + +{% include proof/start.html id="proof-laplace" -%} We expand the left-hand side. Note that the lower integration limit is 0 instead of $$-\infty$$, because we set both $$f(t)$$ and $$g(t)$$ to zero for $$t < 0$$: @@ -106,8 +102,7 @@ $$\begin{aligned} &= \int_0^\infty \tilde{f}(s) \: g(t') \exp(- s t') \dd{t'} = \tilde{f}(s) \: \tilde{g}(s) \end{aligned}$$ -</div> -</div> +{% include proof/end.html id="proof-laplace" %} diff --git a/source/know/concept/curvilinear-coordinates/index.md b/source/know/concept/curvilinear-coordinates/index.md index cb22e43..48a5a72 100644 --- a/source/know/concept/curvilinear-coordinates/index.md +++ b/source/know/concept/curvilinear-coordinates/index.md @@ -48,6 +48,7 @@ we derive general formulae to convert expressions from Cartesian coordinates to the new orthogonal system $$(x_1, x_2, x_3)$$. + ## Basis vectors Consider the the vector form of the line element $$\dd{\ell}$$, @@ -86,6 +87,7 @@ $$\begin{aligned} \end{aligned}$$ + ## Gradient In an orthogonal coordinate system, @@ -102,11 +104,8 @@ $$\begin{gathered} } \end{gathered}$$ -<div class="accordion"> -<input type="checkbox" id="proof-grad"/> -<label for="proof-grad">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-grad">Proof.</label> + +{% include proof/start.html id="proof-grad" -%} For a direction $$\dd{\ell}$$, we know that $$\idv{f}{\ell}$$ is the component of $$\nabla f$$ in that direction: @@ -127,9 +126,8 @@ $$\begin{gathered} + \vu{e}_2 \dv{x_2}{\ell} \pdv{f}{x_2} + \vu{e}_3 \dv{x_3}{\ell} \pdv{f}{x_3} \end{gathered}$$ +{% include proof/end.html id="proof-grad" %} -</div> -</div> ## Divergence @@ -145,11 +143,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-div"/> -<label for="proof-div">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-div">Proof.</label> + +{% include proof/start.html id="proof-div" -%} As preparation, we rewrite $$\vb{V}$$ as follows to introduce the scale factors: @@ -222,8 +217,8 @@ $$\begin{aligned} After repeating this procedure for the other components of $$\vb{V}$$, we get the desired general expression for the divergence. -</div> -</div> +{% include proof/end.html id="proof-div" %} + ## Laplacian @@ -246,6 +241,7 @@ $$\begin{aligned} \end{aligned}$$ + ## Curl The curl of a vector $$\vb{V}$$ is as follows @@ -264,11 +260,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-curl"/> -<label for="proof-curl">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-curl">Proof.</label> + +{% include proof/start.html id="proof-curl" -%} The curl is found in a similar way as the divergence. We rewrite $$\vb{V}$$ like so: @@ -317,8 +310,8 @@ $$\begin{aligned} If we go through the same process for the other components of $$\vb{V}$$ and add up the results, we get the desired expression for the curl. -</div> -</div> +{% include proof/end.html id="proof-curl" %} + ## Differential elements diff --git a/source/know/concept/detailed-balance/index.md b/source/know/concept/detailed-balance/index.md index b89d5da..98f9bd3 100644 --- a/source/know/concept/detailed-balance/index.md +++ b/source/know/concept/detailed-balance/index.md @@ -103,11 +103,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-reversibility"/> -<label for="proof-reversibility">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-reversibility">Proof.</label> + +{% include proof/start.html id="proof-reversibility" -%} Consider the following weighted inner product, whose weight function is a stationary distribution $$\pi$$ satisfying detailed balance, @@ -222,8 +219,7 @@ $$\begin{aligned} Where the integral gave the expectation value at $$X_0$$, since $$\pi$$ does not change in time. -</div> -</div> +{% include proof/end.html id="proof-reversibility" %} diff --git a/source/know/concept/dirac-delta-function/index.md b/source/know/concept/dirac-delta-function/index.md index 518eba1..0185b78 100644 --- a/source/know/concept/dirac-delta-function/index.md +++ b/source/know/concept/dirac-delta-function/index.md @@ -65,11 +65,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-scale"/> -<label for="proof-scale">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-scale">Proof.</label> + +{% include proof/start.html id="proof-scale" -%} Because it is symmetric, $$\delta(s x) = \delta(|s| x)$$. Then by substituting $$\sigma = |s| x$$: @@ -77,9 +74,8 @@ $$\begin{aligned} \int \delta(|s| x) \dd{x} &= \frac{1}{|s|} \int \delta(\sigma) \dd{\sigma} = \frac{1}{|s|} \end{aligned}$$ +{% include proof/end.html id="proof-scale" %} -</div> -</div> An even more impressive property is the behaviour of the derivative of $$\delta(x)$$: @@ -89,11 +85,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-dv1"/> -<label for="proof-dv1">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-dv1">Proof.</label> + +{% include proof/start.html id="proof-dv1" -%} Note which variable is used for the differentiation, and that $$\delta'(x - \xi) = - \delta'(\xi - x)$$: @@ -102,9 +95,8 @@ $$\begin{aligned} &= \dv{}{x}\int f(\xi) \: \delta(x - \xi) \dd{x} = f'(x) \end{aligned}$$ +{% include proof/end.html id="proof-dv1" %} -</div> -</div> This property also generalizes nicely for the higher-order derivatives: diff --git a/source/know/concept/dynkins-formula/index.md b/source/know/concept/dynkins-formula/index.md index c0d20c5..307f098 100644 --- a/source/know/concept/dynkins-formula/index.md +++ b/source/know/concept/dynkins-formula/index.md @@ -39,11 +39,8 @@ $$\begin{aligned} } \end{aligned}$$ -<div class="accordion"> -<input type="checkbox" id="proof-kolmogorov"/> -<label for="proof-kolmogorov">Proof</label> -<div class="hidden" markdown="1"> -<label for="proof-kolmogorov">P |
