https://lunatide.tech/tags/markdown/Recent content in Markdown on LunaTide's BlogHugo -- gohugo.ioen-usLunaTide's BlogThu, 01 Jan 2099 00:00:00 +0000https://lunatide.tech/p/sample-post/Thu, 30 Apr 2026 00:00:00 +0000https://lunatide.tech/p/sample-post/<p>This is a sample post used to validate styling and the sidebar.</p> <aside class="admonition admonition--definition" data-collapsed="false"> <div class="admonition__title">Definition</div> <div class="admonition__content"> An <strong>admonition block</strong> is a styled container for structured content. </div> </aside><aside class="admonition admonition--theorem" data-collapsed="false"> <div class="admonition__title">Theorem</div> <div class="admonition__content"> If the <code>lastmod</code> of a post is the newest, it will appear first in <strong>Recently Updated</strong>. </div> </aside><aside class="admonition admonition--proof" data-collapsed="false"> <div class="admonition__title">Proof</div> <div class="admonition__content"> The sidebar sorts pages by <code>.Lastmod</code> in descending order. </div> </aside>https://lunatide.tech/p/admonition-playground/Wed, 29 Apr 2026 00:00:00 +0000https://lunatide.tech/p/admonition-playground/<h1 id="admonition-playground">Admonition Playground </h1><p>This page demonstrates <strong>colored admonition blocks</strong> using pure Markdown fenced blocks.</p> <p>You can mix:</p> <ul> <li>inline math like \(\hat{\epsilon}_S(h)\)</li> <li>display math</li> <li>lists and nested lists</li> <li>normal code blocks</li> <li>links and references</li> </ul> <hr> <h2 id="1-definition--inline-math">1. Definition + inline math </h2><aside class="admonition admonition--definition" data-collapsed="false"> <div class="admonition__title">Definition (Empirical Risk)</div> <div class="admonition__content"> <p>Given a hypothesis \(h\), a dataset \(S=\{(x^{(i)},y^{(i)})\}_{i=1}^m\), define the empirical error:</p> \[ \hat{\epsilon}_S(h) \;=\; \frac{1}{m}\sum_{i=1}^m \mathbf{1}\{h(x^{(i)}) \neq y^{(i)}\}. \]<p>We will also write \(\hat{\epsilon}(h)\) when \(S\) is clear from context.</p> </div> </aside><hr> <h2 id="2-theorem--proof-with-steps">2. Theorem + proof (with steps) </h2><aside class="admonition admonition--theorem" data-collapsed="false"> <div class="admonition__title">Concentration of Bernoulli Mean</div> <div class="admonition__content"> <p>Let \(Z_1,\dots,Z_m\) be i.i.d. Bernoulli random variables with mean \(\phi\), and let</p> \[ \hat{\phi}=\frac{1}{m}\sum_{i=1}^m Z_i. \]<p>Then for any \(\gamma>0\), \(\hat{\phi}\) concentrates around \(\phi\) as \(m\) grows.</p> </div> </aside><aside class="admonition admonition--proof" data-collapsed="false"> <div class="admonition__title">Proof Sketch</div> <div class="admonition__content"> <ol> <li>View \(\hat{\phi}\) as the average of bounded i.i.d. random variables.</li> <li>Apply a concentration inequality (e.g. Hoeffding).</li> <li>Rearrange the bound into the form “probability of deviation ≤ something”.</li> </ol> <p>This is a sketch; details depend on the exact inequality you choose.</p> </div> </aside><hr> <h2 id="3-proposition--corollary">3. Proposition + Corollary </h2><aside class="admonition admonition--proposition" data-collapsed="false"> <div class="admonition__title">Overfitting Indicator</div> <div class="admonition__content"> If a model has <strong>very low training error</strong> but <strong>high validation error</strong>, it likely has high variance (overfitting). </div> </aside><aside class="admonition admonition--corollary" data-collapsed="false"> <div class="admonition__title">Model Selection</div> <div class="admonition__content"> Cross-validation is a practical tool to estimate generalization error and select model complexity. </div> </aside><hr> <h2 id="4-example--list--nested-list">4. Example + list + nested list </h2><aside class="admonition admonition--example" data-collapsed="false"> <div class="admonition__title">k-fold Cross Validation</div> <div class="admonition__content"> <p>To run \(k\)-fold cross validation:</p> <ul> <li>Split the dataset into \(k\) folds: \(S_1,\dots,S_k\).</li> <li>For each fold \(j\): <ul> <li>Train on \(S \setminus S_j\)</li> <li>Evaluate on \(S_j\)</li> </ul> </li> <li>Average the \(k\) validation errors.</li> </ul> <p>A typical choice is \(k=10\).</p> </div> </aside><hr> <h2 id="5-note--link--inline-code">5. Note + link + inline code </h2><aside class="admonition admonition--note" data-collapsed="false"> <div class="admonition__title">Misc Tips</div> <div class="admonition__content"> <ul> <li>You can include links like <a class="link" href="https://gohugo.io/" target="_blank" rel="noopener" >Hugo</a>.</li> <li>You can include inline code like <code>hugo server -D</code>.</li> <li>You can include inline math: \(m=0.7n\), \(2^n\), \(\log n\).</li> </ul> </div> </aside><hr> <h2 id="6-tip--info">6. Tip / Info </h2><aside class="admonition admonition--tip" data-collapsed="false"> <div class="admonition__title">Best Practice</div> <div class="admonition__content"> Keep your tag set small and consistent. Prefer a few stable tags over many one-off tags. </div> </aside><aside class="admonition admonition--info" data-collapsed="false"> <div class="admonition__title">Implementation Note</div> <div class="admonition__content"> This theme renders these admonitions via a Hugo <code>render-codeblock.html</code> hook.<br> No shortcode is required. </div> </aside><hr> <h2 id="7-warning--caution--danger">7. Warning / Caution / Danger </h2><aside class="admonition admonition--warning" data-collapsed="false"> <div class="admonition__title">Math Rendering</div> <div class="admonition__content"> If one LaTeX expression is malformed, KaTeX may skip rendering that expression. (We configured it to avoid breaking the whole page.) </div> </aside><aside class="admonition admonition--caution" data-collapsed="false"> <div class="admonition__title">Markdown Pitfall</div> <div class="admonition__content"> Avoid mixing <code>$...$</code> math with code spans like <code>`$x$`</code>.<br> KaTeX ignores code blocks by default. </div> </aside><aside class="admonition admonition--danger" data-collapsed="false"> <div class="admonition__title">Security</div> <div class="admonition__content"> Do not publish secrets (tokens/keys) in Markdown posts. </div> </aside><hr> <h2 id="8-admonition-with-a-code-block-inside">8. Admonition with a code block inside </h2><aside class="admonition admonition--example" data-collapsed="false"> <div class="admonition__title">Code Inside Admonition</div> <div class="admonition__content"> <p>Here is a normal code block <strong>inside</strong> an admonition:</p> <div class="highlight"><div class="chroma"> <table class="lntable"><tr><td class="lntd"> <pre tabindex="0" class="chroma"><code><span class="lnt">1 </span><span class="lnt">2 </span><span class="lnt">3 </span><span class="lnt">4 </span></code></pre></td> <td class="lntd"> <pre tabindex="0" class="chroma"><code class="language-python" data-lang="python"><span class="line"><span class="cl"><span class="kn">import</span> <span class="nn">math</span> </span></span><span class="line"><span class="cl"> </span></span><span class="line"><span class="cl"><span class="k">def</span> <span class="nf">sigmoid</span><span class="p">(</span><span class="n">x</span><span class="p">:</span> <span class="nb">float</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">float</span><span class="p">:</span> </span></span><span class="line"><span class="cl"> <span class="k">return</span> <span class="mi">1</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">math</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="p">))</span></span></span></code></pre></td></tr></table> </div> </div> </div> </aside><p>And here is some math below it:</p> \[ \sigma(x) = \frac{1}{1 + e^{-x}} \]<div class="highlight"><div class="chroma"> <table class="lntable"><tr><td class="lntd"> <pre tabindex="0" class="chroma"><code><span class="lnt"> 1 </span><span class="lnt"> 2 </span><span class="lnt"> 3 </span><span class="lnt"> 4 </span><span class="lnt"> 5 </span><span class="lnt"> 6 </span><span class="lnt"> 7 </span><span class="lnt"> 8 </span><span class="lnt"> 9 </span><span class="lnt">10 </span><span class="lnt">11 </span><span class="lnt">12 </span><span class="lnt">13 </span><span class="lnt">14 </span><span class="lnt">15 </span><span class="lnt">16 </span></code></pre></td> <td class="lntd"> <pre tabindex="0" class="chroma"><code class="language-fallback" data-lang="fallback"><span class="line"><span class="cl"> </span></span><span class="line"><span class="cl">--- </span></span><span class="line"><span class="cl"> </span></span><span class="line"><span class="cl">## 9. A plain section after admonitions </span></span><span class="line"><span class="cl"> </span></span><span class="line"><span class="cl">This is normal Markdown text again. </span></span><span class="line"><span class="cl"> </span></span><span class="line"><span class="cl">- Item A </span></span><span class="line"><span class="cl">- Item B </span></span><span class="line"><span class="cl">- Item C </span></span><span class="line"><span class="cl"> </span></span><span class="line"><span class="cl">Inline math still works: </span></span><span class="line"><span class="cl"> </span></span><span class="line"><span class="cl">\[ </span></span><span class="line"><span class="cl">\hat{h}=\arg\min_{h\in\mathcal{H}}\hat{\epsilon}(h) </span></span><span class="line"><span class="cl">\]</span></span></code></pre></td></tr></table> </div> </div>