<?xml version="1.0" encoding="utf-8" standalone="yes"?><rss version="2.0" xmlns:atom="http://www.w3.org/2005/Atom"><channel><title>Shuang Song | Biostatistics and Statistical Genetics | Harvard | Tsinghua</title><link>https://shuangsong0110.github.io/</link><atom:link href="https://shuangsong0110.github.io/index.xml" rel="self" type="application/rss+xml"/><description>Shuang Song | Biostatistics and Statistical Genetics | Harvard | Tsinghua</description><generator>Hugo Blox Builder (https://hugoblox.com)</generator><language>en-us</language><lastBuildDate>Mon, 24 Oct 2022 00:00:00 +0000</lastBuildDate><image><url>https://shuangsong0110.github.io/media/icon_hu68170e94a17a2a43d6dcb45cf0e8e589_3079_512x512_fill_lanczos_center_3.png</url><title>Shuang Song | Biostatistics and Statistical Genetics | Harvard | Tsinghua</title><link>https://shuangsong0110.github.io/</link></image><item><title>Participation bias in the estimation of heritability and genetic correlation</title><link>https://shuangsong0110.github.io/publication/participation-bias-pnas-2025/</link><pubDate>Wed, 31 Dec 2025 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/participation-bias-pnas-2025/</guid><description/></item><item><title>Partitioning and aggregating cross-tissue and tissue-specific genetic effects to identify gene-trait associations</title><link>https://shuangsong0110.github.io/publication/journal-motif/</link><pubDate>Tue, 09 Jul 2024 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/journal-motif/</guid><description/></item><item><title>Leveraging LD eigenvalue regression to improve the estimation of SNP heritability and confounding inflation</title><link>https://shuangsong0110.github.io/publication/lder-ajhg-2022/</link><pubDate>Fri, 30 Dec 2022 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/lder-ajhg-2022/</guid><description/></item><item><title>A data-adaptive Bayesian regression approach for polygenic risk prediction</title><link>https://shuangsong0110.github.io/publication/bayesian-polygenic-bioinformatics-2022/</link><pubDate>Thu, 29 Dec 2022 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/bayesian-polygenic-bioinformatics-2022/</guid><description/></item><item><title>Openness weighted association studies: leveraging personal genome information to prioritize non-coding variants</title><link>https://shuangsong0110.github.io/publication/owas-bioinformatics-2021/</link><pubDate>Fri, 31 Dec 2021 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/owas-bioinformatics-2021/</guid><description/></item><item><title>Leveraging effect size distributions to improve polygenic risk scores derived from summary statistics of genome-wide association studies</title><link>https://shuangsong0110.github.io/publication/ebprs-ploscb-2020/</link><pubDate>Thu, 31 Dec 2020 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/ebprs-ploscb-2020/</guid><description/></item><item><title>Multi-Cell-Type Openness-Weighted Association Studies for Trait-Associated Genomic Segments Prioritization</title><link>https://shuangsong0110.github.io/publication/owasjoint-genes-2022/</link><pubDate>Sat, 31 Dec 2022 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/owasjoint-genes-2022/</guid><description>&lt;h2 id="cite">CITE&lt;/h2>
&lt;p>Click the &lt;strong>Cite&lt;/strong> button above to export this publication in BibTeX format.&lt;/p></description></item><item><title>A set of efficient methods to generate high-dimensional binary data with specified correlation structures</title><link>https://shuangsong0110.github.io/publication/corbin-american-statistician-2021/</link><pubDate>Thu, 30 Dec 2021 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/corbin-american-statistician-2021/</guid><description/></item><item><title>Law of the Iterated Logarithm and Model Selection Consistency for Independent and Dependent GLMs</title><link>https://shuangsong0110.github.io/publication/lil-glm-front-math-2021/</link><pubDate>Wed, 29 Dec 2021 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/lil-glm-front-math-2021/</guid><description/></item><item><title>The predictive value of the orientation and offset of angle α and angle κ for visual outcomes after trifocal intraocular lens implantation in an Asian cohort</title><link>https://shuangsong0110.github.io/publication/ophthalmology-trifocal-asian-2025/</link><pubDate>Tue, 30 Dec 2025 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/ophthalmology-trifocal-asian-2025/</guid><description/></item><item><title>The Effects of Disease‐Modifying Therapies on Optic Nerve Degeneration in Multiple Sclerosis</title><link>https://shuangsong0110.github.io/publication/ms-optic-nerve-dmt-2025/</link><pubDate>Mon, 29 Dec 2025 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/ms-optic-nerve-dmt-2025/</guid><description/></item><item><title>A robust penalized-regression-based method for multivariable Mendelian randomization using GWAS summary statistics</title><link>https://shuangsong0110.github.io/publication/mvmr-penalized-gwas-2023/</link><pubDate>Sun, 31 Dec 2023 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/mvmr-penalized-gwas-2023/</guid><description/></item><item><title>Cofea: correlation-based feature selection for single-cell chromatin accessibility data</title><link>https://shuangsong0110.github.io/publication/cofea-bib-2024/</link><pubDate>Mon, 30 Dec 2024 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/cofea-bib-2024/</guid><description/></item><item><title>Cell type annotation of single-cell chromatin accessibility data via supervised Bayesian embedding</title><link>https://shuangsong0110.github.io/publication/scatac-bayesian-embedding-2022/</link><pubDate>Wed, 28 Dec 2022 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/scatac-bayesian-embedding-2022/</guid><description/></item><item><title>A novel transcriptional risk score for risk prediction of complex human diseases</title><link>https://shuangsong0110.github.io/publication/transcriptional-risk-score-2021/</link><pubDate>Tue, 28 Dec 2021 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/transcriptional-risk-score-2021/</guid><description/></item><item><title>An unbiased kinship estimation method for genetic data analysis</title><link>https://shuangsong0110.github.io/publication/unbiased-kinship-bmc-2022/</link><pubDate>Tue, 27 Dec 2022 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/unbiased-kinship-bmc-2022/</guid><description/></item><item><title>Correcting statistical bias in correlation-based kinship estimators</title><link>https://shuangsong0110.github.io/publication/kinship-bias-biorxiv-2021/</link><pubDate>Mon, 27 Dec 2021 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/kinship-bias-biorxiv-2021/</guid><description/></item><item><title>Blood Glucose Levels Moderate the Associations Between IGF-1 Levels and Choroidal Metrics in Patients With Diabetes With Acromegaly Without Diabetic Retinopathy</title><link>https://shuangsong0110.github.io/publication/igf1-choroidal-diabetes-2024/</link><pubDate>Sun, 29 Dec 2024 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/igf1-choroidal-diabetes-2024/</guid><description/></item><item><title>Reduction of Human Mobility Matters during Early COVID-19 Outbreaks: Evidence from India, Japan and China</title><link>https://shuangsong0110.github.io/publication/covid-mobility-ijerph-2021/</link><pubDate>Sun, 26 Dec 2021 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/covid-mobility-ijerph-2021/</guid><description/></item><item><title>ECR session</title><link>https://shuangsong0110.github.io/event/phd_2024/</link><pubDate>Thu, 12 Mar 2026 10:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/event/phd_2024/</guid><description>&lt;!--
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&lt;span class="dark:text-neutral-300">Click on the &lt;strong>Slides&lt;/strong> button above to view the built-in slides feature.&lt;/span>
&lt;/div>
Slides can be added in a few ways:
- **Create** slides using Hugo Blox Builder's [_Slides_](https://docs.hugoblox.com/reference/content-types/) feature and link using `slides` parameter in the front matter of the talk file
- **Upload** an existing slide deck to `static/` and link using `url_slides` parameter in the front matter of the talk file
- **Embed** your slides (e.g. Google Slides) or presentation video on this page using [shortcodes](https://docs.hugoblox.com/reference/markdown/). -->
&lt;!-- Further event details, including [page elements](https://docs.hugoblox.com/reference/markdown/) such as image galleries, can be added to the body of this page. --></description></item><item><title>🎉 Our paper "Scalable and accurate rare-variant association tests for whole genome sequencing time-to-event analysis in large biobanks" is published on PNAS!</title><link>https://shuangsong0110.github.io/post/mentor/</link><pubDate>Fri, 27 Feb 2026 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/post/mentor/</guid><description>&lt;p>&lt;a href="https://www.pnas.org/doi/10.1073/pnas.2525288123">Read the full paper&lt;/a>&lt;/p></description></item><item><title>Scalable and accurate rare-variant association tests for whole genome sequencing time-to-event analysis in large biobanks</title><link>https://shuangsong0110.github.io/publication/journal-acml/</link><pubDate>Fri, 27 Feb 2026 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/journal-acml/</guid><description>&lt;h2 id="significance">Significance&lt;/h2>
&lt;p>Rare variants (RV) identified through whole genome sequencing hold great promise for elucidating the genetic basis of disease onset, but existing methods for RV association testing are not well suited for time-to-event phenotypes. Here, we develop GATE-STAAR, a scalable and accurate framework integrating frailty modeling with functional annotations. We propose a rare-variant saddlepoint approximation to handle heavy censoring. Through comprehensive simulations and large-scale analysis of approximately 400 K UK Biobank participants, with replication in approximately 230 K All of Us participants, GATE-STAAR uncovers biologically meaningful RV associations while ensuring rigorous control of type I error. This framework is powerful to dissect the genetic architecture of disease onset and progression and advance precision medicine.&lt;/p></description></item><item><title>Scalable and accurate rare variant association tests for whole genome sequencing time-to-event analysis in large biobanks</title><link>https://shuangsong0110.github.io/publication/scalable-rare-variant-pnas-2026/</link><pubDate>Thu, 15 Jan 2026 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/scalable-rare-variant-pnas-2026/</guid><description/></item><item><title>Contributed session</title><link>https://shuangsong0110.github.io/event/symmetry/</link><pubDate>Thu, 07 Aug 2025 13:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/event/symmetry/</guid><description>&lt;!--
&lt;div class="flex px-4 py-3 mb-6 rounded-md bg-primary-100 dark:bg-primary-900">
&lt;span class="pr-3 pt-1 text-primary-600 dark:text-primary-300">
&lt;svg height="24" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24">&lt;path fill="none" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="1.5" d="m11.25 11.25l.041-.02a.75.75 0 0 1 1.063.852l-.708 2.836a.75.75 0 0 0 1.063.853l.041-.021M21 12a9 9 0 1 1-18 0a9 9 0 0 1 18 0m-9-3.75h.008v.008H12z"/>&lt;/svg>
&lt;/span>
&lt;span class="dark:text-neutral-300">Click on the &lt;strong>Slides&lt;/strong> button above to view the built-in slides feature.&lt;/span>
&lt;/div>
Slides can be added in a few ways:
- **Create** slides using Hugo Blox Builder's [_Slides_](https://docs.hugoblox.com/reference/content-types/) feature and link using `slides` parameter in the front matter of the talk file
- **Upload** an existing slide deck to `static/` and link using `url_slides` parameter in the front matter of the talk file
- **Embed** your slides (e.g. Google Slides) or presentation video on this page using [shortcodes](https://docs.hugoblox.com/reference/markdown/). -->
&lt;!-- Further event details, including [page elements](https://docs.hugoblox.com/reference/markdown/) such as image galleries, can be added to the body of this page. --></description></item><item><title>🎉 Our paper "Participation bias in the estimation of heritability and genetic correlation" is published on PNAS!</title><link>https://shuangsong0110.github.io/post/biogen/index-2/</link><pubDate>Tue, 24 Jun 2025 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/post/biogen/index-2/</guid><description>&lt;p>&lt;a href="https://www.pnas.org/doi/abs/10.1073/pnas.2425530122">Read the full paper&lt;/a>&lt;/p></description></item><item><title>Participation bias in the estimation of heritability and genetic correlation</title><link>https://shuangsong0110.github.io/publication/journal-article-llm-review-1/</link><pubDate>Tue, 24 Jun 2025 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/publication/journal-article-llm-review-1/</guid><description>&lt;!-- > [!NOTE]
> Click the _Cite_ button above to demo the feature to enable visitors to import publication metadata into their reference management software.
> [!NOTE]
> Create your slides in Markdown - click the _Slides_ button to check out the example.
Add the publication's **full text** or **supplementary notes** here. You can use rich formatting such as including [code, math, and images](https://docs.hugoblox.com/content/writing-markdown-latex/). --></description></item><item><title>Invited session</title><link>https://shuangsong0110.github.io/event/amia/</link><pubDate>Mon, 07 Apr 2025 13:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/event/amia/</guid><description>&lt;!--
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&lt;/span>
&lt;span class="dark:text-neutral-300">Click on the &lt;strong>Slides&lt;/strong> button above to view the built-in slides feature.&lt;/span>
&lt;/div>
Slides can be added in a few ways:
- **Create** slides using Hugo Blox Builder's [_Slides_](https://docs.hugoblox.com/reference/content-types/) feature and link using `slides` parameter in the front matter of the talk file
- **Upload** an existing slide deck to `static/` and link using `url_slides` parameter in the front matter of the talk file
- **Embed** your slides (e.g. Google Slides) or presentation video on this page using [shortcodes](https://docs.hugoblox.com/reference/markdown/). -->
&lt;!-- Further event details, including [page elements](https://docs.hugoblox.com/reference/markdown/) such as image galleries, can be added to the body of this page. --></description></item><item><title>Projects</title><link>https://shuangsong0110.github.io/projects/</link><pubDate>Sun, 19 May 2024 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/projects/</guid><description/></item><item><title>Pandas</title><link>https://shuangsong0110.github.io/project/pandas/</link><pubDate>Thu, 26 Oct 2023 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/project/pandas/</guid><description>&lt;p>Flexible and powerful data analysis / manipulation library for Python, providing labeled data structures.&lt;/p></description></item><item><title>PyTorch</title><link>https://shuangsong0110.github.io/project/pytorch/</link><pubDate>Thu, 26 Oct 2023 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/project/pytorch/</guid><description>&lt;p>PyTorch is a Python package that provides tensor computation (like NumPy) with strong GPU acceleration.&lt;/p></description></item><item><title>Experience</title><link>https://shuangsong0110.github.io/experience/</link><pubDate>Tue, 24 Oct 2023 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/experience/</guid><description/></item><item><title>Learn JavaScript</title><link>https://shuangsong0110.github.io/teaching/js/</link><pubDate>Tue, 24 Oct 2023 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/teaching/js/</guid><description>&lt;p>&lt;a href="https://hugoblox.com">Hugo Blox Builder&lt;/a> is designed to give technical content creators a seamless experience. You can focus on the content and the Hugo Blox Builder which this template is built upon handles the rest.&lt;/p>
&lt;p>&lt;strong>Embed videos, podcasts, code, LaTeX math, and even test students!&lt;/strong>&lt;/p>
&lt;p>On this page, you&amp;rsquo;ll find some examples of the types of technical content that can be rendered with Hugo Blox.&lt;/p>
&lt;h2 id="video">Video&lt;/h2>
&lt;p>Teach your course by sharing videos with your students. Choose from one of the following approaches:&lt;/p>
&lt;div style="position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden;">
&lt;iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="allowfullscreen" loading="eager" referrerpolicy="strict-origin-when-cross-origin" src="https://www.youtube.com/embed/D2vj0WcvH5c?autoplay=0&amp;controls=1&amp;end=0&amp;loop=0&amp;mute=0&amp;start=0" style="position: absolute; top: 0; left: 0; width: 100%; height: 100%; border:0;" title="YouTube video"
>&lt;/iframe>
&lt;/div>
&lt;p>&lt;strong>Youtube&lt;/strong>:&lt;/p>
&lt;pre>&lt;code>{{&amp;lt; youtube w7Ft2ymGmfc &amp;gt;}}
&lt;/code>&lt;/pre>
&lt;p>&lt;strong>Bilibili&lt;/strong>:&lt;/p>
&lt;pre>&lt;code>{{&amp;lt; bilibili id=&amp;quot;BV1WV4y1r7DF&amp;quot; &amp;gt;}}
&lt;/code>&lt;/pre>
&lt;p>&lt;strong>Video file&lt;/strong>&lt;/p>
&lt;p>Videos may be added to a page by either placing them in your &lt;code>assets/media/&lt;/code> media library or in your &lt;a href="https://gohugo.io/content-management/page-bundles/">page&amp;rsquo;s folder&lt;/a>, and then embedding them with the &lt;em>video&lt;/em> shortcode:&lt;/p>
&lt;pre>&lt;code>{{&amp;lt; video src=&amp;quot;my_video.mp4&amp;quot; controls=&amp;quot;yes&amp;quot; &amp;gt;}}
&lt;/code>&lt;/pre>
&lt;h2 id="podcast">Podcast&lt;/h2>
&lt;p>You can add a podcast or music to a page by placing the MP3 file in the page&amp;rsquo;s folder or the media library folder and then embedding the audio on your page with the &lt;em>audio&lt;/em> shortcode:&lt;/p>
&lt;pre>&lt;code>{{&amp;lt; audio src=&amp;quot;ambient-piano.mp3&amp;quot; &amp;gt;}}
&lt;/code>&lt;/pre>
&lt;p>Try it out:&lt;/p>
&lt;audio controls >
&lt;source src="https://shuangsong0110.github.io/teaching/js/ambient-piano.mp3" type="audio/mpeg">
&lt;/audio>
&lt;h2 id="test-students">Test students&lt;/h2>
&lt;p>Provide a simple yet fun self-assessment by revealing the solutions to challenges with the &lt;code>spoiler&lt;/code> shortcode:&lt;/p>
&lt;div class="highlight">&lt;pre tabindex="0" class="chroma">&lt;code class="language-markdown" data-lang="markdown">&lt;span class="line">&lt;span class="cl">{{&lt;span class="p">&amp;lt;&lt;/span> &lt;span class="nt">spoiler&lt;/span> &lt;span class="na">text&lt;/span>&lt;span class="o">=&lt;/span>&lt;span class="s">&amp;#34;👉 Click to view the solution&amp;#34;&lt;/span> &lt;span class="p">&amp;gt;&lt;/span>}}
&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">You found me!
&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">{{&lt;span class="p">&amp;lt;&lt;/span> &lt;span class="p">/&lt;/span>&lt;span class="nt">spoiler&lt;/span> &lt;span class="p">&amp;gt;&lt;/span>}}
&lt;/span>&lt;/span>&lt;/code>&lt;/pre>&lt;/div>&lt;p>renders as&lt;/p>
&lt;details class="spoiler " id="spoiler-2">
&lt;summary class="cursor-pointer">👉 Click to view the solution&lt;/summary>
&lt;div class="rounded-lg bg-neutral-50 dark:bg-neutral-800 p-2">
You found me 🎉
&lt;/div>
&lt;/details>
&lt;h2 id="math">Math&lt;/h2>
&lt;p>Hugo Blox Builder supports a Markdown extension for $\LaTeX$ math. You can enable this feature by toggling the &lt;code>math&lt;/code> option in your &lt;code>config/_default/params.yaml&lt;/code> file.&lt;/p>
&lt;p>To render &lt;em>inline&lt;/em> or &lt;em>block&lt;/em> math, wrap your LaTeX math with &lt;code>{{&amp;lt; math &amp;gt;}}$...${{&amp;lt; /math &amp;gt;}}&lt;/code> or &lt;code>{{&amp;lt; math &amp;gt;}}$$...$${{&amp;lt; /math &amp;gt;}}&lt;/code>, respectively.&lt;/p>
&lt;div class="flex px-4 py-3 mb-6 rounded-md bg-primary-100 dark:bg-primary-900">
&lt;span class="pr-3 pt-1 text-primary-600 dark:text-primary-300">
&lt;svg height="24" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24">&lt;path fill="none" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="1.5" d="m11.25 11.25l.041-.02a.75.75 0 0 1 1.063.852l-.708 2.836a.75.75 0 0 0 1.063.853l.041-.021M21 12a9 9 0 1 1-18 0a9 9 0 0 1 18 0m-9-3.75h.008v.008H12z"/>&lt;/svg>
&lt;/span>
&lt;span class="dark:text-neutral-300">We wrap the LaTeX math in the Hugo Blox &lt;em>math&lt;/em> shortcode to prevent Hugo rendering our math as Markdown.&lt;/span>
&lt;/div>
&lt;p>Example &lt;strong>math block&lt;/strong>:&lt;/p>
&lt;div class="highlight">&lt;pre tabindex="0" class="chroma">&lt;code class="language-latex" data-lang="latex">&lt;span class="line">&lt;span class="cl">&lt;span class="nb">{{&lt;/span>&amp;lt; math &amp;gt;&lt;span class="nb">}}&lt;/span>
&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="sb">$$&lt;/span>&lt;span class="nb">
&lt;/span>&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="nb">&lt;/span>&lt;span class="nv">\gamma&lt;/span>&lt;span class="nb">_{n} &lt;/span>&lt;span class="o">=&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\frac&lt;/span>&lt;span class="nb">{ &lt;/span>&lt;span class="nv">\left&lt;/span>&lt;span class="nb"> | &lt;/span>&lt;span class="nv">\left&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="o">(&lt;/span>&lt;span class="nv">\mathbf&lt;/span>&lt;span class="nb"> x_{n} &lt;/span>&lt;span class="o">-&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\mathbf&lt;/span>&lt;span class="nb"> x_{n&lt;/span>&lt;span class="o">-&lt;/span>&lt;span class="m">1&lt;/span>&lt;span class="nb">} &lt;/span>&lt;span class="nv">\right&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="o">)&lt;/span>&lt;span class="nb">^T &lt;/span>&lt;span class="nv">\left&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="o">[&lt;/span>&lt;span class="nv">\nabla&lt;/span>&lt;span class="nb"> F &lt;/span>&lt;span class="o">(&lt;/span>&lt;span class="nv">\mathbf&lt;/span>&lt;span class="nb"> x_{n}&lt;/span>&lt;span class="o">)&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="o">-&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\nabla&lt;/span>&lt;span class="nb"> F &lt;/span>&lt;span class="o">(&lt;/span>&lt;span class="nv">\mathbf&lt;/span>&lt;span class="nb"> x_{n&lt;/span>&lt;span class="o">-&lt;/span>&lt;span class="m">1&lt;/span>&lt;span class="nb">}&lt;/span>&lt;span class="o">)&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\right&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="o">]&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\right&lt;/span>&lt;span class="nb"> |}{&lt;/span>&lt;span class="nv">\left&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\|\nabla&lt;/span>&lt;span class="nb"> F&lt;/span>&lt;span class="o">(&lt;/span>&lt;span class="nv">\mathbf&lt;/span>&lt;span class="nb">{x}_{n}&lt;/span>&lt;span class="o">)&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="o">-&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\nabla&lt;/span>&lt;span class="nb"> F&lt;/span>&lt;span class="o">(&lt;/span>&lt;span class="nv">\mathbf&lt;/span>&lt;span class="nb">{x}_{n&lt;/span>&lt;span class="o">-&lt;/span>&lt;span class="m">1&lt;/span>&lt;span class="nb">}&lt;/span>&lt;span class="o">)&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\right&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\|&lt;/span>&lt;span class="nb">^&lt;/span>&lt;span class="m">2&lt;/span>&lt;span class="nb">}
&lt;/span>&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="nb">&lt;/span>&lt;span class="s">$$&lt;/span>
&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="nb">{{&lt;/span>&amp;lt; /math &amp;gt;&lt;span class="nb">}}&lt;/span>
&lt;/span>&lt;/span>&lt;/code>&lt;/pre>&lt;/div>&lt;p>renders as&lt;/p>
$$\gamma_{n} = \frac{ \left | \left (\mathbf x_{n} - \mathbf x_{n-1} \right )^T \left [\nabla F (\mathbf x_{n}) - \nabla F (\mathbf x_{n-1}) \right ] \right |}{\left \|\nabla F(\mathbf{x}_{n}) - \nabla F(\mathbf{x}_{n-1}) \right \|^2}$$
&lt;p>Example &lt;strong>inline math&lt;/strong> &lt;code>{{&amp;lt; math &amp;gt;}}$\nabla F(\mathbf{x}_{n})${{&amp;lt; /math &amp;gt;}}&lt;/code> renders as $\nabla F(\mathbf{x}_{n})$
.&lt;/p>
&lt;p>Example &lt;strong>multi-line math&lt;/strong> using the math linebreak (&lt;code>\\&lt;/code>):&lt;/p>
&lt;div class="highlight">&lt;pre tabindex="0" class="chroma">&lt;code class="language-latex" data-lang="latex">&lt;span class="line">&lt;span class="cl">&lt;span class="nb">{{&lt;/span>&amp;lt; math &amp;gt;&lt;span class="nb">}}&lt;/span>
&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="sb">$$&lt;/span>&lt;span class="nb">f&lt;/span>&lt;span class="o">(&lt;/span>&lt;span class="nb">k;p_{&lt;/span>&lt;span class="m">0&lt;/span>&lt;span class="nb">}^{&lt;/span>&lt;span class="o">*&lt;/span>&lt;span class="nb">}&lt;/span>&lt;span class="o">)&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="o">=&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\begin&lt;/span>&lt;span class="nb">{cases}p_{&lt;/span>&lt;span class="m">0&lt;/span>&lt;span class="nb">}^{&lt;/span>&lt;span class="o">*&lt;/span>&lt;span class="nb">} &amp;amp; &lt;/span>&lt;span class="nv">\text&lt;/span>&lt;span class="nb">{if }k&lt;/span>&lt;span class="o">=&lt;/span>&lt;span class="m">1&lt;/span>&lt;span class="nb">, &lt;/span>&lt;span class="nv">\\&lt;/span>&lt;span class="nb">
&lt;/span>&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="nb">&lt;/span>&lt;span class="m">1&lt;/span>&lt;span class="o">-&lt;/span>&lt;span class="nb">p_{&lt;/span>&lt;span class="m">0&lt;/span>&lt;span class="nb">}^{&lt;/span>&lt;span class="o">*&lt;/span>&lt;span class="nb">} &amp;amp; &lt;/span>&lt;span class="nv">\text&lt;/span>&lt;span class="nb">{if }k&lt;/span>&lt;span class="o">=&lt;/span>&lt;span class="m">0&lt;/span>&lt;span class="nb">.&lt;/span>&lt;span class="nv">\end&lt;/span>&lt;span class="nb">{cases}&lt;/span>&lt;span class="s">$$&lt;/span>
&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="nb">{{&lt;/span>&amp;lt; /math &amp;gt;&lt;span class="nb">}}&lt;/span>
&lt;/span>&lt;/span>&lt;/code>&lt;/pre>&lt;/div>&lt;p>renders as&lt;/p>
$$
f(k;p_{0}^{*}) = \begin{cases}p_{0}^{*} &amp; \text{if }k=1, \\
1-p_{0}^{*} &amp; \text{if }k=0.\end{cases}
$$
&lt;h2 id="code">Code&lt;/h2>
&lt;p>Hugo Blox Builder utilises Hugo&amp;rsquo;s Markdown extension for highlighting code syntax. The code theme can be selected in the &lt;code>config/_default/params.yaml&lt;/code> file.&lt;/p>
&lt;pre>&lt;code>```python
import pandas as pd
data = pd.read_csv(&amp;quot;data.csv&amp;quot;)
data.head()
```
&lt;/code>&lt;/pre>
&lt;p>renders as&lt;/p>
&lt;div class="highlight">&lt;pre tabindex="0" class="chroma">&lt;code class="language-python" data-lang="python">&lt;span class="line">&lt;span class="cl">&lt;span class="kn">import&lt;/span> &lt;span class="nn">pandas&lt;/span> &lt;span class="k">as&lt;/span> &lt;span class="nn">pd&lt;/span>
&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="n">data&lt;/span> &lt;span class="o">=&lt;/span> &lt;span class="n">pd&lt;/span>&lt;span class="o">.&lt;/span>&lt;span class="n">read_csv&lt;/span>&lt;span class="p">(&lt;/span>&lt;span class="s2">&amp;#34;data.csv&amp;#34;&lt;/span>&lt;span class="p">)&lt;/span>
&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="n">data&lt;/span>&lt;span class="o">.&lt;/span>&lt;span class="n">head&lt;/span>&lt;span class="p">()&lt;/span>
&lt;/span>&lt;/span>&lt;/code>&lt;/pre>&lt;/div>&lt;h2 id="inline-images">Inline Images&lt;/h2>
&lt;div class="highlight">&lt;pre tabindex="0" class="chroma">&lt;code class="language-go" data-lang="go">&lt;span class="line">&lt;span class="cl">&lt;span class="p">{{&amp;lt;&lt;/span> &lt;span class="nx">icon&lt;/span> &lt;span class="nx">name&lt;/span>&lt;span class="p">=&lt;/span>&lt;span class="s">&amp;#34;python&amp;#34;&lt;/span> &lt;span class="p">&amp;gt;}}&lt;/span> &lt;span class="nx">Python&lt;/span>
&lt;/span>&lt;/span>&lt;/code>&lt;/pre>&lt;/div>&lt;p>renders as&lt;/p>
&lt;p>
&lt;span class="inline-block pr-1">
&lt;svg style="height: 1em; transform: translateY(0.1em);" xmlns="http://www.w3.org/2000/svg" height="1em" viewBox="0 0 448 512" fill="currentColor">&lt;path d="M439.8 200.5c-7.7-30.9-22.3-54.2-53.4-54.2h-40.1v47.4c0 36.8-31.2 67.8-66.8 67.8H172.7c-29.2 0-53.4 25-53.4 54.3v101.8c0 29 25.2 46 53.4 54.3 33.8 9.9 66.3 11.7 106.8 0 26.9-7.8 53.4-23.5 53.4-54.3v-40.7H226.2v-13.6h160.2c31.1 0 42.6-21.7 53.4-54.2 11.2-33.5 10.7-65.7 0-108.6zM286.2 404c11.1 0 20.1 9.1 20.1 20.3 0 11.3-9 20.4-20.1 20.4-11 0-20.1-9.2-20.1-20.4.1-11.3 9.1-20.3 20.1-20.3zM167.8 248.1h106.8c29.7 0 53.4-24.5 53.4-54.3V91.9c0-29-24.4-50.7-53.4-55.6-35.8-5.9-74.7-5.6-106.8.1-45.2 8-53.4 24.7-53.4 55.6v40.7h106.9v13.6h-147c-31.1 0-58.3 18.7-66.8 54.2-9.8 40.7-10.2 66.1 0 108.6 7.6 31.6 25.7 54.2 56.8 54.2H101v-48.8c0-35.3 30.5-66.4 66.8-66.4zm-6.7-142.6c-11.1 0-20.1-9.1-20.1-20.3.1-11.3 9-20.4 20.1-20.4 11 0 20.1 9.2 20.1 20.4s-9 20.3-20.1 20.3z"/>&lt;/svg>
&lt;/span> Python&lt;/p>
&lt;h2 id="did-you-find-this-page-helpful-consider-sharing-it-">Did you find this page helpful? Consider sharing it 🙌&lt;/h2></description></item><item><title>Learn Python</title><link>https://shuangsong0110.github.io/teaching/python/</link><pubDate>Tue, 24 Oct 2023 00:00:00 +0000</pubDate><guid>https://shuangsong0110.github.io/teaching/python/</guid><description>&lt;p>&lt;a href="https://hugoblox.com">Hugo Blox Builder&lt;/a> is designed to give technical content creators a seamless experience. You can focus on the content and the Hugo Blox Builder which this template is built upon handles the rest.&lt;/p>
&lt;p>&lt;strong>Embed videos, podcasts, code, LaTeX math, and even test students!&lt;/strong>&lt;/p>
&lt;p>On this page, you&amp;rsquo;ll find some examples of the types of technical content that can be rendered with Hugo Blox.&lt;/p>
&lt;h2 id="video">Video&lt;/h2>
&lt;p>Teach your course by sharing videos with your students. Choose from one of the following approaches:&lt;/p>
&lt;div style="position: relative; padding-bottom: 56.25%; height: 0; overflow: hidden;">
&lt;iframe allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen="allowfullscreen" loading="eager" referrerpolicy="strict-origin-when-cross-origin" src="https://www.youtube.com/embed/D2vj0WcvH5c?autoplay=0&amp;controls=1&amp;end=0&amp;loop=0&amp;mute=0&amp;start=0" style="position: absolute; top: 0; left: 0; width: 100%; height: 100%; border:0;" title="YouTube video"
>&lt;/iframe>
&lt;/div>
&lt;p>&lt;strong>Youtube&lt;/strong>:&lt;/p>
&lt;pre>&lt;code>{{&amp;lt; youtube w7Ft2ymGmfc &amp;gt;}}
&lt;/code>&lt;/pre>
&lt;p>&lt;strong>Bilibili&lt;/strong>:&lt;/p>
&lt;pre>&lt;code>{{&amp;lt; bilibili id=&amp;quot;BV1WV4y1r7DF&amp;quot; &amp;gt;}}
&lt;/code>&lt;/pre>
&lt;p>&lt;strong>Video file&lt;/strong>&lt;/p>
&lt;p>Videos may be added to a page by either placing them in your &lt;code>assets/media/&lt;/code> media library or in your &lt;a href="https://gohugo.io/content-management/page-bundles/">page&amp;rsquo;s folder&lt;/a>, and then embedding them with the &lt;em>video&lt;/em> shortcode:&lt;/p>
&lt;pre>&lt;code>{{&amp;lt; video src=&amp;quot;my_video.mp4&amp;quot; controls=&amp;quot;yes&amp;quot; &amp;gt;}}
&lt;/code>&lt;/pre>
&lt;h2 id="podcast">Podcast&lt;/h2>
&lt;p>You can add a podcast or music to a page by placing the MP3 file in the page&amp;rsquo;s folder or the media library folder and then embedding the audio on your page with the &lt;em>audio&lt;/em> shortcode:&lt;/p>
&lt;pre>&lt;code>{{&amp;lt; audio src=&amp;quot;ambient-piano.mp3&amp;quot; &amp;gt;}}
&lt;/code>&lt;/pre>
&lt;p>Try it out:&lt;/p>
&lt;audio controls >
&lt;source src="https://shuangsong0110.github.io/teaching/python/ambient-piano.mp3" type="audio/mpeg">
&lt;/audio>
&lt;h2 id="test-students">Test students&lt;/h2>
&lt;p>Provide a simple yet fun self-assessment by revealing the solutions to challenges with the &lt;code>spoiler&lt;/code> shortcode:&lt;/p>
&lt;div class="highlight">&lt;pre tabindex="0" class="chroma">&lt;code class="language-markdown" data-lang="markdown">&lt;span class="line">&lt;span class="cl">{{&lt;span class="p">&amp;lt;&lt;/span> &lt;span class="nt">spoiler&lt;/span> &lt;span class="na">text&lt;/span>&lt;span class="o">=&lt;/span>&lt;span class="s">&amp;#34;👉 Click to view the solution&amp;#34;&lt;/span> &lt;span class="p">&amp;gt;&lt;/span>}}
&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">You found me!
&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">{{&lt;span class="p">&amp;lt;&lt;/span> &lt;span class="p">/&lt;/span>&lt;span class="nt">spoiler&lt;/span> &lt;span class="p">&amp;gt;&lt;/span>}}
&lt;/span>&lt;/span>&lt;/code>&lt;/pre>&lt;/div>&lt;p>renders as&lt;/p>
&lt;details class="spoiler " id="spoiler-2">
&lt;summary class="cursor-pointer">👉 Click to view the solution&lt;/summary>
&lt;div class="rounded-lg bg-neutral-50 dark:bg-neutral-800 p-2">
You found me 🎉
&lt;/div>
&lt;/details>
&lt;h2 id="math">Math&lt;/h2>
&lt;p>Hugo Blox Builder supports a Markdown extension for $\LaTeX$ math. You can enable this feature by toggling the &lt;code>math&lt;/code> option in your &lt;code>config/_default/params.yaml&lt;/code> file.&lt;/p>
&lt;p>To render &lt;em>inline&lt;/em> or &lt;em>block&lt;/em> math, wrap your LaTeX math with &lt;code>{{&amp;lt; math &amp;gt;}}$...${{&amp;lt; /math &amp;gt;}}&lt;/code> or &lt;code>{{&amp;lt; math &amp;gt;}}$$...$${{&amp;lt; /math &amp;gt;}}&lt;/code>, respectively.&lt;/p>
&lt;div class="flex px-4 py-3 mb-6 rounded-md bg-primary-100 dark:bg-primary-900">
&lt;span class="pr-3 pt-1 text-primary-600 dark:text-primary-300">
&lt;svg height="24" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24">&lt;path fill="none" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="1.5" d="m11.25 11.25l.041-.02a.75.75 0 0 1 1.063.852l-.708 2.836a.75.75 0 0 0 1.063.853l.041-.021M21 12a9 9 0 1 1-18 0a9 9 0 0 1 18 0m-9-3.75h.008v.008H12z"/>&lt;/svg>
&lt;/span>
&lt;span class="dark:text-neutral-300">We wrap the LaTeX math in the Hugo Blox &lt;em>math&lt;/em> shortcode to prevent Hugo rendering our math as Markdown.&lt;/span>
&lt;/div>
&lt;p>Example &lt;strong>math block&lt;/strong>:&lt;/p>
&lt;div class="highlight">&lt;pre tabindex="0" class="chroma">&lt;code class="language-latex" data-lang="latex">&lt;span class="line">&lt;span class="cl">&lt;span class="nb">{{&lt;/span>&amp;lt; math &amp;gt;&lt;span class="nb">}}&lt;/span>
&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="sb">$$&lt;/span>&lt;span class="nb">
&lt;/span>&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="nb">&lt;/span>&lt;span class="nv">\gamma&lt;/span>&lt;span class="nb">_{n} &lt;/span>&lt;span class="o">=&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\frac&lt;/span>&lt;span class="nb">{ &lt;/span>&lt;span class="nv">\left&lt;/span>&lt;span class="nb"> | &lt;/span>&lt;span class="nv">\left&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="o">(&lt;/span>&lt;span class="nv">\mathbf&lt;/span>&lt;span class="nb"> x_{n} &lt;/span>&lt;span class="o">-&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\mathbf&lt;/span>&lt;span class="nb"> x_{n&lt;/span>&lt;span class="o">-&lt;/span>&lt;span class="m">1&lt;/span>&lt;span class="nb">} &lt;/span>&lt;span class="nv">\right&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="o">)&lt;/span>&lt;span class="nb">^T &lt;/span>&lt;span class="nv">\left&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="o">[&lt;/span>&lt;span class="nv">\nabla&lt;/span>&lt;span class="nb"> F &lt;/span>&lt;span class="o">(&lt;/span>&lt;span class="nv">\mathbf&lt;/span>&lt;span class="nb"> x_{n}&lt;/span>&lt;span class="o">)&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="o">-&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\nabla&lt;/span>&lt;span class="nb"> F &lt;/span>&lt;span class="o">(&lt;/span>&lt;span class="nv">\mathbf&lt;/span>&lt;span class="nb"> x_{n&lt;/span>&lt;span class="o">-&lt;/span>&lt;span class="m">1&lt;/span>&lt;span class="nb">}&lt;/span>&lt;span class="o">)&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\right&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="o">]&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\right&lt;/span>&lt;span class="nb"> |}{&lt;/span>&lt;span class="nv">\left&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\|\nabla&lt;/span>&lt;span class="nb"> F&lt;/span>&lt;span class="o">(&lt;/span>&lt;span class="nv">\mathbf&lt;/span>&lt;span class="nb">{x}_{n}&lt;/span>&lt;span class="o">)&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="o">-&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\nabla&lt;/span>&lt;span class="nb"> F&lt;/span>&lt;span class="o">(&lt;/span>&lt;span class="nv">\mathbf&lt;/span>&lt;span class="nb">{x}_{n&lt;/span>&lt;span class="o">-&lt;/span>&lt;span class="m">1&lt;/span>&lt;span class="nb">}&lt;/span>&lt;span class="o">)&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\right&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\|&lt;/span>&lt;span class="nb">^&lt;/span>&lt;span class="m">2&lt;/span>&lt;span class="nb">}
&lt;/span>&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="nb">&lt;/span>&lt;span class="s">$$&lt;/span>
&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="nb">{{&lt;/span>&amp;lt; /math &amp;gt;&lt;span class="nb">}}&lt;/span>
&lt;/span>&lt;/span>&lt;/code>&lt;/pre>&lt;/div>&lt;p>renders as&lt;/p>
$$\gamma_{n} = \frac{ \left | \left (\mathbf x_{n} - \mathbf x_{n-1} \right )^T \left [\nabla F (\mathbf x_{n}) - \nabla F (\mathbf x_{n-1}) \right ] \right |}{\left \|\nabla F(\mathbf{x}_{n}) - \nabla F(\mathbf{x}_{n-1}) \right \|^2}$$
&lt;p>Example &lt;strong>inline math&lt;/strong> &lt;code>{{&amp;lt; math &amp;gt;}}$\nabla F(\mathbf{x}_{n})${{&amp;lt; /math &amp;gt;}}&lt;/code> renders as $\nabla F(\mathbf{x}_{n})$
.&lt;/p>
&lt;p>Example &lt;strong>multi-line math&lt;/strong> using the math linebreak (&lt;code>\\&lt;/code>):&lt;/p>
&lt;div class="highlight">&lt;pre tabindex="0" class="chroma">&lt;code class="language-latex" data-lang="latex">&lt;span class="line">&lt;span class="cl">&lt;span class="nb">{{&lt;/span>&amp;lt; math &amp;gt;&lt;span class="nb">}}&lt;/span>
&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="sb">$$&lt;/span>&lt;span class="nb">f&lt;/span>&lt;span class="o">(&lt;/span>&lt;span class="nb">k;p_{&lt;/span>&lt;span class="m">0&lt;/span>&lt;span class="nb">}^{&lt;/span>&lt;span class="o">*&lt;/span>&lt;span class="nb">}&lt;/span>&lt;span class="o">)&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="o">=&lt;/span>&lt;span class="nb"> &lt;/span>&lt;span class="nv">\begin&lt;/span>&lt;span class="nb">{cases}p_{&lt;/span>&lt;span class="m">0&lt;/span>&lt;span class="nb">}^{&lt;/span>&lt;span class="o">*&lt;/span>&lt;span class="nb">} &amp;amp; &lt;/span>&lt;span class="nv">\text&lt;/span>&lt;span class="nb">{if }k&lt;/span>&lt;span class="o">=&lt;/span>&lt;span class="m">1&lt;/span>&lt;span class="nb">, &lt;/span>&lt;span class="nv">\\&lt;/span>&lt;span class="nb">
&lt;/span>&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="nb">&lt;/span>&lt;span class="m">1&lt;/span>&lt;span class="o">-&lt;/span>&lt;span class="nb">p_{&lt;/span>&lt;span class="m">0&lt;/span>&lt;span class="nb">}^{&lt;/span>&lt;span class="o">*&lt;/span>&lt;span class="nb">} &amp;amp; &lt;/span>&lt;span class="nv">\text&lt;/span>&lt;span class="nb">{if }k&lt;/span>&lt;span class="o">=&lt;/span>&lt;span class="m">0&lt;/span>&lt;span class="nb">.&lt;/span>&lt;span class="nv">\end&lt;/span>&lt;span class="nb">{cases}&lt;/span>&lt;span class="s">$$&lt;/span>
&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="nb">{{&lt;/span>&amp;lt; /math &amp;gt;&lt;span class="nb">}}&lt;/span>
&lt;/span>&lt;/span>&lt;/code>&lt;/pre>&lt;/div>&lt;p>renders as&lt;/p>
$$
f(k;p_{0}^{*}) = \begin{cases}p_{0}^{*} &amp; \text{if }k=1, \\
1-p_{0}^{*} &amp; \text{if }k=0.\end{cases}
$$
&lt;h2 id="code">Code&lt;/h2>
&lt;p>Hugo Blox Builder utilises Hugo&amp;rsquo;s Markdown extension for highlighting code syntax. The code theme can be selected in the &lt;code>config/_default/params.yaml&lt;/code> file.&lt;/p>
&lt;pre>&lt;code>```python
import pandas as pd
data = pd.read_csv(&amp;quot;data.csv&amp;quot;)
data.head()
```
&lt;/code>&lt;/pre>
&lt;p>renders as&lt;/p>
&lt;div class="highlight">&lt;pre tabindex="0" class="chroma">&lt;code class="language-python" data-lang="python">&lt;span class="line">&lt;span class="cl">&lt;span class="kn">import&lt;/span> &lt;span class="nn">pandas&lt;/span> &lt;span class="k">as&lt;/span> &lt;span class="nn">pd&lt;/span>
&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="n">data&lt;/span> &lt;span class="o">=&lt;/span> &lt;span class="n">pd&lt;/span>&lt;span class="o">.&lt;/span>&lt;span class="n">read_csv&lt;/span>&lt;span class="p">(&lt;/span>&lt;span class="s2">&amp;#34;data.csv&amp;#34;&lt;/span>&lt;span class="p">)&lt;/span>
&lt;/span>&lt;/span>&lt;span class="line">&lt;span class="cl">&lt;span class="n">data&lt;/span>&lt;span class="o">.&lt;/span>&lt;span class="n">head&lt;/span>&lt;span class="p">()&lt;/span>
&lt;/span>&lt;/span>&lt;/code>&lt;/pre>&lt;/div>&lt;h2 id="inline-images">Inline Images&lt;/h2>
&lt;div class="highlight">&lt;pre tabindex="0" class="chroma">&lt;code class="language-go" data-lang="go">&lt;span class="line">&lt;span class="cl">&lt;span class="p">{{&amp;lt;&lt;/span> &lt;span class="nx">icon&lt;/span> &lt;span class="nx">name&lt;/span>&lt;span class="p">=&lt;/span>&lt;span class="s">&amp;#34;python&amp;#34;&lt;/span> &lt;span class="p">&amp;gt;}}&lt;/span> &lt;span class="nx">Python&lt;/span>
&lt;/span>&lt;/span>&lt;/code>&lt;/pre>&lt;/div>&lt;p>renders as&lt;/p>
&lt;p>
&lt;span class="inline-block pr-1">
&lt;svg style="height: 1em; transform: translateY(0.1em);" xmlns="http://www.w3.org/2000/svg" height="1em" viewBox="0 0 448 512" fill="currentColor">&lt;path d="M439.8 200.5c-7.7-30.9-22.3-54.2-53.4-54.2h-40.1v47.4c0 36.8-31.2 67.8-66.8 67.8H172.7c-29.2 0-53.4 25-53.4 54.3v101.8c0 29 25.2 46 53.4 54.3 33.8 9.9 66.3 11.7 106.8 0 26.9-7.8 53.4-23.5 53.4-54.3v-40.7H226.2v-13.6h160.2c31.1 0 42.6-21.7 53.4-54.2 11.2-33.5 10.7-65.7 0-108.6zM286.2 404c11.1 0 20.1 9.1 20.1 20.3 0 11.3-9 20.4-20.1 20.4-11 0-20.1-9.2-20.1-20.4.1-11.3 9.1-20.3 20.1-20.3zM167.8 248.1h106.8c29.7 0 53.4-24.5 53.4-54.3V91.9c0-29-24.4-50.7-53.4-55.6-35.8-5.9-74.7-5.6-106.8.1-45.2 8-53.4 24.7-53.4 55.6v40.7h106.9v13.6h-147c-31.1 0-58.3 18.7-66.8 54.2-9.8 40.7-10.2 66.1 0 108.6 7.6 31.6 25.7 54.2 56.8 54.2H101v-48.8c0-35.3 30.5-66.4 66.8-66.4zm-6.7-142.6c-11.1 0-20.1-9.1-20.1-20.3.1-11.3 9-20.4 20.1-20.4 11 0 20.1 9.2 20.1 20.4s-9 20.3-20.1 20.3z"/>&lt;/svg>
&lt;/span> Python&lt;/p>
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