doc/implementation/kmer/index.html


<!doctype html>
<html lang="en" class="no-js">
  <head>
    
      <meta charset="utf-8">
      <meta name="viewport" content="width=device-width,initial-scale=1">
      
      
        <link rel="prev" href="../superkmer/">
      
      
        <link rel="next" href="../chunkreader/">
      
      
      <link rel="icon" href="../../assets/images/favicon.png">
      <meta name="generator" content="mkdocs-1.6.1, mkdocs-material-9.7.6">
    
    
        <title>Kmer - obikmer</title>
      
    
      <link rel="stylesheet" href="../../assets/stylesheets/main.484c7ddc.min.css">
      
      
        <link rel="preconnect" href="https://fonts.gstatic.com" crossorigin>
        <link rel="stylesheet" href="https://fonts.googleapis.com/css?family=Roboto:300,300i,400,400i,700,700i%7CRoboto+Mono:400,400i,700,700i&display=fallback">
        <style>:root{--md-text-font:"Roboto";--md-code-font:"Roboto Mono"}</style>
      
    
    <script>__md_scope=new URL("../..",location),__md_hash=e=>[...e].reduce(((e,_)=>(e<<5)-e+_.charCodeAt(0)),0),__md_get=(e,_=localStorage,t=__md_scope)=>JSON.parse(_.getItem(t.pathname+"."+e)),__md_set=(e,_,t=localStorage,a=__md_scope)=>{try{t.setItem(a.pathname+"."+e,JSON.stringify(_))}catch(e){}}</script>
    
      
  </head>
  
  
    <body dir="ltr">
  
    
    <input class="md-toggle" data-md-toggle="drawer" type="checkbox" id="__drawer" autocomplete="off">
    <input class="md-toggle" data-md-toggle="search" type="checkbox" id="__search" autocomplete="off">
    <label class="md-overlay" for="__drawer"></label>
    <div data-md-component="skip">
      
        
        <a href="#kmer-implementation" class="md-skip">
          Skip to content
        </a>
      
    </div>
    <div data-md-component="announce">
      
    </div>
    
    
<header class="md-header md-header--shadow" data-md-component="header">
  <nav class="md-header__inner md-grid" aria-label="Header">
    <a href="../.." title="obikmer" class="md-header__button md-logo" aria-label="obikmer" data-md-component="logo">
      
  
  <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 8a3 3 0 0 0 3-3 3 3 0 0 0-3-3 3 3 0 0 0-3 3 3 3 0 0 0 3 3m0 3.54C9.64 9.35 6.5 8 3 8v11c3.5 0 6.64 1.35 9 3.54 2.36-2.19 5.5-3.54 9-3.54V8c-3.5 0-6.64 1.35-9 3.54"/></svg>

    </a>
    <label class="md-header__button md-icon" for="__drawer">
      
      <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M3 6h18v2H3zm0 5h18v2H3zm0 5h18v2H3z"/></svg>
    </label>
    <div class="md-header__title" data-md-component="header-title">
      <div class="md-header__ellipsis">
        <div class="md-header__topic">
          <span class="md-ellipsis">
            obikmer
          </span>
        </div>
        <div class="md-header__topic" data-md-component="header-topic">
          <span class="md-ellipsis">
            
              Kmer
            
          </span>
        </div>
      </div>
    </div>
    
    
      <script>var palette=__md_get("__palette");if(palette&&palette.color){if("(prefers-color-scheme)"===palette.color.media){var media=matchMedia("(prefers-color-scheme: light)"),input=document.querySelector(media.matches?"[data-md-color-media='(prefers-color-scheme: light)']":"[data-md-color-media='(prefers-color-scheme: dark)']");palette.color.media=input.getAttribute("data-md-color-media"),palette.color.scheme=input.getAttribute("data-md-color-scheme"),palette.color.primary=input.getAttribute("data-md-color-primary"),palette.color.accent=input.getAttribute("data-md-color-accent")}for(var[key,value]of Object.entries(palette.color))document.body.setAttribute("data-md-color-"+key,value)}</script>
    
    
  </nav>
  
</header>
    
    <div class="md-container" data-md-component="container">
      
      
      <main class="md-main" data-md-component="main">
        <div class="md-main__inner md-grid">
          
            
              <div class="md-sidebar md-sidebar--primary" data-md-component="sidebar" data-md-type="navigation" >
                <div class="md-sidebar__scrollwrap">
                  <div class="md-sidebar__inner">
                    

<nav class="md-nav md-nav--primary" aria-label="Navigation" data-md-level="0">
  <label class="md-nav__title" for="__drawer">
    <a href="../.." title="obikmer" class="md-nav__button md-logo" aria-label="obikmer" data-md-component="logo">
      
  
  <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 24 24"><path d="M12 8a3 3 0 0 0 3-3 3 3 0 0 0-3-3 3 3 0 0 0-3 3 3 3 0 0 0 3 3m0 3.54C9.64 9.35 6.5 8 3 8v11c3.5 0 6.64 1.35 9 3.54 2.36-2.19 5.5-3.54 9-3.54V8c-3.5 0-6.64 1.35-9 3.54"/></svg>

    </a>
    obikmer
  </label>
  
  <ul class="md-nav__list" data-md-scrollfix>
    
      
    <li class="md-nav__item">
      <a href="../.." class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    Home
  

  </span>
  
  
      </a>
    </li>
  

    <li class="md-nav__item md-nav__item--nested">
      
        
        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_2" >
        
          
          <label class="md-nav__link" for="__nav_2" id="__nav_2_label" tabindex="0">
            
  
  <span class="md-ellipsis">
    
  
    Theory
  

  </span>
  
  
            <span class="md-nav__icon md-icon"></span>
          </label>
        
        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_2_label" aria-expanded="false">
          <label class="md-nav__title" for="__nav_2">
            <span class="md-nav__icon md-icon"></span>
            
  
    Theory
  

          </label>
          <ul class="md-nav__list" data-md-scrollfix>
            
              
    <li class="md-nav__item">
      <a href="../../kmers/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    Kmers and super-kmers
  

  </span>
  
  
      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="../../theory/encoding/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    DNA encoding
  

  </span>
  
  
      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="../../theory/entropy/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    Entropy filter
  

  </span>
  
  
      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="../../theory/minimizer/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    Minimizer selection
  

  </span>
  
  
      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="../../theory/indexing/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    Partitioning architecture
  

  </span>
  
  
      </a>
    </li>
  

          </ul>
        </nav>
      
    </li>
  

    <li class="md-nav__item md-nav__item--active md-nav__item--nested">
      
        
        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_3" checked>
        
          
          <label class="md-nav__link" for="__nav_3" id="__nav_3_label" tabindex="0">
            
  
  <span class="md-ellipsis">
    
  
    Implementation
  

  </span>
  
  
            <span class="md-nav__icon md-icon"></span>
          </label>
        
        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_3_label" aria-expanded="true">
          <label class="md-nav__title" for="__nav_3">
            <span class="md-nav__icon md-icon"></span>
            
  
    Implementation
  

          </label>
          <ul class="md-nav__list" data-md-scrollfix>
            
              
    <li class="md-nav__item">
      <a href="../superkmer/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    SuperKmer
  

  </span>
  
  
      </a>
    </li>
  

    <li class="md-nav__item md-nav__item--active">
      
      <input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
      
      
        <label class="md-nav__link md-nav__link--active" for="__toc">
          
  
  <span class="md-ellipsis">
    
  
    Kmer
  

  </span>
  
  
          <span class="md-nav__icon md-icon"></span>
        </label>
      
      <a href="./" class="md-nav__link md-nav__link--active">
        
  
  <span class="md-ellipsis">
    
  
    Kmer
  

  </span>
  
  
      </a>
      
        
<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
  
  
    <label class="md-nav__title" for="__toc">
      <span class="md-nav__icon md-icon"></span>
      Table of contents
    </label>
    <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
      
        <li class="md-nav__item">
  <a href="#memory-layout" class="md-nav__link">
    <span class="md-ellipsis">
      
        Memory layout
      
    </span>
  </a>
  
</li>
      
        <li class="md-nav__item">
  <a href="#encoding" class="md-nav__link">
    <span class="md-ellipsis">
      
        Encoding
      
    </span>
  </a>
  
</li>
      
        <li class="md-nav__item">
  <a href="#decoding" class="md-nav__link">
    <span class="md-ellipsis">
      
        Decoding
      
    </span>
  </a>
  
</li>
      
        <li class="md-nav__item">
  <a href="#reverse-complement" class="md-nav__link">
    <span class="md-ellipsis">
      
        Reverse complement
      
    </span>
  </a>
  
</li>
      
        <li class="md-nav__item">
  <a href="#canonical-form" class="md-nav__link">
    <span class="md-ellipsis">
      
        Canonical form
      
    </span>
  </a>
  
</li>
      
    </ul>
  
</nav>
      
    </li>
  

    <li class="md-nav__item">
      <a href="../chunkreader/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    Chunk reader
  

  </span>
  
  
      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="../pipeline/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    Construction pipeline
  

  </span>
  
  
      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="../obipipeline/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    obipipeline library
  

  </span>
  
  
      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="../storage/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    On-disk storage
  

  </span>
  
  
      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="../mphf/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    MPHF selection
  

  </span>
  
  
      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="../unitig_evidence/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    Unitig evidence encoding
  

  </span>
  
  
      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="../obilayeredmap/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    obilayeredmap crate
  

  </span>
  
  
      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="../persistent_compact_int_vec/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    PersistentCompactIntVec
  

  </span>
  
  
      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="../persistent_bit_vec/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    PersistentBitVec
  

  </span>
  
  
      </a>
    </li>
  

          </ul>
        </nav>
      
    </li>
  

    <li class="md-nav__item md-nav__item--nested">
      
        
        <input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_4" >
        
          
          <label class="md-nav__link" for="__nav_4" id="__nav_4_label" tabindex="0">
            
  
  <span class="md-ellipsis">
    
  
    Architecture
  

  </span>
  
  
            <span class="md-nav__icon md-icon"></span>
          </label>
        
        <nav class="md-nav" data-md-level="1" aria-labelledby="__nav_4_label" aria-expanded="false">
          <label class="md-nav__title" for="__nav_4">
            <span class="md-nav__icon md-icon"></span>
            
  
    Architecture
  

          </label>
          <ul class="md-nav__list" data-md-scrollfix>
            
              
    <li class="md-nav__item">
      <a href="../../architecture/sequences/invariant/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    Sequences
  

  </span>
  
  
      </a>
    </li>
  

    <li class="md-nav__item">
      <a href="../../architecture/index_architecture/" class="md-nav__link">
        
  
  <span class="md-ellipsis">
    
  
    Kmer index
  

  </span>
  
  
      </a>
    </li>
  

          </ul>
        </nav>
      
    </li>
  

  </ul>
</nav>
                  </div>
                </div>
              </div>
            
            
              <div class="md-sidebar md-sidebar--secondary" data-md-component="sidebar" data-md-type="toc" >
                <div class="md-sidebar__scrollwrap">
                  <div class="md-sidebar__inner">
                    

<nav class="md-nav md-nav--secondary" aria-label="Table of contents">
  
  
    <label class="md-nav__title" for="__toc">
      <span class="md-nav__icon md-icon"></span>
      Table of contents
    </label>
    <ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
      
        <li class="md-nav__item">
  <a href="#memory-layout" class="md-nav__link">
    <span class="md-ellipsis">
      
        Memory layout
      
    </span>
  </a>
  
</li>
      
        <li class="md-nav__item">
  <a href="#encoding" class="md-nav__link">
    <span class="md-ellipsis">
      
        Encoding
      
    </span>
  </a>
  
</li>
      
        <li class="md-nav__item">
  <a href="#decoding" class="md-nav__link">
    <span class="md-ellipsis">
      
        Decoding
      
    </span>
  </a>
  
</li>
      
        <li class="md-nav__item">
  <a href="#reverse-complement" class="md-nav__link">
    <span class="md-ellipsis">
      
        Reverse complement
      
    </span>
  </a>
  
</li>
      
        <li class="md-nav__item">
  <a href="#canonical-form" class="md-nav__link">
    <span class="md-ellipsis">
      
        Canonical form
      
    </span>
  </a>
  
</li>
      
    </ul>
  
</nav>
                  </div>
                </div>
              </div>
            
          
            <div class="md-content" data-md-component="content">
              
              <article class="md-content__inner md-typeset">
                
                  
<h1 id="kmer-implementation">Kmer — implementation</h1>
<h2 id="memory-layout">Memory layout</h2>
<p><code>Kmer</code> is a <code>#[repr(transparent)]</code> newtype over <code>u64</code>:</p>
<div class="highlight"><pre><span></span><code><span class="cp">#[repr(transparent)]</span>
<span class="k">pub</span><span class="w"> </span><span class="k">struct</span><span class="w"> </span><span class="nc">Kmer</span><span class="p">(</span><span class="kt">u64</span><span class="p">);</span>
</code></pre></div>
<p>Nucleotides are packed 2 bits each, <strong>left-aligned</strong>, MSB-first. Nucleotide 0 occupies bits 63–62; nucleotide i occupies bits 63−2i and 62−2i. The low 64−2k bits are always zero. k is <strong>not stored</strong> — it is a parameter of every operation that needs it, and will be owned by the future collection-level indexer.</p>
<table>
<thead>
<tr>
<th>63–62</th>
<th>61–60</th>
<th>…</th>
<th>63−2(k−1)−1 to 63−2(k−1)</th>
<th>63−2k down to 0</th>
</tr>
</thead>
<tbody>
<tr>
<td>nt 0</td>
<td>nt 1</td>
<td>…</td>
<td>nt k−1</td>
<td>zero padding</td>
</tr>
</tbody>
</table>
<h2 id="encoding">Encoding</h2>
<p><code>Kmer::from_ascii(ascii, k)</code> encodes the first k bytes of an ASCII slice using the shared <code>ENC</code> table (see <a href="../superkmer/#ascii-encoding-and-decoding">SuperKmer — ASCII encoding</a>):</p>
<div class="highlight"><pre><span></span><code><span class="k">for</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="k">in</span><span class="w"> </span><span class="mi">0</span><span class="o">..</span><span class="n">k</span><span class="w"> </span><span class="p">{</span>
<span class="w">    </span><span class="n">val</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="n">encode_base</span><span class="p">(</span><span class="n">ascii</span><span class="p">[</span><span class="n">i</span><span class="p">])</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="kt">u64</span><span class="p">;</span>
<span class="p">}</span>
<span class="n">Kmer</span><span class="p">(</span><span class="n">val</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="p">(</span><span class="mi">64</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">k</span><span class="p">))</span>
</code></pre></div>
<p>Zero allocation — result lives on the stack.</p>
<h2 id="decoding">Decoding</h2>
<p><code>write_ascii(k, buf)</code> appends k ASCII characters to a caller-supplied <code>Vec&lt;u8&gt;</code> using the shared <code>DEC4</code> table: one lookup per 4 nucleotides, two partial-byte lookups for the remainder. No allocation in the hot path.</p>
<p><code>to_ascii(k)</code> is a convenience wrapper that allocates and returns a <code>Vec&lt;u8&gt;</code>; intended for tests and display only.</p>
<h2 id="reverse-complement">Reverse complement</h2>
<p>Computed as pure arithmetic — no lookup table, no memory access:</p>
<div class="highlight"><pre><span></span><code><span class="kd">let</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="o">!</span><span class="bp">self</span><span class="p">.</span><span class="mi">0</span><span class="p">;</span><span class="w">                                                               </span><span class="c1">// complement</span>
<span class="kd">let</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">x</span><span class="p">.</span><span class="n">swap_bytes</span><span class="p">();</span><span class="w">                                                        </span><span class="c1">// reverse bytes</span>
<span class="kd">let</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">((</span><span class="n">x</span><span class="w"> </span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="mi">4</span><span class="p">)</span><span class="w"> </span><span class="o">&amp;</span><span class="w"> </span><span class="mh">0x0F0F0F0F0F0F0F0F</span><span class="p">)</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="p">((</span><span class="n">x</span><span class="w"> </span><span class="o">&amp;</span><span class="w"> </span><span class="mh">0x0F0F0F0F0F0F0F0F</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">4</span><span class="p">);</span><span class="w"> </span><span class="c1">// swap nibbles</span>
<span class="kd">let</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">((</span><span class="n">x</span><span class="w"> </span><span class="o">&gt;&gt;</span><span class="w"> </span><span class="mi">2</span><span class="p">)</span><span class="w"> </span><span class="o">&amp;</span><span class="w"> </span><span class="mh">0x3333333333333333</span><span class="p">)</span><span class="w"> </span><span class="o">|</span><span class="w"> </span><span class="p">((</span><span class="n">x</span><span class="w"> </span><span class="o">&amp;</span><span class="w"> </span><span class="mh">0x3333333333333333</span><span class="p">)</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="mi">2</span><span class="p">);</span><span class="w">   </span><span class="c1">// swap 2-bit groups</span>
<span class="n">Kmer</span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="o">&lt;&lt;</span><span class="w"> </span><span class="p">(</span><span class="mi">64</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">k</span><span class="p">))</span>
</code></pre></div>
<p>After complementing, bytes are reversed (<code>swap_bytes</code>), then nibbles, then 2-bit groups — restoring 2-bit nucleotides to their correct positions in reverse order. A final left-shift realigns to MSB. Zero allocation — result lives on the stack.</p>
<h2 id="canonical-form">Canonical form</h2>
<div class="highlight"><pre><span></span><code><span class="k">pub</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="nf">canonical</span><span class="p">(</span><span class="o">&amp;</span><span class="bp">self</span><span class="p">,</span><span class="w"> </span><span class="n">k</span><span class="p">:</span><span class="w"> </span><span class="kt">usize</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="nc">Self</span><span class="w"> </span><span class="p">{</span>
<span class="w">    </span><span class="kd">let</span><span class="w"> </span><span class="n">rc</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="n">revcomp</span><span class="p">(</span><span class="n">k</span><span class="p">);</span>
<span class="w">    </span><span class="k">if</span><span class="w"> </span><span class="bp">self</span><span class="p">.</span><span class="mi">0</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">rc</span><span class="p">.</span><span class="mi">0</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="o">*</span><span class="bp">self</span><span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span><span class="w"> </span><span class="n">rc</span><span class="w"> </span><span class="p">}</span>
<span class="p">}</span>
</code></pre></div>
<p>Lexicographic minimum of forward and reverse-complement, comparing the raw <code>u64</code> values directly (left-aligned encoding makes this equivalent to nucleotide-wise comparison). Zero allocation — result lives on the stack.</p>


              </article>
            </div>
          
          
<script>var target=document.getElementById(location.hash.slice(1));target&&target.name&&(target.checked=target.name.startsWith("__tabbed_"))</script>
        </div>
        
      </main>
      
        <footer class="md-footer">
  
  <div class="md-footer-meta md-typeset">
    <div class="md-footer-meta__inner md-grid">
      <div class="md-copyright">
  
  
    Made with
    <a href="https://squidfunk.github.io/mkdocs-material/" target="_blank" rel="noopener">
      Material for MkDocs
    </a>
  
</div>
      
    </div>
  </div>
</footer>
      
    </div>
    <div class="md-dialog" data-md-component="dialog">
      <div class="md-dialog__inner md-typeset"></div>
    </div>
    
    
      <script id="__config" type="application/json">{"annotate": null, "base": "../..", "features": [], "search": "../../assets/javascripts/workers/search.2c215733.min.js", "tags": null, "translations": {"clipboard.copied": "Copied to clipboard", "clipboard.copy": "Copy to clipboard", "search.result.more.one": "1 more on this page", "search.result.more.other": "# more on this page", "search.result.none": "No matching documents", "search.result.one": "1 matching document", "search.result.other": "# matching documents", "search.result.placeholder": "Type to start searching", "search.result.term.missing": "Missing", "select.version": "Select version"}, "version": null}</script>
    
    
      <script src="../../assets/javascripts/bundle.79ae519e.min.js"></script>
      
        <script src="https://unpkg.com/mathjax@3/es5/tex-mml-chtml.js"></script>
      
    
  </body>
</html>