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| author | Yuchen Pei <me@ypei.me> | 2021-06-18 12:58:44 +1000 |
|---|---|---|
| committer | Yuchen Pei <me@ypei.me> | 2021-06-18 12:58:44 +1000 |
| commit | 147a19e84a743f1379f05bf2f444143b4afd7bd6 (patch) | |
| tree | 3127395250cb958f06a98b86f73e77658150b43c /site-from-md/notations.html | |
| parent | 4fa26fec8b7e978955e5630d3f820ba9c53be72c (diff) | |
Updated.
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| -rw-r--r-- | site-from-md/notations.html | 67 |
1 files changed, 67 insertions, 0 deletions
diff --git a/site-from-md/notations.html b/site-from-md/notations.html new file mode 100644 index 0000000..753cff7 --- /dev/null +++ b/site-from-md/notations.html @@ -0,0 +1,67 @@ +<!doctype html> +<html lang="en"> + <head> + <meta charset="utf-8"> + <title>List of Notations</title> + <link rel="stylesheet" href="../assets/css/default.css" /> + <script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script> + <script src="../assets/js/analytics.js" type="text/javascript"></script> + </head> + <body> + <header> + <span class="logo"> + <a href="index.html">Yuchen Pei</a> + </span> + <nav> + <a href="blog.html">Blog</a><a href="microblog.html">Microblog</a><a href="links.html">Links</a> + </nav> + </header> + + <div class="main"> + <div class="bodyitem"> + <!DOCTYPE html> +<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang=""> +<head> + <meta charset="utf-8" /> + <meta name="generator" content="pandoc" /> + <meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" /> + <title>Untitled</title> + <style> + code{white-space: pre-wrap;} + span.smallcaps{font-variant: small-caps;} + span.underline{text-decoration: underline;} + div.column{display: inline-block; vertical-align: top; width: 50%;} + </style> + <script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/MathJax.js?config=TeX-AMS_CHTML-full" type="text/javascript"></script> + <!--[if lt IE 9]> + <script src="//cdnjs.cloudflare.com/ajax/libs/html5shiv/3.7.3/html5shiv-printshiv.min.js"></script> + <![endif]--> +</head> +<body> +<p>Here I list meanings of notations that may have not been explained elsewhere.</p> +<ul> +<li><span class="math inline">\(\text{ty}\)</span>: type. Given a word <span class="math inline">\(w \in [n]^\ell\)</span>, <span class="math inline">\(\text{ty} w = (m_1, m_2, ..., m_n)\)</span> where <span class="math inline">\(m_i\)</span> is the number of <span class="math inline">\(i\)</span>'s in <span class="math inline">\(w\)</span>. For example <span class="math inline">\(\text{ty} (1, 2, 2, 1, 4, 2) = (2, 3, 0, 1)\)</span>. The definition of <span class="math inline">\(\text{ty} T\)</span> for a tableau <span class="math inline">\(T\)</span> is similar.</li> +<li><span class="math inline">\([n]\)</span>: for <span class="math inline">\(n \in \mathbb N_{>0}\)</span>, <span class="math inline">\([n]\)</span> stands for the set <span class="math inline">\(\{1, 2, ..., n\}\)</span>.</li> +<li><span class="math inline">\(i : j\)</span>: for <span class="math inline">\(i, j \in \mathbb Z\)</span>, <span class="math inline">\(i : j\)</span> stands for the set <span class="math inline">\(\{i, i + 1, ..., j\}\)</span>, or the sequence <span class="math inline">\((i, i + 1, ..., j)\)</span>, depending on the context.</li> +<li><span class="math inline">\(k = i : j\)</span>: means <span class="math inline">\(k\)</span> iterates over <span class="math inline">\(i\)</span>, <span class="math inline">\(i + 1\)</span>,..., <span class="math inline">\(j\)</span>. For example <span class="math inline">\(\sum_{k = 1 : n} a_k := \sum_{k = 1}^n a_k\)</span>.</li> +<li><span class="math inline">\(x_{i : j}\)</span>: stands for the set <span class="math inline">\(\{x_k: k = i : j\}\)</span> or the sequence <span class="math inline">\((x_i, x_{i + 1}, ..., x_j)\)</span>, depending on the context. So are notations like <span class="math inline">\(f(i : j)\)</span>, <span class="math inline">\(y^{i : j}\)</span> etc.</li> +<li><span class="math inline">\(\mathbb N\)</span>: the set of natural numbers / nonnegative integer numbers <span class="math inline">\(\{0, 1, 2,...\}\)</span>, whereas</li> +<li><span class="math inline">\(\mathbb N_{>0}\)</span> or <span class="math inline">\(\mathbb N^+\)</span>: Are the set of positive integer numbers.</li> +<li><span class="math inline">\(x^w\)</span>: when both <span class="math inline">\(x\)</span> and <span class="math inline">\(w\)</span> are tuples of objects, this means <span class="math inline">\(\prod_i x_{w_i}\)</span>. For example say <span class="math inline">\(w = (1, 2, 2, 1, 4, 2)\)</span>, and <span class="math inline">\(x = x_{1 : 7}\)</span>, then <span class="math inline">\(x^w = x_1^2 x_2^3 x_4\)</span>.</li> +<li><span class="math inline">\(LHS\)</span>, LHS, <span class="math inline">\(RHS\)</span>, RHS: left hand side and right hand side of a formula</li> +<li><span class="math inline">\(e_i\)</span>: the <span class="math inline">\(i\)</span>th standard basis in a vector space: <span class="math inline">\(e_i = (0, 0, ..., 0, 1, 0, 0, ...)\)</span> where the sequence is finite or infinite depending on the dimension of the vector space and the <span class="math inline">\(1\)</span> is the <span class="math inline">\(i\)</span>th entry and all other entries are <span class="math inline">\(0\)</span>.</li> +<li><span class="math inline">\(1_{A}(x)\)</span> where <span class="math inline">\(A\)</span> is a set: an indicator function, which evaluates to <span class="math inline">\(1\)</span> if <span class="math inline">\(x \in A\)</span>, and <span class="math inline">\(0\)</span> otherwise.</li> +<li><span class="math inline">\(1_{p}\)</span>: an indicator function, which evaluates to <span class="math inline">\(1\)</span> if the predicate <span class="math inline">\(p\)</span> is true and <span class="math inline">\(0\)</span> otherwise. Example: <span class="math inline">\(1_{x \in A}\)</span>, same as <span class="math inline">\(1_A(x)\)</span>.</li> +<li><span class="math inline">\(\xi \sim p\)</span>: the random variable <span class="math inline">\(xi\)</span> is distributed according to the probability density function / probability mass function / probability measure <span class="math inline">\(p\)</span>.</li> +<li><span class="math inline">\(\xi \overset{d}{=} \eta\)</span>: the random variables <span class="math inline">\(\xi\)</span> and <span class="math inline">\(\eta\)</span> have the same distribution.</li> +<li><span class="math inline">\(\mathbb E f(\xi)\)</span>: expectation of <span class="math inline">\(f(\xi)\)</span>.</li> +<li><span class="math inline">\(\mathbb P(A)\)</span>: probability of event <span class="math inline">\(A\)</span>.</li> +</ul> +</body> +</html> + + </div> + </div> + + </body> +</html> |