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# Copyright (C) 2013-2021 Yuchen Pei.
# Permission is granted to copy, distribute and/or modify this
# document under the terms of the GNU Free Documentation License,
# Version 1.3 or any later version published by the Free Software
# Foundation; with no Invariant Sections, no Front-Cover Texts, and
# no Back-Cover Texts. You should have received a copy of the GNU
# Free Documentation License. If not, see <https://www.gnu.org/licenses/>.
# This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA.
#+title: List of Notations
#+date: 2019-03-15
Here I list meanings of notations that may have not been explained
elsewhere.
- \(\text{ty}\): type. Given a word \(w \in [n]^\ell\),
\(\text{ty} w = (m_1, m_2, ..., m_n)\) where \(m_i\) is the number of
\(i\)'s in \(w\). For example
\(\text{ty} (1, 2, 2, 1, 4, 2) = (2, 3, 0, 1)\). The definition of
\(\text{ty} T\) for a tableau \(T\) is similar.
- \([n]\): for \(n \in \mathbb N_{>0}\), \([n]\) stands for the set
\(\{1, 2, ..., n\}\).
- \(i : j\): for \(i, j \in \mathbb Z\), \(i : j\) stands for the set
\(\{i, i + 1, ..., j\}\), or the sequence \((i, i + 1, ..., j)\),
depending on the context.
- \(k = i : j\): means \(k\) iterates over \(i\), \(i + 1\),..., \(j\).
For example \(\sum_{k = 1 : n} a_k := \sum_{k = 1}^n a_k\).
- \(x_{i : j}\): stands for the set \(\{x_k: k = i : j\}\) or the
sequence \((x_i, x_{i + 1}, ..., x_j)\), depending on the context. So
are notations like \(f(i : j)\), \(y^{i : j}\) etc.
- \(\mathbb N\): the set of natural numbers / nonnegative integer
numbers \(\{0, 1, 2,...\}\), whereas
- \(\mathbb N_{>0}\) or \(\mathbb N^+\): Are the set of positive integer
numbers.
- \(x^w\): when both \(x\) and \(w\) are tuples of objects, this means
\(\prod_i x_{w_i}\). For example say \(w = (1, 2, 2, 1, 4, 2)\), and
\(x = x_{1 : 7}\), then \(x^w = x_1^2 x_2^3 x_4\).
- \(LHS\), LHS, \(RHS\), RHS: left hand side and right hand side of a
formula
- \(e_i\): the \(i\)th standard basis in a vector space:
\(e_i = (0, 0, ..., 0, 1, 0, 0, ...)\) where the sequence is finite or
infinite depending on the dimension of the vector space and the \(1\)
is the \(i\)th entry and all other entries are \(0\).
- \(1_{A}(x)\) where \(A\) is a set: an indicator function, which
evaluates to \(1\) if \(x \in A\), and \(0\) otherwise.
- \(1_{p}\): an indicator function, which evaluates to \(1\) if the
predicate \(p\) is true and \(0\) otherwise. Example: \(1_{x \in A}\),
same as \(1_A(x)\).
- \(\xi \sim p\): the random variable \(xi\) is distributed according to
the probability density function / probability mass function /
probability measure \(p\).
- \(\xi \overset{d}{=} \eta\): the random variables \(\xi\) and \(\eta\)
have the same distribution.
- \(\mathbb E f(\xi)\): expectation of \(f(\xi)\).
- \(\mathbb P(A)\): probability of event \(A\).
- \(a \wedge b\): \(\min\{a, b\}\).
- \(a \vee b\): \(\max\{a, b\}\).
- \((\alpha)_+\): the positive part of \(\alpha\),
i.e. \(\alpha \vee 0\).
- \((\alpha)_-\): the negative part of \(\alpha\),
i.e. \((- \alpha)_+\).
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