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diff --git a/pages/notations.md b/pages/notations.md new file mode 100644 index 0000000..2dac4fd --- /dev/null +++ b/pages/notations.md @@ -0,0 +1,48 @@ +--- +date: 2019-03-15 +title: List of Notations +template: default +--- + +Here I list meanings of notations that may have not been explained +elsewhere. + +- $\text{ty}$: type. Given a word $w \in [n]^\ell$, + $\ty w = (m_1, m_2, ..., m_n)$ where $m_i$ is the number of $i$\'s + in $w$. For example $\ty (1, 2, 2, 1, 4, 2) = (2, 3, 0, 1)$. The + definition of $\ty T$ for a tableau $T$ is similar. +- $[n]$: for $n \in \mathbb N_{>0}$, $[n]$ stands for the set + $\{1, 2, ..., n\}$. +- $i : j$: for $i, j \in \intg$, $i : j$ 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 $A$. |