diff options
Diffstat (limited to 'pages/notations.md')
-rw-r--r-- | pages/notations.md | 53 |
1 files changed, 0 insertions, 53 deletions
diff --git a/pages/notations.md b/pages/notations.md deleted file mode 100644 index 2980a08..0000000 --- a/pages/notations.md +++ /dev/null @@ -1,53 +0,0 @@ ---- -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 $w \in [n]^\ell$, - $\text{ty} w = (m_1, m_2, ..., m_n)$ 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. $(- \alpha)_+$. |