author: niplav, created: 2021-03-23, modified: 2023-03-24, language: english, status: notes, importance: 1, confidence: draft
People write mathematics in a specific way, and use different conventions. Here I collect mine, mainly for myself.
Zum Schein nämlich steht das Ausdruckslose, wiewohl im Gegensatz, doch in derart notwendigem Verhältnis, daß eben das Schöne, ob auch selber nicht Schein, aufhört ein wesentlich Schönes zu sein, wenn der Schein von ihm schwindet.
— Walter Benjamin, “Goethes Wahlverwandschaften”, 1925
$:=$ for assignment, not $\rightarrow$ or $\leftarrow$$\{x|x \in P(x) \land \forall y \in S: x \succ y\}$ (some people use a colon)$\bigcup/\bigvee/\sum/\prod$ etc, use $i$ and $j$ as the variables. If there are more, use $i_1, i_2, \dots$.$\text{abcd}$, $abcd$ is the product of four variables $a, b, c$ and $d$.$0 \in \mathbb{N}$, $\mathbb{N}^+:=\mathbb{N}\backslash \{0\}$.
$0 \in \mathbb{N}$. I know it's kind of dumb, but standards are nice & there for a reason.$(\mathbb{N}, +)$ and $(\mathbb{N}^+, \cdot)$ are both commutative monoids.$(\mathbb{N}, +, \cdot)$ is a semiring.$(\mathbb{N}^+, +, \cdot)$ is…nothing in particular?
$\cdot$ distributes over $+$$+$, so it can't be a rng or a semiring$+$ and $\cdot$$\cdot$ has an identity, so it can't be a near-ring (but we also can't make $(\mathbb{N}^+, \cdot, +)$ a rng or semiring because $+$ doesn't distribute over $\cdot$)$(\mathbb{N}, +, \cdot)$ is the nicer structure, $0 \in \mathbb{N}$.$()$, sets are denoted using $\{\}$, and $[]$ is sometimes used in the context of statistics (variance of a variable, mean of a variable, and so on). These are not mixed.$[n] = \{x \in \mathbb{N} | 1 \le x \le n \}$ for $n \in \mathbb{N}$.$\mapsto$ instead of $\rightarrow$, e.g. $f: ℝ^n \mapsto ℝ$ instead of $f: ℝ^n \rightarrow ℝ$$a \cdot b \cdot c$, and sometimes concatenation $abcd$. Rarely asterisks $a * b * c * d$, but I try to avoid them.$f(x)=x^p$ is "quadratic" iff $p=2$, and "radical" iff $p \in (0,1)$$A \subset B$ if definitely $A \not =B$, otherwise write $A \subseteq B$.$\backslash$, not with a minus $-$.$e \not \in S$ (and not something like $e \in' S$).$A$ is $|A|$, not $\#(A)$$\Rightarrow$ or $\Leftarrow$ for implication and $\Leftrightarrow$ for the biconditional instead of $\leftarrow$ or $\rightarrow$ and $\leftrightarrow$.$X$ as $\mathbb{E}[X]$ and the variance of $X$ as $\mathbb{V}[X]$. Unfortunately, $\mathbb{C}$ is already taken for the complex numbers, so I am forced to write $\text{cov}[X,Y]$ for the covariance, and $\text{cor}[X,Y]$ for the correlation.(Or social choice theory/decision theory/utility theory…)
$P, R, I$ instead of $\prec, \preceq, \sim$. I often don't.| (Optimizing one's own value) | Maximum | Minimum |
| maximize | maximax | maximin |
| minimize | minimax | minimin |
| (Optimizing the other player's value) | Maximum | Minimum |
| maximize | maxmaxi | maxmini |
| minimize | minmaxi | minmini |
(This is not the terminology I will use, but I would if I were brave enough)
$\mathbb{M_{1}}, \mathbb{M_{2}}, \mathbb{M_{3}}$.$p$-norm.$\newcommand{\bigplus}{\mathop{\Large+\normalsize}} \bigplus_{i \in I}V_i$ and $\newcommand{\bigdot}{\mathop{\Large•\normalsize}} \bigdot_{i \in I}V_i$ or $\newcommand{\bigmult}{\mathop{\Large*\normalsize}} \bigmult_{i \in I}V_i$ instead of $\sum_{i \in I} V_i$ and $\prod_{i \in I} V_i$.$\mathbb{Z}$, and then specify when one wants only the positive numbers ($\mathbb{Z}^+$), the positive numbers with 0 ($\mathbb{Z}^+_0$), the negative numbers ($\mathbb{Z}^-$) and the negative numbers with 0 ($\mathbb{Z}^-_0$). That would be much nicer than using $\mathbb{N}$, since $\mathbb{Z}$ is a commutative ring under addition and multiplication.$-$ and $+$ as idempotent operators for making expressions negative and positive, and sign flipping being done by explicitely multiplying with $-1$.
$\ell$ feels weird to me (except for the $\ell_p$ norm and $\ell^p$ space maybe). It often seems too important/out of place for a simple variable (or, god forbid, an index).