If you're at a Math 3 level, you'll probably be familiar with at least some of these sentences and formulas, or you would be able to understand what they meant on a surface level if you were to look them up. Note that you don't necessarily have to understand the *proofs* of these statements (that's what we're here for, to teach you what they mean), but your eyes shouldn't gloss over them either.

In a group $~$G$~$, the Conjugacy class of an element $~$g$~$ is the set of elements that can be written as $~$hgh^{-1}$~$ for all $~$h \in G$~$.

The [ rank-nullity theorem] states that for any [-linear_mapping] $~$f: V \to W$~$, the [-dimension] of the [-image] of $~$f$~$ plus the dimension of the [-kernel] of $~$f$~$ is equal to the dimension of $~$V$~$.

A [ Baire space] is a space that satisfies [ Baire's Theorem] on [complete_metric_space complete metric spaces]: For a [-topological_space] $~$X$~$, if $~${F_1, F_2, F_3, \ldots}$~$ is a [countable_set countable] collection of open sets that are [dense_set dense] in $~$X$~$, then $~$\bigcap_{n=1}^\infty F_n$~$ is also dense in $~$X$~$.

The [riemann_hypothesis] asserts that every non-trivial zero of the [riemann_zeta_function] $~$\zeta(s) = \sum_{n=1}^\infty \frac{1}{s^n}$~$ when $~$s$~$ is a complex number has a real part equal to $~$\frac12$~$.

$~$\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}$~$ The [jacobian_matrix] of a [vector_valued_function vector-valued function] $~$f: \mathbb{R}^m \to \mathbb{R}^n$~$ is the matrix of [-partial_derivatives] $~$\left[ \begin{matrix} \pd{y_1}{x_1} & \pd{y_1}{x_2} & \cdots & \pd{y_1}{x_m} \\ \pd{y_2}{x_1} & \pd{y_2}{x_2} & \cdots & \pd{y_2}{x_m} \\ \vdots & \vdots & \ddots & \vdots \\ \pd{y_n}{x_1} & \pd{y_n}{x_2} & \cdots & \pd{y_n}{x_m} \end{matrix} \right]$~$ between each component of the argument vector $~$x = (x_1, x_2, \ldots, x_m)$~$ and each component of the result vector $~$y = f(x) = (y_1, y_2, \ldots, y_n)$~$. It is notated as $~$\displaystyle \frac{d\mathbf{y}}{d\mathbf{x}}$~$ or $~$\displaystyle \frac{d(y_1, y_2, \ldots, y_n)}{d(x_1, x_2, \ldots, x_m)}$~$.