Let $D$ be a parameter-free definable strongly minimal set in a monster model $\mathfrak{M}$ of a complete theory. For $A \subset D$, define \begin{align*} \operatorname{cl}: \mathcal{P}(D) \to \mathcal{P}(D), \qquad \operatorname{cl}(A)=D \cap \operatorname{acl}_{\mathfrak{M}}(A). \end{align*} Then $\operatorname{cl}$ is a pregeometry on $D$: it is extensive, increasing, idempotent, has finite character, and satisfies exchange.

Let $L$ be a first-order language, let $M$ be an $L$-structure, and let $\kappa$ be an infinite cardinal such that $|L|<\kappa$, $\kappa^{<\kappa}=\kappa$, and $|M|\le \kappa$. Then there is an elementary extension $N\succeq M$ such that $|N|\le \kappa$ and $N$ is $\kappa$-saturated.

Let $T$ be a complete first-order theory in a countable language. Suppose that there exists an uncountable cardinal $\kappa$ such that $T$ is $\kappa$-categorical, meaning that any two models of $T$ of cardinality $\kappa$ are isomorphic. Then no formula $\varphi(x;y)$ has the order property modulo $T$: there do not exist a model $M \models T$ and sequences $(a_i)_{i \in \mathbb{N}}$ and $(b_j)_{j \in \mathbb{N}}$ of finite tuples from $M$, with $|a_i| = |x|$ and $|b_j| = |y|$, such that for all $i,j \in \mathbb{N}$, \begin{align*} M \models \varphi(a_i;b_j) \quad \iff \quad i < j. \end{align*}

Let $L$ be a first-order language, and let $T$ be a complete $L$-theory with quantifier elimination. Let $M,N \models T$. Let $A$ be a parameter set equipped with interpretations $i_M:A \to M$ and $i_N:A \to N$ whose named constants have matching quantifier-free diagram in $M$ and $N$; for example, $A$ may be a common substructure of both models with the same interpretation of all $L$-terms. If tuples $b \in M^n$ and $c \in N^n$ have the same quantifier-free type over $A$ via the interpretations $i_M$ and $i_N$, then they have the same complete type over $A$ via those interpretations.

Let $\Omega\subset\mathbb C$ be open, let $f:\Omega\to\mathbb C$ be continuous, and let $\gamma:[a,b]\to\Omega$ be a piecewise $C^1$ path. If $\phi:[c,d]\to[a,b]$ is a piecewise $C^1$ increasing bijection, then \begin{align*} \int_{\gamma\circ\phi} f(z)\,dz = \int_\gamma f(z)\,dz. \end{align*} Moreover, \begin{align*} \int_{-\gamma} f(z)\,dz = -\int_\gamma f(z)\,dz. \end{align*}

Let $\Omega \subset \mathbb{C}$ be open, let $a,b \in \mathbb{R}$ with $a < b$, let $f: \Omega \to \mathbb{C}$ be continuous, and let $\gamma: [a,b] \to \Omega$ be a piecewise $C^1$ path. Let $M \in [0,\infty)$ satisfy $|f(z)| \le M$ for every $z \in \gamma([a,b])$. Then \begin{align*} \left|\int_\gamma f(z)\,dz\right| \le M L(\gamma), \end{align*} where the path length is \begin{align*} L(\gamma) := \int_a^b |\gamma'(t)|\,d\mathcal{L}^1(t). \end{align*}

Let $\Omega\subset\mathbb C$ be open, let $z_1,\ldots,z_n\in\Omega$, and let $f:\Omega\setminus\{z_1,\ldots,z_n\}\to\mathbb C$ be holomorphic, with $z_1,\ldots,z_n$ the only isolated singularities of $f$ in $\Omega$. Let $\gamma$ be a closed piecewise $C^1$ path in $\Omega\setminus\{z_1,\ldots,z_n\}$. Suppose $\gamma$ is null-homologous in $\Omega$, meaning \begin{align*} n(\gamma,w)=0 \end{align*} for every $w\in\mathbb C\setminus\Omega$. Then \begin{align*} \oint_\gamma f(z)\,dz = 2\pi i\sum_{j=1}^n n(\gamma,z_j)\operatorname{Res}(f,z_j). \end{align*}

Let $a,b \in \mathbb{R}$ with $a < b$, let $\gamma: [a,b] \to \mathbb{C}$ be a closed piecewise $C^1$ path, and let $z_0 \in \mathbb{C}$ satisfy $z_0 \notin \gamma([a,b])$. Define the [winding number](/page/Winding%20Number) of $\gamma$ about $z_0$ by \begin{align*} N(\gamma,z_0) := \frac{1}{2\pi i}\int_\gamma \frac{1}{z-z_0}\,dz = \frac{1}{2\pi i}\int_a^b \frac{\gamma'(t)}{\gamma(t)-z_0}\,d\mathcal{L}^1(t), \end{align*} where the last integral is interpreted piecewise on the $C^1$ subintervals of $\gamma$. Then \begin{align*} N(\gamma,z_0) \in \mathbb{Z}. \end{align*}

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, let $\mathcal G \subset \mathcal F$ be a sub-$\sigma$-algebra, and let $X: (\Omega,\mathcal F) \to (\mathbb R,\mathcal B(\mathbb R))$ be a real-valued [random variable](/page/Random%20Variable) such that $\mathbb E[X^2] < \infty$. Define the conditional variance of $X$ given $\mathcal G$ by \begin{align*} \operatorname{Var}(X \mid \mathcal G) := \mathbb E\left[\left(X - \mathbb E[X \mid \mathcal G]\right)^2 \mid \mathcal G\right]. \end{align*} Then \begin{align*} \operatorname{Var}(X) = \mathbb E[\operatorname{Var}(X \mid \mathcal G)] + \operatorname{Var}(\mathbb E[X \mid \mathcal G]). \end{align*}

Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space, and let $X:(\Omega,\mathcal F)\to(\mathbb R,\mathcal B(\mathbb R))$ be a real-valued [random variable](/page/Random%20Variable) such that \begin{align*} \mathbb E[X^2]=\int_\Omega X(\omega)^2\,d\mathbb P(\omega)<\infty. \end{align*} Then $X$ is integrable, $\operatorname{Var}(X):=\mathbb E[(X-\mathbb E[X])^2]$ is finite, and \begin{align*} \operatorname{Var}(X)=\mathbb E[X^2]-(\mathbb E[X])^2. \end{align*}