Let $T$ be a complete first-order theory, let $\mathfrak C \models T$ be a sufficiently saturated and strongly homogeneous monster model, and let $M \preceq \mathfrak C$. Let $p(x) \in S_x(M)$ be a complete type over $M$. Suppose that $p$ has a coherent definitional scheme over $M$: for every formula $\varphi(x;y)$ there is an $M$-definable formula $d_p\varphi(y)$ such that, for every $b \in M^{|y|}$, \begin{align*} \varphi(x;b) \in p \quad \iff \quad M \models d_p\varphi(b), \end{align*} and the assignment $\varphi \mapsto d_p\varphi$ commutes with Boolean combinations, in particular with negation and finite conjunctions. Then there exists a complete global type $q(x) \in S_x(\mathfrak C)$ such that $p \subseteq q$ and $q$ is $M$-invariant.

Let $T$ be a strongly minimal theory and let $M \models T$. Then the [algebraic closure](/page/Algebraic%20Closure) operator $A \mapsto \operatorname{acl}_M(A)$ defines a pregeometry on the underlying set of $M$.

Let $T$ be a complete theory in a language $L$, let $M\models T$, 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 $N$ realizes every complete type in finitely many variables over every parameter set $A \subset N$ with $|A| < \kappa$.

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 $\Omega \subset \mathbb{C}$ 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 $f:\Omega\to\mathbb C$ 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 $a,b \in \mathbb{R}$ 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^2\, d\mathbb P < \infty. \end{align*} Define \begin{align*} \operatorname{Var}(X) := \mathbb E\left[(X - \mathbb E[X])^2\right]. \end{align*} Then \begin{align*} \operatorname{Var}(X) \geq 0. \end{align*} Moreover, \begin{align*} \operatorname{Var}(X) = 0 \end{align*} if and only if \begin{align*} \mathbb P\left(\{\omega \in \Omega : X(\omega) = \mathbb E[X]\}\right) = 1. \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*}