Let $T$ be strongly minimal and let $M\models T$. For every $A\subseteq M$, every set $I\subseteq M$ that is independent over $A$ extends to a basis of $M$ over $A$, and any two bases of $M$ over $A$ have the same cardinality.

Let $T$ be a complete first-order theory in a language $L$, let $\varphi(x,z)$ be an $L$-formula, where $x$ and $z$ are finite tuples of variables, and let $p(z)\in S_z(\varnothing)$ be a complete type. Let $\kappa$ be a cardinal such that $\kappa>|T|+|z|$. Assume that there exist a $\kappa$-saturated model $S\models T$ and a tuple $e\in S^{|z|}$ realizing $p(z)$ such that $\varphi(S,e)$ is infinite and, for every $L$-formula $\psi(x,y)$ and every tuple $b\in S^{|y|}$, the subset \begin{align*} \{a\in \varphi(S,e): S\models \psi(a,b)\} \end{align*} is finite or cofinite in $\varphi(S,e)$. Then for every model $M\models T$ and every tuple $e'\in M^{|z|}$ realizing $p(z)$, the definable set $\varphi(M,e')$ is strongly minimal in the following sense: for every elementary extension $N\succeq M$…

Let $A$ be a parameter set, let $x$ be a tuple of variables, and let $S_x(A)$ denote the Stone space of complete $x$-types over $A$, equipped with the Stone topology whose basic clopen sets are \begin{align*} [\psi] := \{p \in S_x(A) : \psi \in p\} \end{align*} for $A$-formulas $\psi(x)$. For every $A$-formula $\varphi(x)$, define \begin{align*} f_\varphi: S_x(A) &\to \{0,1\} \\ p &\mapsto \begin{cases} 1, & \varphi \in p, \\ 0, & \varphi \notin p. \end{cases} \end{align*} where $\{0,1\}$ is equipped with the discrete topology. Then $f_\varphi$ is continuous.

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 $T$ be a complete first-order theory in a language $L$, and work inside a sufficiently saturated monster model $\mathfrak{C} \models T$. Suppose that for every parameter set $A \subseteq \mathfrak{C}$ with $A$ infinite and \begin{align*} |A| \geq |L| + \aleph_0, \end{align*} there exist a set $D_A$ and an injective map \begin{align*} \Phi_A: S_1(A) &\to D_A \end{align*} such that \begin{align*} |D_A| \leq |A|. \end{align*} Then, for every infinite cardinal $\kappa$ with $\kappa \geq |L| + \aleph_0$, the theory $T$ is $\kappa$-stable, in the sense that for every parameter set $A \subseteq \mathfrak{C}$ with $|A| = \kappa$, \begin{align*} |S_1(A)| \leq \kappa. \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 $\Omega\subset\mathbb C$ 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 $(\Omega,\mathcal F,\mathbb P)$ be a probability space, and let $X,Y:(\Omega,\mathcal F)\to(\mathbb R,\mathcal B(\mathbb R))$ be real-valued random variables such that $\mathbb E[X^2]<\infty$ and $\mathbb E[Y^2]<\infty$. Define \begin{align*} \operatorname{Cov}(X,Y)&:=\mathbb E\!\left[(X-\mathbb E[X])(Y-\mathbb E[Y])\right],\\ \operatorname{Var}(X)&:=\mathbb E\!\left[(X-\mathbb E[X])^2\right],\qquad \operatorname{Var}(Y):=\mathbb E\!\left[(Y-\mathbb E[Y])^2\right],\\ \sigma_X&:=\sqrt{\operatorname{Var}(X)},\qquad \sigma_Y:=\sqrt{\operatorname{Var}(Y)}. \end{align*} Then \begin{align*} |\operatorname{Cov}(X,Y)|\le \sigma_X\sigma_Y. \end{align*} If $\operatorname{Var}(X)>0$ and $\operatorname{Var}(Y)>0$, and if \begin{align*} \rho(X,Y):=\frac{\operatorname{Cov}(X,Y)}{\sigma_X\sigma_Y}, \end{align*}…