hcju/proofwikips huggingface dataset:
Definition:Gradient Operator
Let $\struct {M, g}$ be a Riemannian manifold equiped with a metric $g$. Let $f \in \map {\CC^\infty} M$ be a smooth mapping on $M$. The '''gradient''' of $f$ is defined as: {{begin-eqn}} {{eqn | l = \grad f        | o = :=        | r = \nabla f }} {{eqn | r = g^{-1} \d_{\d R} f }} {{end-eqn}} where $\d_{\d R}$ is de Rham differential.
Definition:Beta Distribution
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\Img X = \closedint 0 1$. $X$ is said to have a '''Beta distribution'''  it has probability density function:  :$\map {f_X} X = \dfrac {x^{\alpha - 1} \paren {1 - x}^{\beta - 1} } {\map \Beta {\alpha, \beta} }$ for $\alpha, \beta > 0$, where $\Beta$ denotes the Beta function. This is written:  :$X \sim \BetaDist \alpha \beta$
Definition:Divergence Operator
Let $\struct {M, g}$ be a Riemannian manifold equiped with a metric $g$. Let $\mathbf X : \map {\CC^\infty} M \to \map {\CC^\infty} M$ be a smooth vector field. The '''divergence''' of $\mathbf X$ is defined as: {{begin-eqn}} {{eqn | l = \operatorname {div} \mathbf X        | o = :=        | r = \nabla \cdot \mathbf X }} {{eqn | r = \star^{−1}_g \d_{\d R} \star_g \map g {\mathbf X} }} {{end-eqn}} where: :$\star_g$ is the Hodge star operator of $\struct {M, g}$ :$\d_{\d R}$ is de Rham differential.
Definition:Divergence Operator/Real Cartesian Space
Let $\R^n \left({x_1, x_2, \ldots, x_n}\right)$ denote the real Cartesian space of $n$ dimensions. Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis on $\R^n$. Let $\mathbf f = \left({f_1 \left({\mathbf x}\right), f_2 \left({\mathbf x}\right), \ldots, f_n \left({\mathbf x}\right)}\right): \R^n \to \R^n$ be a vector-valued function on $\R^n$. Let the partial derivative of $\mathbf f$ with respect to $x_k$ exist for all $f_k$. The '''divergence of $\mathbf f$''' is defined as: {{begin-eqn}} {{eqn | l = \operatorname {div} \mathbf f       | o = :=       | r = \nabla \cdot \mathbf f }} {{eqn | r = \left({\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial u_k} }\right) \cdot \mathbf f       | c = {{Defof|Del Operator}} }} {{eqn | r = \sum_{k \mathop = 1}^n \dfrac {\partial f_k} {\partial x_k}       | c =  }} {{end-eqn}}
Definition:Curl Operator
Let $\map {\R^3} {x, y, z}$ denote the real Cartesian space of $3$ dimensions.. Let $\tuple {\mathbf i, \mathbf j, \mathbf k}$ be the standard ordered basis on $\R^3$. Let $\mathbf f := \tuple {\map {f_x} {\mathbf x}, \map {f_y} {\mathbf x}, \map {f_z} {\mathbf x} }: \R^3 \to \R^3$ be a vector-valued function on $\R^3$. The '''curl''' of $\mathbf f$ is defined as: {{begin-eqn}} {{eqn | l = \curl \mathbf f       | o = :=       | r = \nabla \times \mathbf f       | c = where $\nabla$ denotes the del operator }} {{eqn | r = \paren {\mathbf i \dfrac \partial {\partial x} + \mathbf j \dfrac \partial {\partial y} + \mathbf k \dfrac \partial {\partial z} } \times \paren {f_x \mathbf i + f_y \mathbf j + f_z \mathbf k}       | c = {{Defof|Del Operator}} }} {{eqn | r = \begin {vmatrix} \mathbf i & \mathbf j & \mathbf k \\ \dfrac \partial {\partial x} & \dfrac \partial {\partial y} & \dfrac \partial {\partial z} \\ f_x & f_y & f_z \end{vmatrix}       | c = {{Defof|Vector Cross Product}} }} {{eqn | r = \paren {\dfrac {\partial f_z} {\partial y} - \dfrac {\partial f_y} {\partial z} } \mathbf i + \paren {\dfrac {\partial f_x} {\partial z} - \dfrac {\partial f_z} {\partial x} } \mathbf j + \paren {\dfrac {\partial f_y} {\partial x} - \dfrac {\partial f_x} {\partial y} } \mathbf k       | c =  }} {{end-eqn}}
Definition:Laplacian
=== Real-Valued Function === {{:Definition:Laplacian/Real-Valued Function}} === Vector-Valued Function === {{:Definition:Laplacian/Vector-Valued Function}}
Definition:Laplacian/Real-Valued Function
Let $\R^n$ denote the real Cartesian space of $n$ dimensions. Let $\map f {x_1, x_2, \ldots, x_n}$ denote a real-valued function on $\R^n$. Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$. Let the partial derivative of $f$ with respect to $x_k$ exist for all $x_k$. The '''Laplacian of $f$''' is defined as: {{begin-eqn}} {{eqn | l = \nabla^2 f       | o = :=       | r = \nabla \cdot \left({\nabla f}\right) }} {{eqn | r = \paren {\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial x_k} } \cdot \paren {\sum_{k \mathop = 1}^n \dfrac {\partial f} {\partial x_k} \mathbf e_k}       | c = {{Defof|Del Operator}} }} {{eqn | r = \sum_{k \mathop = 1}^n \dfrac {\partial^2 f} {\partial {x_k}^2}       | c =  }} {{end-eqn}}
Definition:Laplacian/Vector-Valued Function
Let $\R^n \left({x_1, x_2, \ldots, x_n}\right)$ denote the real Cartesian space of $n$ dimensions. Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis on $\mathbf V$. Let $\mathbf f = \left({f_1 \left({\mathbf x}\right), f_2 \left({\mathbf x}\right), \ldots, f_n \left({\mathbf x}\right)}\right): \mathbf V \to \mathbf V$ be a vector-valued function on $\mathbf V$. Let the partial derivative of $\mathbf f$ with respect to $x_k$ exist for all $f_k$. The '''Laplacian of $\mathbf f$''' is defined as: {{begin-eqn}} {{eqn | l = \nabla^2 \mathbf f       | o = :=       | r = \left({\sum_{k \mathop = 1}^n \dfrac {\partial^2 \mathbf f} {\partial {x_k}^2} }\right) }} {{end-eqn}}
Definition:Biharmonic Operator
Let $\R^n$ denote the real Cartesian space of $n$ dimensions. Let $f \left({x_1, x_2, \ldots, x_n}\right)$ denote a real-valued function on $\R^n$. Let $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ be the standard ordered basis on $\R^n$. Let the partial derivative of $f$ with respect to $x_k$ exist for all $x_k$. The '''biharmonic operator on $f$''' is defined as: {{begin-eqn}} {{eqn | l = \nabla^4 f       | o = :=       | r = \nabla^2 \left({\nabla^2 f}\right) }} {{eqn | r = \left({\sum_{k \mathop = 1}^n \dfrac {\partial^2} {\partial x_k^2} }\right) \left({\sum_{k \mathop = 1}^n \dfrac {\partial^2} {\partial x_k^2} }\right) f       | c = {{Defof|Del Operator}} }} {{end-eqn}}
Definition:Primitive (Calculus)/Vector-Valued Function
Let $U \subset \R$ be an open set in $\R$. Let $\mathbf f: U \to \R^n$ be a vector-valued function on $U$: :$\forall x \in U: \map {\mathbf f} x = \displaystyle \sum_{k \mathop = 1}^n \map {f_k} x \mathbf e_k$ where: :$f_1, f_2, \ldots, f_n$ are real functions from $U$ to $\R$ :$\tuple {e_1, e_2, \ldots, e_k}$ denotes the standard ordered basis on $\R^n$. Let $\mathbf f$ be differentiable on $U$. Let $\map {\mathbf g} x := \dfrac \d {\d x} \map {\mathbf f} x$ be the derivative of $\mathbf f$ {{WRT|Differentiation}} $x$. The '''primitive of $\mathbf g$ {{WRT|Integration}} $x$''' is defined as: :$\displaystyle \int \map {\mathbf g} x \rd x := \map {\mathbf f} x + \mathbf c$ where $\mathbf c$ is an arbitrary constant vector.
Definition:Definite Integral of Vector-Valued Function
Let $I \:= \closedint a b \subset \R$ be a closed real interval. Let $\mathbf f: I \to \R^n$ be a vector-valued function on $I$: :$\forall x \in I: \map {\mathbf f} x = \displaystyle \sum_{k \mathop = 1}^n \map {f_k} x \mathbf e_k$ where: :$f_1, f_2, \ldots, f_n$ are real functions from $U$ to $\R$ :$\tuple {e_1, e_2, \ldots, e_k}$ denotes the standard ordered basis on $\R^n$. Let $\mathbf f$ be differentiable on $I$. Let $\map {\mathbf g} x := \dfrac \d {\d x} \map {\mathbf f} x$ be the derivative of $\mathbf f$ {{WRT|Differentiation}} $x$. The '''definite integral of $\mathbf g$ {{WRT|Integration}} $x$ from $a$ to $b$''' is defined as: :$\displaystyle \int_a^b \map {\mathbf g} x \rd x := \map {\mathbf f} b - \map {\mathbf f} a$
Definition:Derivative of Smooth Path
=== Real Cartesian Space === {{:Definition:Derivative of Smooth Path/Real Cartesian Space}} === Complex Plane === {{:Definition:Derivative of Smooth Path/Complex Plane}}
Definition:Smooth Path
=== Real Cartesian Space === {{:Definition:Smooth Path/Real Cartesian Space}} === Complex Analysis === {{:Definition:Smooth Path/Complex}}
Definition:Smooth Path/Real Cartesian Space
Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval. Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a path in $\R^n$. That is, let $\rho$ be a continuous real-valued function from $\left[{a \,.\,.\, b}\right]$ to $\R^n$. For each $k \in \left\{ {1, 2, \ldots, n}\right\}$, define the real function $\rho_k: \left[{a \,.\,.\, b}\right] \to \R$ by: :$\forall t \in \left[{a \,.\,.\, b}\right]: \rho_k \left({t}\right) = \pr_k \left({\rho \left({t}\right)}\right)$ where $\pr_k$ denotes the $k$th projection from the image $\operatorname{Im} \left({\rho}\right)$ of $\rho$ to $\R$. Then $\rho$ is a '''smooth path (in $\R^n$)''' : :$(1): \quad$ all of $\pr_k$ are continuously differentiable :$(2): \quad$ for all $t \in \left[{a \,.\,.\, b}\right]$, at least one $\rho_k' \left({t}\right) \ne 0$, where $\rho_k'$ denotes the derivative of $\rho_k$ {{WRT|Differentiation}} $t$.
Definition:Smooth Path/Closed/Real Cartesian Space
Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a '''smooth path in $\R^n$'''. $\rho$ is a '''closed smooth path'''  $\rho$ is a closed path. That is,  $\rho \left({a}\right) = \rho \left({b}\right)$.
Definition:Smooth Path/Simple/Real Cartesian Space
Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a '''smooth path in $\R^n$'''. $\rho$ is a '''simple smooth path (in $\R^n$)''' : :$(1): \quad \rho$ is injective on the half-open interval $\left[{a \,.\,.\, b}\right)$ :$(2): \quad \forall t \in \left({a \,.\,.\, b}\right): \rho \left({t}\right) \ne \rho \left({b}\right)$ That is,  $t_1, t_2 \in \left({a \,.\,.\, b}\right)$ with $t_1 \ne t_2$, then $\gamma \left({a}\right) \ne \gamma \left({t_1}\right) \ne \gamma \left({t_2}\right) \ne \gamma \left({b}\right)$.
Definition:Derivative of Smooth Path/Real Cartesian Space
Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval. Let $\rho: \left[{a \,.\,.\, b}\right] \to \R^n$ be a '''smooth path in $\R^n$'''. For each $k \in \left\{ {1, 2, \ldots, n}\right\}$, define the real function $\rho_k: \left[{a \,.\,.\, b}\right] \to \R$ by: :$\forall t \in \left[{a \,.\,.\, b}\right]: \rho_k \left({t}\right) = \pr_k \left({\rho \left({t}\right)}\right)$ where $\pr_k$ denotes the $k$th projection from the image $\operatorname{Im} \left({\rho}\right)$ of $\rho$ to $\R$. It follows from the definition of a  smooth path that $\rho_k$ is continuously differentiable for all $k$. Let $\rho_k' \left({t}\right)$ denote the derivative of $\rho_k$ {{WRT|Differentiation}} $t$. The '''derivative of $\rho$''' is the continuous vector-valued function $\rho': \left[{a \,.\,.\, b}\right] \to \R^n$ defined by: :$\forall t \in \left[{a \,.\,.\, b}\right]: \rho' \left({t}\right) = \displaystyle \sum_{k \mathop = 1}^n \rho_k' \left({t}\right) \mathbf e_k$ where $\left({\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}\right)$ denotes the standard ordered basis of $\R^n$.
Definition:Contour
Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $C_1, \ldots, C_n$ be directed smooth curves in $\R^n$. For each $i \in \left\{ {1, \ldots, n}\right\}$, let $C_i$ be parameterized by the smooth path $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$. For each  $i \in \left\{ {1, \ldots, n-1}\right\}$, let the end point of $\rho_i$ equal the start point of $\rho_{i + 1}$: :$\rho_i \left({b_i}\right) = \rho_{i + 1} \left({a_{i + 1} }\right)$ Then the finite sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ is called a '''contour''' (in $\R^n$).
Definition:Image of Contour
=== Real Cartesian Space === {{:Definition:Image of Contour (Real Cartesian Space)}} === Complex Plane === {{:Definition:Image of Contour (Complex Plane)}}
Definition:Student's t-Distribution
Let $X$ be a continuous random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. Let $\Img X = \R$. $X$ is said to have a '''$t$-distribution''' with $k$ degrees of freedom  it has probability density function:  :$\map {f_X} x = \dfrac {\map \Gamma {\frac {k + 1} 2} } {\sqrt {\pi k} \map \Gamma {\frac k 2} } \paren {1 + \dfrac {x^2} k}^{-\frac {k + 1} 2}$ for some $k \in \R_{> 0}$. This is written:  :$X \sim \StudentT k$
Definition:Estimator
Let $X_1, X_2, \ldots, X_n$ be random variables.  Let the joint distribution of $X_1, X_2, \ldots, X_n$ be indexed by a population parameter $\theta$.  Let $\delta: \R^n \to \R$ be a real-valued function.  The random variable $\hat \theta = \map \delta {X_1, X_2, \ldots, X_n}$ is called an '''estimator''' of $\theta$.  A particular realization of $\hat \theta$ is called an '''estimate''' of $\theta$.
Definition:Directed Smooth Curve/Complex Plane
Let $\gamma : \left[{ a \,.\,.\, b }\right] \to \C$ be a smooth path in $\C$. The '''directed smooth curve''' with '''parameterization''' $\gamma$ is defined as an equivalence class of smooth paths as follows: A smooth path $\sigma: \left[{ a \,.\,.\, b }\right] \to \C$ belongs to the equivalence class of $\gamma$ : : there exists a bijective differentiable strictly increasing real function: :: $\phi: \left[{c \,.\,.\, d}\right] \to \left[{a \,.\,.\, b}\right]$ : such that $\sigma = \gamma \circ \phi$.
Definition:Directed Smooth Curve/Parameterization
Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ be a directed smooth curve in $\R^n$. Let $\rho: \left[{a \,.\,.\, b}\right] \to \C$ be a smooth path in $\R^n$. Then $\rho$ is a '''parameterization''' of $C$  $\rho$ is an element of the equivalence class that constitutes $C$.
Definition:Parameterization
=== Parameterization of Directed Smooth Curve === {{:Definition:Parameterization of Directed Smooth Curve}} === Parameterization of Contour === {{:Definition:Parameterization of Contour}}
Definition:Directed Smooth Curve/Endpoints
Let $C$ be parameterized by a smooth path $\rho: \left[{a \,.\,.\, b}\right] \to \C$. Then: : $\rho \left({a}\right)$ is the '''start point''' of $C$ : $\rho \left({b}\right)$ is the '''end point''' of $C$. Collectively, $\rho \left({a}\right)$ and $\rho \left({b}\right)$ are known as the '''endpoints''' of $\rho$.
Definition:Bias of Estimator
Let $\theta$ be a population parameter of some statistical model.  Let $\mathbf X$ be a random sample from this population. Let $\delta$ be an estimator of $\theta$.  The '''bias''' of $\delta$ is defined as:  :$\map {\operatorname{bias} } \delta = \expect {\map \delta {\mathbf X} } - \theta$
Definition:Contour/Parameterization
Let $C_1, \ldots, C_n$ be directed smooth curves in $\R^n$. Let $C_i$ be parameterized by the smooth path $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$. The '''parameterization of $C$''' is defined as the function $\rho: \left[{a_1 \,.\,.\, c_n}\right] \to \R^n$ with: :$\rho \restriction_{\left[{c_i \,.\,.\, c_{i + 1} }\right] } \left({t}\right) = \rho_i \left({t}\right)$ where: : $\displaystyle c_i = a_1 + \sum_{j \mathop = 1}^i b_j - \sum_{j \mathop = 1}^i a_j$ for $i \in \left\{ {0, \ldots, n}\right\}$ : $\rho \restriction_{\left[{c_i \,.\,.\, c_{i + 1} }\right] }$ denotes the restriction of $\rho$ to $\left[{c_i \,.\,.\, c_{i + 1} }\right]$.
Definition:Contour/Closed
Let $C$ be the '''contour''' in $\R^n$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of directed smooth curves in $\R^n$. Let $C_i$ be parameterized by the smooth path $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$ for all $i \in \left\{ {1, \ldots, n}\right\}$. $C$ is a '''closed contour'''  the start point of $C$ is equal to the end point of $C$: :$\rho_1 \left({a_1}\right) = \rho_n \left({b_n}\right)$
Definition:Contour/Simple
Let $C_1, \ldots, C_n$ be directed smooth curves in $\R^n$. Let $C_i$ be parameterized by the smooth path $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $C$ be the contour in $\R^n$ defined by the finite sequence $C_1, \ldots, C_n$. $C$ is a '''simple contour''' : :$(1): \quad$ For all $i, j \in \left\{ {1, \ldots, n}\right\}, t_1 \in \left[{a_i \,.\,.\, b_i}\right), t_2 \in \left[{a_j \,.\,.\, b_j}\right)$ with $t_1 \ne t_2$, we have $\rho_i \left({t_1}\right) \ne \rho_j \left({t_2}\right)$ :$(2): \quad$ For all $k \in \left\{ {1, \ldots, n}\right\}, t \in \left[{a_k \,.\,.\, b_k}\right)$ where either $k \ne 1$ or $t \ne a_1$, we have $\rho_k \left({t}\right) \ne \rho_n \left({b_n}\right)$.
Definition:Contour/Length
Let $C$ be a '''contour''' in $C$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of directed smooth curves in $\R^n$. Let $C_i$ be parameterized by the smooth path $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$ for all $i \in \left\{ {1, \ldots, n}\right\}$. The '''length''' of $C$ is defined as: :$\displaystyle L \left({C}\right) := \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \left\vert{\rho_i' \left({t}\right) }\right\vert \rd t$
Definition:Contour/Image
Let $C$ be a '''contour''' in $\R^n$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of directed smooth curves in $\R^n$. Let $C_i$ be parameterized by the smooth path $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$ for all $i \in \left\{ {1, \ldots, n}\right\}$. The '''image of $C$''' is defined as: :$\displaystyle \operatorname{Im} \left({C}\right) := \bigcup_{i \mathop = 1}^n \operatorname{Im} \left({\rho_i}\right)$ where $\operatorname{Im} \left({\rho_i}\right)$ denotes the image of $\rho_i$.
Definition:Contour/Endpoints
Let $C_1, \ldots, C_n$ be directed smooth curves in $\R^n$. Let $C_i$ be parameterized by the smooth path $\rho_i: \left[{a_i \,.\,.\, b_i}\right] \to \R^n$ for all $i \in \left\{ {1, \ldots, n}\right\}$. Let $C$ be the contour defined by the finite sequence $C_1, \ldots, C_n$. The '''start point''' of $C$ is $\rho_1 \left({a_1}\right)$. The '''end point''' of $C$ is $\rho_n \left({b_n}\right)$. Collectively, $\rho_1 \left({a_1}\right)$ and $\rho_n \left({b_n}\right)$ are referred to as the '''endpoints''' of $C$.
Definition:Concatenation of Contours/Complex Plane
Let $C$ and $D$ be contours in the complex plane $\C$. Thus: :$C$ is a (finite) sequence of directed smooth curves $\sequence {C_1, \ldots, C_n}$ :$D$ is a (finite) sequence of directed smooth curves $\sequence {D_1, \ldots, D_m}$. Let $C_i$ be parameterized by the smooth path: :$\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$ Let $D_i$ be parameterized by the smooth path: :$\sigma_i: \closedint {c_i} {d_i} \to \C$ for all $i \in \set {1, \ldots, m}$ Let $\map {\gamma_n} {b_n} = \map {\sigma_1} {c_1}$. Then the '''concatenation of the contours''' $C$ and $D$, denoted $C \cup D$, is the contour defined by the (finite) sequence: :$\sequence {C_1, \ldots, C_n, D_1, \ldots, D_m}$
Definition:Pointwise Operation/Real-Valued Functions/Multiary
For ease of notation, write $\left[{S \to \R}\right]$ for $\R^S$. Let $I$ be some index set. Let $\oplus^I: \R^I \to \R$ be an $I$-ary operation on $\R$. Then $\oplus^I: \left[{S \to \R}\right]^I \to \left[{S \to \R}\right]$, referred to as '''pointwise $\oplus^I$''', is defined as: :$\forall \left({f_i}\right)_{i \mathop \in I} \in \left[{S \to \R}\right]^I: \forall s \in S: \left({\oplus^I \left({f_i}\right)_{i \in I} }\right) \left({s}\right) := \oplus^I \left({f_i \left({s}\right) }\right)_{i \in I}$
Definition:Confidence Interval
Let $\theta$ be a population parameter of some population. Let $\mathbf X$ be a random sample from this population. Let $I = \openint {\map f {\mathbf X} } {\map g {\mathbf X} }$ for some real-valued functions $f$, $g$. $I$ is said to be a '''$100\gamma \%$ confidence interval''' for $\theta$ if:  :$\map \Pr {\theta \in I} = \gamma$ where $0 < \gamma < 1$.
Definition:Random Vector
Let $X_1, X_2, \ldots, X_n$ be random variables on a probability space $\left({\Omega, \Sigma, \Pr}\right)$. Then the vector $\mathbf X = \left({X_1, X_2, \ldots, X_n}\right)$ is referred to as a '''random vector'''.
Definition:Mean Squared Error of Estimator
Let $\theta$ be a population parameter of some population. Let $\mathbf X$ be a random sample from this population. Let $\hat \theta$ be an estimator of $\theta$.  The '''mean squared error''' of $\hat \theta$ is defined by: :$\map {\operatorname {MSE} } {\hat \theta} = \expect {\paren {\map {\hat \theta} {\mathbf X} - \theta}^2} $
Definition:Density (Physics)/Area
The '''area density''' of a two-dimensional body is its mass per unit area.
Definition:Density (Physics)/Linear
The '''linear density''' of a one-dimensional body is its mass per unit length.
Definition:Expectation of Random Vector
Let $X_1, X_2, \ldots, X_n$ be random variables on a probability space $\left({\Omega, \Sigma, \Pr}\right)$. Let $\mathbf X = \left({X_1, X_2, \ldots, X_n}\right)$ be a random vector.  Then the expected value of $\mathbf X$, $\mathbb E \left[{\mathbf X}\right]$, is defined by:  :$\mathbb E \left[{\mathbf X}\right] = \left({\mathbb E \left[{X_1}\right], \mathbb E \left[{X_2}\right], \ldots, \mathbb E \left[{X_n}\right]}\right)$
Definition:Field of Rational Functions
Let $K$ be a field. Let $K \sqbrk x$ be the integral domain of polynomial forms on $K$. Let $\map K x$ be the set of rational functions on $K$: :$\map K x := \set {\forall f \in K \sqbrk x, g \in K \sqbrk x^*: \dfrac {\map f x} {\map g x} }$ where $K \sqbrk x^* = K \sqbrk x \setminus \set {\text {the null polynomial} }$. Then $\map K x$ is the '''field of rational functions''' on $K$.
Definition:Sylow p-Subgroup/Definition 3
Let $n$ be the largest integer such that: :$p^n \divides \order G$ where $\divides$ denotes divisibility. A '''Sylow $p$-subgroup''' is a $p$-subgroup of $G$ which has $p^n$ elements.
Definition:Cross-Covariance Matrix
Let $\mathbf X = \left({X_1, X_2, \ldots, X_n}\right)^T$ and $\mathbf Y = \left({Y_1, Y_2, \ldots, Y_m}\right)^T$ be random vectors.   Then the '''cross-covariance matrix''' of $\mathbf X$ and $\mathbf Y$ is defined by:  :$\operatorname{cov} \left({\mathbf X, \mathbf Y}\right) = \mathbb E \left[{\left({\mathbf X - \mathbb E \left[{\mathbf X}\right]}\right) \left({\mathbf Y - \mathbb E \left[{\mathbf Y}\right]}\right)^T}\right]$ where this expectation exists.
Definition:Doubleton/Class Theory
Let $a$ and $b$ be sets. The class $\set {a, b}$ is a '''doubleton (class)'''. It is defined as the class of all $x$ such that $x = a$ or $x = b$: :$\set {a, b} = \set {x: x = a \lor x = b: a \ne b}$
Definition:Variance of Random Vector
Let $\mathbf X$ be a random vector. The '''variance''' of $\mathbf X$ is defined by:  :$\operatorname{var} \left({\mathbf X}\right) = \operatorname{cov} \left({\mathbf X, \mathbf X}\right)$ where $\operatorname{cov}$ is the cross-covariance matrix.
Definition:Even Permutation
$\rho$ is an '''even permutation''' : :$\map \sgn \rho = 1$
Definition:Odd Permutation
$\rho$ is an '''odd permutation''' : :$\map \sgn \rho = -1$
Definition:Zero Residue Class
Let $m \in \Z$. Let $\mathcal R_m$ be the congruence relation modulo $m$ on the set of all $a, b \in \Z$: :$\mathcal R_m = \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$ Let $\eqclass 0 m$ be the residue class of $0$ (modulo $m$): :$\eqclass 0 m = \set {x \in \Z: \exists k \in \Z: x = k m}$ Then $\eqclass 0 m$ is known as the '''zero residue class (modulo $m$)'''.
Definition:Binomial Coefficient/Complex Numbers
Let $z, w \in \C$. Then $\dbinom z w$ is defined as: :$\dbinom z w := \displaystyle \lim_{\zeta \mathop \to z} \lim_{\omega \mathop \to w} \dfrac {\Gamma \left({\zeta + 1}\right)} {\Gamma \left({\omega + 1}\right) \Gamma \left({\zeta - \omega + 1}\right)}$ where $\Gamma$ denotes the Gamma function. When $z$ is a negative integer and $w$ is not an integer, $\dbinom z w$ is infinite.