hcju/proofwikips huggingface dataset:
Definition:Concave Real Function/Definition 2
$f$ is '''concave on $I$''' : :$\displaystyle \forall x_1, x_2, x_3 \in I: x_1 < x_2< x_3: \frac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \ge \frac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$
Definition:Concave Real Function/Definition 3
$f$ is '''concave on $I$''' : :$\forall x_1, x_2, x_3 \in I: x_1 < x_2< x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \ge \dfrac {f \left({x_3}\right) - f \left({x_1}\right)} {x_3 - x_1}$
Definition:Convex Real Function/Definition 1/Strictly
$f$ is '''strictly convex on $I$''' : :$\forall x, y \in I, x \ne y: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: f \left({\alpha x + \beta y}\right) < \alpha f \left({x}\right) + \beta f \left({y}\right)$
Definition:Convex Real Function/Definition 2/Strictly
$f$ is '''strictly convex on $I$''' : :$\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} < \dfrac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$
Definition:Convex Real Function/Definition 3/Strictly
$f$ is '''strictly convex on $I$''' : :$\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} < \dfrac {f \left({x_3}\right) - f \left({x_1}\right)} {x_3 - x_1}$
Definition:Concave Real Function/Definition 1/Strictly
$f$ is '''strictly concave on $I$''' : :$\forall x, y \in I, x \ne y: \forall \alpha, \beta \in \R_{>0}, \alpha + \beta = 1: f \left({\alpha x + \beta y}\right) > \alpha f \left({x}\right) + \beta f \left({y}\right)$
Definition:Concave Real Function/Definition 2/Strictly
$f$ is '''strictly concave on $I$''' : :$\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} > \dfrac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$
Definition:Concave Real Function/Definition 3/Strictly
$f$ is '''strictly concave on $I$''' : :$\forall x_1, x_2, x_3 \in I: x_1 < x_2 < x_3: \dfrac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} > \dfrac {f \left({x_3}\right) - f \left({x_1}\right)} {x_3 - x_1}$
Definition:Beta Function/Definition 1
:$\displaystyle \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$
Definition:Beta Function/Definition 2
:$\displaystyle \Beta \left({x, y}\right) := 2 \int_0^{\pi / 2} \left({\sin \theta}\right)^{2 x - 1} \left({\cos \theta}\right)^{2 y - 1} \rd \theta$
Definition:Beta Function/Definition 3
:$\map \Beta {x, y} := \dfrac {\map \Gamma x \, \map \Gamma y} {\map \Gamma {x + y} }$ where $\Gamma$ is the Gamma function.
Definition:Improper Integral/Half Open Interval
==== Open Above ==== {{:Definition:Improper Integral/Half Open Interval/Open Above}} ==== Open Below ==== {{:Definition:Improper Integral/Half Open Interval/Open Below}}
Definition:Improper Integral/Open Interval
Let $f$ be a real function which is continuous on the open interval $\openint a b$. Then the improper integral of $f$ over $\openint a b$ is defined as: :$\displaystyle \int_{\mathop \to a}^{\mathop \to b} \map f t \rd t := \lim_{\gamma \mathop \to a} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to b} \int_c^\gamma \map f t \rd t$ for some $c \in \openint a b$.
Definition:Improper Integral/Unbounded Closed Interval
==== Unbounded Above ==== {{:Definition:Improper Integral/Unbounded Closed Interval/Unbounded Above}} ==== Unbounded Below ==== {{:Definition:Improper Integral/Unbounded Closed Interval/Unbounded Below}}
Definition:Improper Integral/Unbounded Open Interval/Unbounded Above
Let $f$ be a real function which is continuous on the unbounded open interval $\openint a {+\infty}$. Then the improper integral of $f$ over $\openint a {+\infty}$ is defined as: :$\displaystyle \int_{\mathop \to a}^{\mathop \to +\infty} \map f t \rd t := \lim_{\gamma \mathop \to a} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to +\infty} \int_c^\gamma \map f t \rd t$ for some $c \in \openint a {+\infty}$.
Definition:Improper Integral/Unbounded Open Interval/Unbounded Below
Let $f$ be a real function which is continuous on the unbounded open interval $\openint {-\infty} b$. Then the improper integral of $f$ over $\openint {-\infty} b$ is defined as: :$\displaystyle \int_{\mathop \to -\infty}^{\mathop \to b} \map f t \rd t := \lim_{\gamma \mathop \to -\infty} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to b} \int_c^\gamma \map f t \rd t$ for some $c \in \openint {-\infty} b$.
Definition:Improper Integral/Unbounded Open Interval/Unbounded Above and Below
Let $f$ be a real function which is continuous everywhere. Then the improper integral of $f$ over $\R$ is defined as: :$\displaystyle \int_{\mathop \to -\infty}^{\mathop \to +\infty} \map f t \rd t := \lim_{\gamma \mathop \to -\infty} \int_\gamma^c \map f t \rd t + \lim_{\gamma \mathop \to +\infty} \int_c^\gamma \map f t \rd t$ for some $c \in \R$. Usually $c$ is taken to be $0$ as this usually simplifies the evaluation of the expressions.
Definition:Improper Integral/Unbounded Open Interval
==== Unbounded Above ==== {{:Definition:Improper Integral/Unbounded Open Interval/Unbounded Above}} ==== Unbounded Below ==== {{:Definition:Improper Integral/Unbounded Open Interval/Unbounded Below}} A specific and important instance of this occurs when the interval in question is the set of all real numbers: ==== Unbounded Above and Below ==== {{:Definition:Improper Integral/Unbounded Open Interval/Unbounded Above and Below}}
Definition:Improper Integral/Half Open Interval/Open Above
Let $f$ be a real function which is continuous on the half open interval $\hointr a b$. Then the improper integral of $f$ over $\hointr a b$ is defined as: :$\displaystyle \int_a^{\mathop \to b} \map f t \rd t := \lim_{\gamma \mathop \to b} \int_a^\gamma \map f t \rd t$
Definition:Improper Integral/Half Open Interval/Open Below
Let $f$ be a real function which is continuous on the half open interval $\hointl a b$. Then the improper integral of $f$ over $\hointl a b$ is defined as: :$\displaystyle \int_{\mathop \to a}^b \map f t \rd t := \lim_{\gamma \mathop \to a} \int_\gamma^b \map f t \rd t$
Definition:Improper Integral/Unbounded Closed Interval/Unbounded Above
Let $f$ be a real function which is continuous on the unbounded closed interval $\hointr a {+\infty}$. Then the improper integral of $f$ over $\hointr a {+\infty}$ is defined as: :$\displaystyle \int_a^{\mathop \to + \infty} \map f t \rd t := \lim_{\gamma \mathop \to +\infty} \int_a^\gamma \map f t \rd t$
Definition:Improper Integral/Unbounded Closed Interval/Unbounded Below
Let $f$ be a real function which is continuous on the unbounded closed interval $\hointl {-\infty} b$. Then the improper integral of $f$ over $\hointl {-\infty} b$ is defined as: :$\displaystyle \int_{\mathop \to -\infty}^b \map f t \rd t := \lim_{\gamma \mathop \to -\infty} \int_\gamma^b \map f t \rd t$
Definition:Positive Real Function
Let $I$ be a real interval. Let $f$ be a real function. Then $f$ is '''positive (on $I$)''' : :$\forall x \in I: f \left({x}\right) \ge 0$
Definition:Negative Real Function
Let $I$ be a real interval. Let $f$ be a real function. Then $f$ is '''negative (on $I$)''' : :$\forall x \in I: \map f x \le 0$
Definition:Strictly Positive Real Function
Let $I$ be a real interval. Let $f$ be a real function. Then $f$ is '''strictly positive (on $I$)''' iff: :$\forall x \in I: f \left({x}\right) > 0$
Definition:Strictly Negative Real Function
Let $I$ be a real interval. Let $f$ be a real function. Then $f$ is '''strictly negative (on $I$)''' : :$\forall x \in I: \map f x < 0$
Definition:Horizontal Line
=== Definition 1 === {{Definition:Horizontal Line/Definition 1}} === Definition 2 === {{Definition:Horizontal Line/Definition 2}} === Definition 3 === {{Definition:Horizontal Line/Definition 3}} File:Horizontal.png
Definition:Vertical Line
=== Definition 1 === {{:Definition:Vertical Line/Definition 1}} === Definition 2 === {{:Definition:Vertical Line/Definition 2}} :::File:Vertical.png
Definition:Laplace Transform
Let $f: \R_{\ge 0} \to \mathbb F$ be a function of a real variable $t$, where $\mathbb F \in \set {\R, \C}$. The '''Laplace transform''' of $f$, denoted $\laptrans f$ or $F$, is defined as: :$\laptrans {\map f t} = \map F s = \displaystyle \int_0^{\to +\infty} e^{-s t} \map f t \rd t$ whenever this improper integral converges. If this improper integral does not converge, then $\laptrans {\map f t}$ does not exist.
Definition:Euler Numbers
The '''Euler Numbers''' $E_n$ are a sequence of integers defined by the exponential generating function: :$\displaystyle \sech x = \frac {2 e^x} {e^{2 x} + 1} = \sum_{n \mathop = 0}^\infty \frac {E_n x^n} {n!}$ where $\size x < \dfrac \pi 2$.
Definition:Catalan Number
The '''Catalan Numbers''' $C_n$ are a sequence of natural numbers defined by: :$C_n = \dfrac 1 {n + 1} \dbinom {2 n} n $
Definition:Population/Infinite
An '''infinite population''' is a population which is infinite.
Definition:Population/Finite
A '''finite population''' is a population which is finite.
Definition:Statistics
'''Statistics''' is the branch of science which studies the collection and analysis of data, and the presentation of organized inferences from that analysis. It is also concerned with the design of experiments and sample surveys, data reduction, data processing, and various other things. As such, it can be considered as a form of applied probability theory. It has been defined as '''the technology of the scientific method'''.
Definition:Inductive Statistics
'''Inductive statistics''' is the phase of statistics which is concerned with the conditions under which conclusions about populations can be drawn from analysis of particular samples.
Definition:Descriptive Statistics
'''Descriptive statistics''' is the phase of statistics whose task is to collect, analyze and summarize data about statistical samples, without drawing conclusions or inferences about the populations from which those samples are taken.
Definition:Variable/Domain
The collection of all possible objects that a variable may refer to has to be specified. This collection is the '''domain''' of the variable.
Definition:Variable/Continuous
A '''continuous variable''' is a variable which can take any value between two given values.
Definition:Variable/Discrete
A '''discrete variable''' is a variable which is not continuous.
Definition:Sample Statistic/Discrete
Data which can be described with a discrete variable are known as '''discrete data'''.
Definition:Sample Statistic/Continuous
Data which can be described with a continuous variable are known as '''continuous data'''.
Definition:Measurable Property/Measurement
'''Measurement''' is the process of determining the quantity of a measurable property. A '''measurement''' is reported as a (real) number multiplied by a unit of measurement for that quantity.
Definition:Rounding
'''Rounding''' is the process of approximation of a value of a variable to a multiple of a given power of whatever number base one is working in (usually decimal). Let $n \in \Z$ be an integer. Let $x \in \R$ be a real number. Let $y \in \R$ such that: :$y = 10^n \floor {\dfrac x {10^n} + \dfrac 1 2}$ or: :$y = 10^n \ceiling {\dfrac x {10^n} - \dfrac 1 2}$ where $\floor {\, \cdot \,}$ denotes the floor function and $\ceiling {\, \cdot \,}$ denotes the ceiling function. Then $y$ is defined as '''$x$ rounded to the nearest $n$th power of $10$'''. Both of these definitions amount to the same thing, except for when $\dfrac x {10^n}$ is exactly halfway between $\floor {\dfrac x {10^n} }$ and $\ceiling {\dfrac x {10^n} }$. How these instances is treated is known as the '''treatment of the half'''.
Definition:Derivative/Higher Derivatives/Second Derivative
Let $f$ be a real function which is differentiable on an open interval $I$. Hence $f'$ is defined on $I$ as the derivative of $f$. Let $\xi \in I$ be a point in $I$. Let $f'$ be differentiable at the point $\xi$. Then the '''second derivative''' $\map {f''} \xi$ is defined as: :$\displaystyle f'' := \lim_{x \mathop \to \xi} \dfrac {\map {f'} x - \map {f'} \xi} {x - \xi}$
Definition:Derivative/Higher Derivatives/Third Derivative
Let $f$ be a real function which is twice differentiable on an open interval $I$. Let $f''$ denote the second derivate. Then the '''third derivative''' $f'''$ is defined as: :$f''' := \dfrac {\d} {\d x} f'' = \map {\dfrac {\d} {\d x} } {\dfrac {\d^2} {\d x^2} f}$
Definition:Derivative/Higher Derivatives/Higher Order
The $n$th derivative of a function $y = \map f x$ is defined as: :$\map {f^{\paren n} } x = \dfrac {\d^n y} {\d x^n} := \begin {cases} \map {\dfrac \d {\d x} } {\dfrac {\d^{n - 1} y} {\d x^{n - 1} } } & : n > 0 \\ y & : n = 0 \end {cases}$ assuming appropriate differentiability for a given $f^{\paren {n - 1} }$.
Definition:Derivative/Higher Derivatives/Zeroth Derivative
The '''zeroth derivative''' of a real function $f$ is defined as $f$ itself: :$f^{\paren 0} := f$ where $f^{\paren n}$ denotes the $n$th derivative of $f$.
Definition:Dirichlet Eta Function
The '''Dirichlet $\eta$ (eta) function''' $\eta$ is the complex function defined on the half-plane $\map \Re s > 0$ as the series: :$\displaystyle \map \eta s = \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} n^{-s}$
Definition:Partial Derivative/Second Derivative
Let $\map f {x, y}$ be a function of the two independent variables $x$ and $y$. The '''second partial derivatives of $f$ with respect to $x$ and $y$''' are defined and denoted by: {{begin-eqn}} {{eqn | n = 1       | l = \dfrac {\partial^2 f} {\partial x^2}       | r = \map {\dfrac \partial {\partial x} } {\dfrac {\partial f} {\partial x} }       | rr= =: \map {f_{1 1} } {x, y}       | c =  }} {{eqn | n = 2       | l = \dfrac {\partial^2 f} {\partial y^2}       | r = \map {\dfrac \partial {\partial y} } {\dfrac {\partial f} {\partial y} }       | rr= =: \map {f_{2 2} } {x, y}       | c =  }} {{eqn | n = 3       | l = \quad \dfrac {\partial^2 f} {\partial x \partial y}       | r = \map {\dfrac \partial {\partial x} } {\dfrac {\partial f} {\partial y} }       | rr= =: \map {f_{2 1} } {x, y}       | c =  }} {{eqn | n = 4       | l = \dfrac {\partial^2 f} {\partial y \partial x}       | r = \map {\dfrac \partial {\partial y} } {\dfrac {\partial f} {\partial x} }       | rr= =: \map {f_{1 2} } {x, y}       | c =  }} {{end-eqn}}
Definition:Primitive (Calculus)/Integration
The process of finding a primitive for a function is known as '''integration'''.
Definition:Bernoulli Numbers/Generating Function
:$\displaystyle \frac x {e^x - 1} = \sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!}$
Definition:Bernoulli Numbers/Recurrence Relation
:$B_n = \begin{cases} 1 & : n = 0 \\ \displaystyle - \sum_{k \mathop = 0}^{n - 1} \binom n k \frac {B_k} {n + 1 - k} & : n > 0 \end{cases}$ or equivalently: :$B_n = \begin{cases} 1 & : n = 0 \\ \displaystyle - \frac 1 {n+1} \sum_{k \mathop = 0}^{n - 1} \binom {n+1} k B_k & : n > 0 \end{cases}$
Definition:Logarithmic Integral/Eulerian
The '''Eulerian logarithmic integral''' is defined as: :$\displaystyle \map \Li x = \int_2^x \frac {\d t} {\map \ln t}$
Definition:Heaviside Step Function
Let $c \ge 0$ be a constant real number. The '''Heaviside step function on $c$''' is the real function $u_c: \R \to \R$ defined as: :$\map {u_c} t := \begin{cases} 1 & : t > c \\ 0 & : t < c \end{cases}$ If $c = 0$, the subscript is often omitted: :$\map u t := \begin{cases} 1 & : t > 0 \\ 0 & : t < 0 \end{cases}$
Definition:Doubly Periodic Function
Let $f: \C \to \C$ be a complex function.  Then $f \left({z}\right)$ is a '''doubly-periodic function''' if there exist $\omega_1, \omega_2 \in \C$ such that: : $(1): \quad \omega_1, \omega_2 \ne 0$ : $(2): \quad \dfrac {\omega_1} {\omega_2} \notin \R$ : $(3): \quad \forall z \in \C: f \left({z}\right) = f \left({z + \omega_1}\right) = f \left({z + \omega_2}\right)$
Definition:Elliptic Function
Let $\displaystyle y \left({x}\right) = \int_0^x \dfrac {\d t} {\sqrt {P \left({t}\right)} }$ be an elliptic integral, where $P \left({t}\right)$ is a polynomial of degree $3$ or $4$. Consider the inverse of $y \left({x}\right)$: :$x = \phi \left({y}\right)$ Then $\phi$ is an '''elliptic function'''.
Definition:Quadratic Function
A '''quadratic function''' is an expression of the form: :$\map Q x := a_0 + a_1 x + a_2 x^2$ where $a_0, a_1, a_2$ are constants.
Definition:Exponential Order/Real Index
Let $e^{a t}$ be the exponential function, where $a \in \R$ is constant. Then $\map f t$ is said to be of '''exponential order''' $a$, denoted $f \in \mathcal E_a$,  there exist strictly positive real numbers $M, K$ such that: :$\forall t \ge M: \size {\map f t} < K e^{a t}$
Definition:Minimal Infinite Successor Set
=== Definition 1 === {{:Definition:Minimal Infinite Successor Set/Definition 1}} === Definition 2 === {{:Definition:Minimal Infinite Successor Set/Definition 2}} === Definition 3 === {{:Definition:Minimal Infinite Successor Set/Definition 3}}
Definition:Step Function
A real function $f: \R \to \R$ is a '''step function'''  it can be expressed as a finite linear combination of the form: :$\map f x = \lambda_1 \chi_{\mathbb I_1} + \lambda_2 \chi_{\mathbb I_2} + \cdots + \lambda_n \chi_{\mathbb I_n}$ where: :$\lambda_1, \lambda_2, \ldots, \lambda_n$ are real constants :$\mathbb I_1, \mathbb I_2, \ldots, \mathbb I_n$ are open intervals, where these intervals partition $\R$ (except for the endpoints) :$\chi_{\mathbb I_1}, \chi_{\mathbb I_2}, \ldots, \chi_{\mathbb I_n}$ are characteristic functions of $\mathbb I_1, \mathbb I_2, \ldots, \mathbb I_n$.
Definition:Limit Inferior of Sequence of Sets/Notation
The limit inferior of $E_n$ can also be seen denoted as: :$\displaystyle {\underline {\lim} }_{n \mathop \to \infty} \ E_n$ but this notation is not used on  because it does not render well. Some sources merely present this as: :$\ds \underline \lim E_n$
Definition:Limit Superior of Sequence of Sets/Notation
The limit superior of $E_n$ can also be seen denoted as: :$\ds \overline {\lim}_{n \mathop \to \infty} \ E_n$ but this notation is not used on  because it does not render well. Some sources merely present this as: :$\ds \overline \lim E_n$
Definition:Gamma Function/Graph
The graph of the Gamma function is illustrated here for real arguments. <!--thumbright400pxThe Gamma function: $\Gamma \left({z}\right)$ (red solid line) and $\dfrac 1 {\Gamma\left({z}\right)}$ (blue broken line)--> :800pxThe Gamma function: $\Gamma \left({z}\right)$ (red solid line) and $\dfrac 1 {\Gamma\left({z}\right)}$ (blue broken line) The Gamma function: :$\Gamma \left({z}\right)$ (red solid line) :$\dfrac 1 {\Gamma \left({z}\right)}$ (blue broken line)
Definition:O Notation/Big-O Notation/Implied Constant
From the definition of the limit of a function, it can be seen that this is also equivalent to: :$\exists c \in \R: c > 0, k \ge 0: \forall n > k, f \left({n}\right) \le c g \left({n}\right)$ For some fixed $k$ (appropriate to the function under consideration) the infimum of such $c$ is called the '''implied constant'''.
Definition:Arithmetic Series
An '''arithmetic series''' is a series whose underlying sequence is an arithmetic sequence: {{begin-eqn}} {{eqn | l = S_n       | r = \sum_{k \mathop = 0}^{n - 1} a + k d       | c =  }} {{eqn | r = a + \paren {a + d} + \paren {a + 2 d} + \cdots + \paren {a + \paren {n - 1} d}       | c =  }} {{end-eqn}}
Definition:Arithmetic Sequence/Common Difference
The term $d$ is the '''common difference''' of $\sequence {a_k}$.
Definition:Arithmetic-Geometric Sequence
An '''arithmetic-geometric sequence''' is a sequence $\sequence {a_k}$ in $\R$ defined as: :$a_k = \paren {a_0 + k d} r^k$ for $k = 0, 1, 2, \ldots$ Thus its general form is: :$a_0, \paren {a_0 + d} r, \paren {a_0 + 2 d} r^2, \paren {a_0 + 3 d} r^3, \ldots$
Definition:Arithmetic-Geometric Series
An '''arithmetic-geometric series''' is a series whose underlying sequence is an arithmetic-geometric sequence: {{begin-eqn}} {{eqn | l = S_n       | r = \sum_{k \mathop = 0}^{n - 1} \paren {a + k d} r^k       | c =  }} {{eqn | r = a + \paren {a + d} r + \paren {a + 2 d} r^2 + \cdots + \paren {a + \paren {n - 1} d}r^{n-1}       | c =  }} {{end-eqn}}
Definition:Arithmetic Sequence/Last Term
The term $a_{n-1} = a_0 + \paren {n - 1} d$ is the '''last term''' of $\sequence {a_k}$.
Definition:Geometric Mean/Mean Proportional
In the language of {{AuthorRef|Euclid}}, the geometric mean of two magnitudes is called the '''mean proportional'''. Thus the '''mean proportional''' of $a$ and $b$ is defined as that magnitude $c$ such that: :$a : c = c : b$ where $a : c$ denotes the ratio between $a$ and $c$.
Definition:SI Units/Base Units
{| class="wikitable" style="margin:1em auto 1em auto" |+ SI base units |- !Name ! Unit symbol ! Dimension ! Symbol |- ! metre | $\mathrm m$ | $\mathsf L$: Length | $l$ |- ! kilogram | $\mathrm {kg}$ | $\mathsf M$: Mass | $m$ |- ! second | $\mathrm s$ | $\mathsf T$: Time | $t$ |- ! ampere | $\mathrm A$ | $\mathsf I$: Electric Current | $I$ |- ! kelvin | $\mathrm K$ | $\Theta$: Temperature | $T$ |- ! candela | $\mathrm {cd}$ | $\mathsf J$: Luminous Intensity | $I_v$ |- ! mole | $\mathrm {mol}$ | $\mathsf N$: Amount of Substance | $n$ |- |}
Definition:Bernoulli Numbers/Archaic Form
A different definition of the Bernoulli numbers can be found in older literature. Usually denoted with the symbol ${B_n}^*$, they are considered archaic, and will not be used on . === Definition 1 === {{:Definition:Bernoulli Numbers/Archaic Form/Definition 1}} === Definition 2 === {{:Definition:Bernoulli Numbers/Archaic Form/Definition 2}}
Definition:Planar Graph/Face
The '''faces''' of a planar graph are the areas which are surrounded by edges. In the above, the '''faces''' are $ABHC$, $CEGH$, $ACD$, $CDFE$ and $ADFEGHIHB$. === Incident === {{:Definition:Incident (Graph Theory)/Planar Graph}} === Adjacent === {{:Definition:Adjacent (Graph Theory)/Faces}} In the above diagram, $ABHC$ and $ACD$ are adjacent, but $ABHC$ and $CDFE$ are ''not'' adjacent.
Definition:Tangent Map/Affine Transformation
Let $\mathcal E$ and $\mathcal F$ be affine spaces with difference spaces $E$ and $F$ respectively. Let $\mathcal L : \mathcal E \to \mathcal F$ be an affine transformation. The associated linear transformation $L: E \to F$ of difference spaces is called the '''tangent map''' of $\mathcal L$.
Definition:Cube
=== Geometry === {{:Definition:Cube/Geometry}} === Algebra === {{:Definition:Cube/Algebra}}
Definition:Cube/Algebra
The '''cube''' of a quantity is the third power of that quantity.
Definition:Baire Space (Topology)/Definition 1
$T$ is a '''Baire space'''  the union of any countable set of closed sets of $T$ whose interiors are empty also has an empty interior.
Definition:Baire Space (Topology)/Definition 2
$T$ is a '''Baire space'''  the intersection of any countable set of open sets of $T$ which are everywhere dense is everywhere dense.
Definition:Baire Space (Topology)/Definition 3
$T$ is a '''Baire space'''  the interior of the union of any countable set of closed sets of $T$ which are nowhere dense is empty.
Definition:Baire Space (Topology)/Definition 4
$T$ is a '''Baire space''' , whenever the union of any countable set of closed sets of $T$ has an interior point, then one of those closed sets must have an interior point.
Definition:Lychrel Number
A '''Lychrel number''' is a natural number which cannot form a palindromic number through repeated iteration of the reverse-and add process.
Definition:Angle/Types
Angles can be divided into categories: === Zero Angle === {{:Definition:Zero Angle}} === Acute Angle === {{:Definition:Acute Angle}} === Right Angle === {{:Definition:Right Angle}} === Obtuse Angle === {{:Definition:Obtuse Angle}} === Straight Angle === {{:Definition:Straight Angle}} === Reflex Angle === {{:Definition:Reflex Angle}} === Full Angle === {{:Definition:Full Angle}} It is possible to consider angles outside the range $\closedint {0 \degrees} {360 \degrees}$, that is, $\closedint 0 {2 \pi}$. However, in geometric contexts it is usually preferable to convert these to angles inside this range by adding or subtracting multiples of a full angle.