hcju/proofwikips huggingface dataset:
Definition:Sufficient Statistic
Let $X_1, X_2, \ldots, X_n$ form a random sample from a population indexed by a parameter $\theta$. Let $T$ be a sample statistic. Let $I = \Img {\map T {X_1, X_2, \ldots, X_n} }$. Let $D$ be the conditional joint distribution of $X_1, X_2, \ldots, X_n$ given $T = t$ and $\theta$.   We call $T$ a '''sufficient statistic''' for $\theta$ if $D$ is independent of the value of $\theta$ for all $t \in I$.
Definition:Error/Absolute
The '''absolute error''' $\varepsilon$ is the difference between $x$ and a $X$, and can be defined in one of three ways: {{begin-eqn}} {{eqn | n = 1       | l = \varepsilon       | o = :=       | r = X - x }} {{eqn | n = 2       | l = \varepsilon       | o = :=       | r = x - X }} {{eqn | n = 3       | l = \varepsilon       | o = :=       | r = \size {X - x} }} {{end-eqn}} where $\size {X - x}$ denotes the absolute value of $X - x$. Different sources use different conventions.
Definition:Error/Relative
The '''relative error''' in $x$ is defined as: :$\dfrac {\size {X - x} } X$ :where $\size {X - x}$ denotes the absolute value of $X - x$.
Definition:Abstraction
'''Abstraction''' is the process of making a general statement from a collection of particular statements so as to summarise a property that applies to the subject of those particular statements.
Definition:Imperial/Area/Square Mile
One '''square mile''' is equal to a square whose side measures $1$ mile. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''square mile''' }} {{eqn | r = 640       | c = acres }} {{end-eqn}}
Definition:Hectare
One '''hectare''' is equal to a square whose side measures $100$ metres. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''hectare''' }} {{eqn | r = 100       | c = ares }} {{eqn | r = 10 \, 000       | c = square metres }} {{eqn | o = \approx       | r = 0.4       | c = acres }} {{end-eqn}}
Definition:Angle/Adjacent/Also defined as
Some sources give that '''adjacent angles''' are the angles on a straight line at the intersection of that line and another. Under such a definition, $\angle \alpha$ and $\angle \beta$ are '''adjacent''' in the diagram below: :250px Such '''adjacent angles''' are seen to be supplementary.
Definition:The Algebra of Sets
Let $E$ be a universal set. The power set $\powerset E$, together with: :the binary operations union $\cup$ and intersection $\cap$ :the unary operation complement $\complement$ is referred to as '''the algebra of sets''' on $E$.
Definition:Absolute Geometry
'''Absolute geometry''' is the study of '''Euclidean geometry''' without the parallel postulate.
Definition:Normal Real Number
A real number $r$ is '''normal''' with respect to a number base $b$  its basis expansion in number base $b$ is such that: :no finite sequence of digits of $r$ of length $n$ occurs more frequently than any other such finite sequence of length $n$.
Definition:Disjunction/Disjunct
The substatements $p$ and $q$ are known as the '''disjuncts''', or the '''members of the disjunction'''.
Definition:Dirichlet Beta Function
The '''Dirichlet beta function''' $\beta$ is the complex function defined on the half-plane $\map \Re s > 0$ by the series:  :$\displaystyle \map \beta s = \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {\paren {2 n + 1}^s}$
Definition:Convergent Sequence/Metric Space/Definition 4
$\sequence {x_k}$ '''converges to the limit $l \in A$''' : :for every $\epsilon \in \R{>0}$, the open $\epsilon$-ball about $l$ contains all but finitely many of the $p_n$.
Definition:Alexandroff Extension of Real Number Line
The '''Alexandroff extension of the real number line''' $\R^*$ is defined as: :$\R^* := \R \cup \set \infty$ that is, the set of real numbers together with an element $\infty$ which is not in $\R$.
Definition:Möbius Transformation/Real Numbers
A '''Möbius transformation''' is a mapping $f: \R^* \to \R^*$ of the form: :$\map f x = \dfrac {a x + b} {c x + d}$ where: :$\R^*$ denotes the Alexandroff extension of the real number line :$a, b, c, d \in \R$ such that $a d - b c \ne 0$ We define: :$\map f {-\dfrac d c} = \infty$ if $c \ne 0$, and: :$\map f \infty = \begin{cases} \dfrac a c & : c \ne 0 \\ \infty & : c = 0 \end{cases}$
Definition:P-adic Expansion
Let $p$ be a prime number. A '''$p$-adic expansion''' is a power series in the rational numbers $\Q$ of the form: :$\displaystyle \sum_{n \mathop = m}^\infty d_n p^n$ where: :$m \in \Z_{\le 0}$ :$\forall n \in \Z_{\ge m}: d_n \in \N \mathop {\text{ and } } 0 \le d_n < p$ :$m < 0 \implies d_m \ne 0$
Definition:Composition of Mappings/Definition 2
The '''composite of $f_1$ and $f_2$''' is defined and denoted as: :$f_2 \circ f_1 := \set {\tuple {x, z} \in S_1 \times S_3: \tuple {\map {f_1} x, z} \in f_2}$
Definition:Composition of Mappings/Definition 1
The '''composite mapping''' $f_2 \circ f_1$ is defined as: :$\forall x \in S_1: \map {\paren {f_2 \circ f_1} } x := \map {f_2} {\map {f_1} x}$
Definition:Generated Ring Extension
Let $S$ be a commutative rings with unity. Let $R$ be a subring of $S$ with unity such that the unity of $R$ is the unity of $S$. That is, $S$ is a ring extension of $R$. Let $T \subseteq S$ be a subset of $S$.
Definition:Generated Ring Extension/Smallest Subring
The '''ring extension $R \sqbrk T$ generated by $T$''' is the smallest subring of $S$ containing $T$ and $R$, that is, the intersection of all subrings of $S$ containing $T$ and $R$. Thus $T$ is a '''generator''' of $R \sqbrk T$  $R \sqbrk T$ has no proper subring containing $T$ and $R$.
Definition:Generated Ring Extension/Evaluation of Polynomial Ring
Let $R \sqbrk {\set {X_t} }$ be the polynomial ring in $T$ variables $X_t$. Let $\operatorname {ev} : R \sqbrk {\set {X_t} } \to S$ be the evaluation homomorphism associated with the inclusion $T \hookrightarrow S$. The '''ring extension $R \sqbrk T$ generated by $T$''' is $\map {\operatorname {Img}} {\operatorname {ev}}$, the image  of $\operatorname {ev}$. $T$ is said to be a '''generator''' of $R \sqbrk T$.
Definition:Composition of Mappings/Commutative Diagram
The concept of '''composition of mappings''' can be illustrated by means of a commutative diagram. This diagram illustrates the specific example of $f_2 \circ f_1$: ::$\begin{xy}\xymatrix@+1em{  S_1   \ar[r]^*+{f_1}   \ar@{-->}[rd]_*[l]+{f_2 \mathop \circ f_1} &  S_2   \ar[d]^*+{f_2} \\ &  S_3 }\end{xy}$
Definition:Composition of Mappings/Binary Operation
Let $\sqbrk {S \to S}$ be the set of all mappings from a set $S$ to itself. Then the concept of '''composite mapping''' defines a binary operation on $\sqbrk {S \to S}$: :$\forall f, g \in \sqbrk {S \to S}: g \circ f = \set {\tuple {s, t}: s \in S, \tuple {f \paren s, t} \in g} \in \sqbrk {S \to S}$ Thus, for every pair $\tuple {f, g}$ of mappings in $\sqbrk {S \to S}$, the composition $g \circ f$ is another element of $\sqbrk {S \to S}$.
Definition:Composition of Mappings/Warning
Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$. If $\Dom {f_2} \ne \Cdm {f_1}$, then the '''composite mapping''' $f_2 \circ f_1$ is '''not defined'''. This definition is directly analogous to that of composition of relations owing to the fact that a mapping is a special kind of relation.
Definition:Composition of Mappings/Also known as
In the context of analysis, this is often found referred to as a '''function of a function''', which (according to some sources) makes set theorists wince, as it is technically defined as a '''function on the codomain of a function'''. Some sources call $f_2 \circ f_1$ the '''resultant of $f_1$ and $f_2$''' or the '''product of $f_1$ and $f_2$'''. Some authors write $f_2 \circ f_1$ as $f_2 f_1$. Some use the notation $f_2 \cdot f_1$ or $f_2 . f_1$. Some use the notation $f_2 \bigcirc f_1$. Others, particularly in books having ties with computer science, write $f_1; f_2$ or $f_1 f_2$ (note the reversal of order), which is read as '''(apply) $f_1$, then $f_2$'''.
Definition:Composition of Mappings/Definition 3
The '''composite of $f_1$ and $f_2$''' is defined and denoted as: :$f_2 \circ f_1 := \set {\tuple {x, z} \in S_1 \times S_3: \exists y \in S_2: \map {f_1} x = y \land \map {f_2} y = z}$
Definition:Cartesian Product/Cartesian Space/Family of Sets/Definition 1
The '''Cartesian space of $S$ indexed by $I$''' is the set of all families $\family {s_i}_{i \mathop \in I}$ with $s_i \in S$ for each $i \in I$: :$S_I := \displaystyle \prod_I S = \set {\family {s_i}_{i \mathop \in I}: s_i \in S}$
Definition:Cartesian Product/Cartesian Space/Family of Sets/Definition 2
The '''Cartesian space of $S$ indexed by $I$''' is defined and denoted as: :$\displaystyle S^I := \set {f: \paren {f: I \to S} \land \paren {\forall i \in I: \paren {\map f i \in S} } }$
Definition:Definition/If or Iff
It is a standard convention, when making a definition in mathematics, to use '''if''' to introduce the definiens, when in fact the intent is generally '''iff''', that is: ''''''. This convention is specifically '''not followed''' on , where the mandatory style is to use ''''''.
Definition:Characteristic Property of Set
Let $S$ be a set which is specified by means of such a propositional function: :$S = \set {a: \map P a}$ The open statement $\map P x$ is known as a '''characteristic property''' of $S$.
Definition:Fresnel Sine Integral Function
The '''Fresnel sine integral function''' is the real function $\R \to \R$ defined by:  :$\displaystyle \map {\operatorname S} x = \sqrt {\frac 2 \pi} \int_0^x \sin u^2 \rd u$
Definition:Fresnel Cosine Integral Function
The '''Fresnel cosine integral function''' is the real function $\R \to \R$ defined by:  :$\displaystyle \map {\operatorname C} x = \sqrt {\frac 2 \pi} \int_0^x \cos u^2 \rd u$
Definition:Ring of Mappings
Let $\struct {R, +, \circ}$ be a ring. Let $S$ be a set. Let $R^S$ be the set of all mappings from $S$ to $R$. The '''ring of mappings''' from $S$ to $R$ is the algebraic structure $\struct {R^S, +', \circ'}$ where $+'$ and $\circ'$ are the (pointwise) operations induced on $R^S$ by $+$ and $\circ$.
Definition:Hypergeometric Differential Equation
A '''hypergeometric differential equation''' is a second order ODE of the form: :$\displaystyle x \paren {1 - x} \frac {\d^2 y} {\d x^2} + \paren {c - \paren {a + b + 1} x} \frac {\d y} {\d x} - a b y = 0$
Definition:Legendre's Associated Differential Equation
'''Legendre's associated differential equation''' is a second order ODE of the form: :$\displaystyle \paren {1 - x^2} \frac {\d^2 y} {\d x^2} - 2 x \frac {\d y} {\d x} + \paren {n \paren {n + 1} - \frac {m^2} {1 - x^2} } y = 0$
Definition:Ring of Mappings/Zero
The zero of the ring of mappings is the constant mapping $f_0 : S \to R$ defined by: :$\quad \forall s \in S : \map {f_0} x = 0$ where $0$ is the zero in $R$
Definition:Ring of Mappings/Additive Inverse
The additive inverse in the ring of mappings is defined by: :$\forall f \in R^S : -f \in R^S : \forall s \in S : \map {\paren {-f} } x = -\map f x$
Definition:Ring of Mappings/Unity
Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1$.
Definition:Ring of Mappings/Units
Let $\struct {R, +, \circ}$ be a ring with unity $1$.
Definition:Ring of Mappings/Commutativity
Let $\struct {R, +, \circ}$ be a commutative ring.
Definition:Coproduct/Sets
Let $S_1$ and $S_2$ be sets. A '''coproduct''' $\struct {C, i_1, i_2}$ of $S_1$ and $S_2$ comprises a set $C$ together with mappings $i_1: S_1 \to C$, $i_2: S_2 \to C$ such that: :for all sets $X$ and mappings $f_1: S_1 \to X$ and $f_2: S_2 \to X$: ::there exists a unique mapping $h: C \to X$ such that: :::$h \circ i_1 = f_1$ :::$h \circ i_2 = f_2$ Hence: ::$\begin{xy} \xymatrix@L+2mu@+1em{ & C \ar@{-->}[dd]_*{h} & \\ S_1 \ar[ru]^*{i_1} \ar[rd]_*{f_1} & & S_2 \ar[lu]_*{i_2} \ar[ld]^*{f_2} \\ & X & }\end{xy}$ :is a commutative diagram.
Definition:Ring of Mappings/Pointwise Addition
Let $\struct {R, +, \circ}$ be a ring. Let $S$ be a set. Let $\struct {R^S, +', \circ'}$ be the ring of mappings from $S$ to $R$. <section notitle=true name=PointwiseAddition> The pointwise operation $+'$ induced by $+$ on the ring of mappings from $S$ to $R$ is called '''pointwise addition''' and is defined as: :$\forall f, g \in R^S: f +’ g \in R^S :$ :::$\forall s \in S : \map {\paren {f +’ g}} x = \map f x + \map g x$ </section>
Definition:Ring of Mappings/Pointwise Multiplication
Let $\struct {R, +, \circ}$ be a ring. Let $S$ be a set. Let $\struct {R^S, +', \circ'}$ be the ring of mappings from $S$ to $R$. <section notitle=true name=PointwiseMultiplication> The pointwise operation $\circ'$ induced by $\circ$ on the ring of mappings from $S$ to $R$ is called '''pointwise multiplication''' and is defined as: :$\forall f, g \in R^S: f \circ’ g \in R^S :$ :::$\forall s \in S : \map {\paren {f \circ’ g}} x = \map f x \circ \map g x$ </section>
Definition:Ring of Sequences/Pointwise Addition
Let $\struct {R, +, \circ}$ be a ring. Let $\struct {R^\N, +', \circ'}$ be the ring of sequences over $R$. <section notitle=true name=PointwiseAddition> The pointwise operation $+'$ induced by $+$ on the ring of sequences is called '''pointwise addition''' and is defined as: :$\forall \sequence {x_n}, \sequence {y_n} \in R^\N: \sequence {x_n} +' \sequence {y_n} = \sequence {x_n + y_n}$ </section>
Definition:Ring of Sequences/Pointwise Multiplication
Let $\struct {R, +, \circ}$ be a ring. Let $\struct {R^\N, +', \circ'}$ be the ring of sequences over $R$. <section notitle=true name=PointwiseMultiplication> The pointwise operation $\circ'$ induced by $\circ$ on the ring of sequences is called '''pointwise multiplication''' and is defined as: :$\forall \sequence {x_n}, \sequence {y_n} \in R^{\N}: \sequence {x_n} \circ' \sequence {y_n} = \sequence {x_n \circ y_n}$ </section>
Definition:Ring of Sequences/Zero
The zero of the ring of sequences is the constant sequence $\tuple {0, 0, 0, \dots}$, where $0$ is the zero of $R$.
Definition:Ring of Sequences/Unity
Let $\struct {R, +, \circ}$ be a ring with unity $1$.
Definition:Ring of Sequences/Commutativity
Let $\struct {R, +, \circ}$ be a commutative ring.
Definition:Ring of Sequences/Additive Inverse
The additive inverse in the ring of sequences is defined by: :$\forall \sequence {x_n} \in R^\N: -\sequence {x_n} = \sequence {-x_n}$
Definition:Ring of Sequences/Units
Let $\struct {R, +, \circ}$ be a ring with unity $1$.
Definition:Euclidean Number
A '''Euclid number''' is a perfect number which is even.
Definition:Friction/Coefficient
Let $B$ be a body at rest on a plane surface $S$ on which friction acts. Let $N$ be the normal reaction of $S$ on $B$. Let a force be applied to $B$ parallel to $S$. === Coefficient of Static Friction === {{:Definition:Friction/Coefficient/Static}} === Coefficient of Kinetic Friction === {{:Definition:Friction/Coefficient/Kinetic}} These '''coefficients of friction''' depend upon the materials out of which $B$ and $S$ are made. It is usual for $\mu_s$ to be greater than $\mu_k$.
Definition:Friction/Coefficient/Static
Let $F$ be the magnitude of that force in the limiting case when $B$ is just about to move. Then the '''coefficient of static friction''' is defined and denoted: :$\mu_s = \dfrac F N$
Definition:Friction/Coefficient/Kinetic
Let $F$ be the magnitude of that force needed to keep $B$ moving at a constant velocity. Then the '''coefficient of kinetic friction''' is defined and denoted: :$\mu_k = \dfrac F N$
Definition:Bimodule
Let $\struct {R, +_R, \times_R}$ and $\struct {S, +_S, \times_S}$ be rings. Let $\struct {G, +_G}$ be an abelian group. Let $\circ_R : R \times G \to G$ and $\circ_S : G \times S \to G$ be binary operations such that: :$(1): \quad \struct {G, +_G, \circ_R}$ is a left module :$(2): \quad \struct {G, +_G, \circ_S}$ is a right module :$(3): \quad \forall \lambda \in R: \forall \mu \in S: \forall x \in G: \paren {\lambda \circ_R x} \circ_S \mu = \lambda \circ_R \paren {x \circ_S \mu}$ Then $\struct {G, +_G, \circ_R, \circ_S}$ is a '''bimodule over $\tuple {R, S}$'''. If $\struct {S, +_S, \times_S} = \struct {R, +_R, \times_R}$ then a '''bimodule over $\tuple {R, R}$''' is simply called a '''bimodule over $R$'''
Definition:Weight (Physics)/Warning
There is a certain amount of confusion in the common mind between weight and mass. The latter is usually determined by measuring its weight. But while the mass of a body is (under normal circumstances) constant, its weight varies according to its position relative to the gravitational field it is in, and so is not a constant property of that body. However, under usual terrestrial conditions the gravitational field is more or less constant (any differences being detectable only by instruments). This means that the weight and mass of a body are commonly considered "the same". Thus a weighing machine, while indicating the mass of a body, does so by measuring its weight.
Definition:Weight (Physics)/Dimension
The dimension of '''weight''' is $M L T^{-2}$: mass times acceleration, that is, a force.
Definition:Whole Number
The phrase '''whole number''' is an ambiguous term which could mean either: :an integer :a natural number :a strictly positive integer. Hence it is recommended that a specific instance of the above is used instead.
Definition:Are
One '''are''' is equal to a square whose side measures $10$ metres. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''are''' }} {{eqn | r = 100       | c = square metres }} {{eqn | o = \approx       | r = 119 \cdot 60       | c = square yards }} {{end-eqn}}
Definition:Finite Difference Operator/Forward Difference
The '''forward difference operator''' on $f$ is defined as: :$\map {\Delta f} x := \map f {x + 1} - \map f x$
Definition:Finite Difference Operator/Backward Difference
The '''backward difference operator''' on $f$ is defined as: :$\map {\nabla f} x := \map f x - \map f {x - 1}$
Definition:Bernoulli Trial/Bernoulli Variable
Let $X$ be a discrete random variable whose sample space is $\Omega$ in such a Bernoulli trial. Then $X$ is known as a '''Bernoulli variable'''.
Definition:Borromean Rings
The '''Borromean rings''' is an arrangement of $3$ interlinked rings such that no $2$ of the rings are themselves linked. That is, if one of them is removed, then the other $2$ are unlinked. :600px
Definition:Basis Representation
Let $b \in \Z$ be an integer such that $b > 1$. Let $n \in \Z$ be an integer such that $n \ne 0$. The '''representation of $n$ to the base $b$''' is the unique string of digits: :$\pm \sqbrk {r_m r_{m - 1} \ldots r_2 r_1 r_0}_b$ where $\pm$ is: :the negative sign $-$  $n < 0$ :the positive sign $+$ (or omitted)  $n > 0$ If $n = 0$, then $n$ is represented simply as $0$.
Definition:Lagrangian Mechanics
'''Lagrangian Mechanics''' is a reformulation of mechanics, where the main object of study is the Lagrangian function.
Definition:Action of Physical System
Let $S$ be an integral functional. Suppose that equations of motion of a physical system are to be derived from $S$ by the Principle of Stationary Action. Then $S$ is the '''action''' of a '''physical system'''.
Definition:Lagrangian
Let $P$ be a physical system composed of $n \in \N$ particles. Let the real variable $t$ be the time of $P$. $\forall i \le n$ let $\map { {\mathbf x}_i} t$ be the position of the $i$th particle. Suppose, the action $S$ of $P$ is of the following form: :$\displaystyle S = \int_{t_1}^{t_2} L \rd t$ where $L$ is a mapping of (possibly) $t$, $\map {{\mathbf x}_i} t$ and their derivatives. Then $L$ is the '''Lagrangian''' of $P$. {{NamedforDef|Joseph Louis Lagrange|cat = Lagrange}}
Definition:Free Group/Definition 1
A group $G$ is a '''free group'''  it is isomorphic to the free group on some set.
Definition:Free Group/Definition 2
A group $G$ is a '''free group'''  it has a presentation of the form $\gen S$, where $S$ is a set. That is, it has a presentation without relators. In this context, '''free''' means '''free of non-trivial relations'''.
Definition:Standard Lagrangian
Let $P$ be a physical system composed of countable number of particles. Let $L$ be a Lagrangian of $P$. Let: :$L = T - V$ where $T$ is the kinetic energy and $V$ the potential energy of $P$. Then $L$ is the '''Standard Lagrangian''' of $P$. {{NamedforDef|Joseph Louis Lagrange|cat = Lagrange}}
Definition:Kinetic Energy of Classical Particle
Let $\MM$ be an $n$-dimensional Euclidean manifold. Let $P$ be a particle with an inertial mass $m_i$. Let $t$ be the time variable of $P$. Suppose the position of $P$ is a real differentiable $n$-dimensional vector-valued mapping $\mathbf x = \map {\mathbf x} t$. Then the '''kinetic energy''' of a '''classical particle''' $P$ is: :$T = \dfrac {m_i} 2 \paren {\dfrac {\d \mathbf x} {\d t} }^2$ where $\paren {\dfrac {\d \mathbf x} {\d t} }^2$ is the dot product of the vector $\dfrac {\d \mathbf x} {\d t}$ with itself.
Definition:Pedal Triangle
=== Pedal Triangle of Point with respect to Triangle === {{:Definition:Pedal Triangle/Point}} === Orthic (Pedal) Triangle === {{:Definition:Orthic Triangle}} {{:Definition:Orthic Triangle/Also known as}}
Definition:Orthic Triangle
Let $\triangle ABC$ be a triangle. Let $\triangle DEF$ be the triangle formed by the feet of the altitudes $AD$, $BC$ and $ED$ of $\triangle ABC$. :400px $\triangle DEF$ is known as the '''orthic triangle of $\triangle ABC$'''. That is, the '''orthic triangle of $\triangle ABC$''' is the pedal triangle of its orthocenter.
Definition:Pedal Triangle/Point
Let $\triangle ABC$ be a triangle. Let $P$ be a point in the plane of $\triangle ABC$. Let $PD$, $PE$ and $PF$ be perpendiculars dropped from $P$ to $BC$, $AC$ and $AB$ respectively. Let $\triangle DEF$ be the triangle formed by the feet of the perpendiculars $PD$, $PE$ and $PF$. :400px $\qquad$ 400px $\triangle DEF$ is known as the '''pedal triangle of $P$ with respect to $\triangle ABC$'''.
Definition:Total Energy of Particles
Let $P$ be a physical system composed of a countable number of particles. Suppose the kinetic and potential energy of $P$ is respectively $T$ and $U$. Denote: :$E = T + U$ Then $E$ is known as the '''total energy''' of $P$.
Definition:Annulus (Geometry)
An '''annulus''' is a plane figure whose boundary consists of $2$ concentric circles: :400px In the above diagram, the '''annulus''' is the colored area.
Definition:Annulus (Geometry)/Center
The '''center''' of an '''annulus''' is the common center of the $2$ concentric circles that form its boundary. In the above diagram, the '''center''' is the point $O$.
Definition:Annulus (Geometry)/Inner Radius
The '''inner radius''' of an '''annulus''' is the radius of the smaller of the $2$ concentric circles that form its boundary. In the above diagram, the '''inner radius''' is denoted $r$.
Definition:Annulus (Geometry)/Outer Radius
The '''outer radius''' of an '''annulus''' is the radius of the larger of the $2$ concentric circles that form its boundary. In the above diagram, the '''outer radius''' is denoted $R$.
Definition:Power of Element/Ring
Let $\struct {R, \circ, +}$ be a ring. Let $r \in R$. Let $n \in \Z_{>0}$ be the set of strictly positive integers. The '''$n$th power of $r$ in $R$''' is defined as the $n$th power of $r$ with respect to the semigroup $\struct {R, \circ}$: :$\forall n \in \Z_{>0}: r^n = \begin {cases} r & : n = 1 \\ r^{n - 1} \circ r & : n > 1  \end {cases}$ If $R$ is a ring with unity where $1_R$ is that unity, the definition extends to $n \in \Z_{\ge 0}$: :$\forall n \in \Z_{\ge 0}: r^n = \begin {cases} 1_R & : n = 0 \\ r^{n - 1} \circ r & : n > 0 \end {cases}$
Definition:Half-Range Fourier Series
=== Half-Range Fourier Cosine Series === {{:Definition:Half-Range Fourier Cosine Series/Formulation 1}} === Half-Range Fourier Sine Series === {{:Definition:Half-Range Fourier Sine Series/Formulation 1}}
Definition:Half-Plane/Edge
Let $H$ denote one of the half-planes into which $L$ divides $P$. Then $L$ is called the '''edge''' of $H$.
Definition:Complex Half-Plane
Let $\C$ denote the complex plane. A '''complex half-plane''' is a '''half-plane''' of $\C$ whose edge is an infinite straight line of $\C$.