hcju/proofwikips huggingface dataset:
Definition:Closed Ball/Normed Division Ring/Center
In $\map { {B_\epsilon}^-} a$, the value $a$ is referred to as the '''center''' of the closed $\epsilon$-ball.
Definition:Lattice Point
Let $\CC$ be a cartesian coordinate system of $n$ dimensions. Let $P = \tuple {a_1, a_2, \ldots, a_n}$ be a point in $\CC$. Then $P$ is a '''lattice point''' of $\CC$  $a_1, a_2, \ldots, a_n$ are all integers. === Rational Lattice Point === {{:Definition:Lattice Point/Rational}}
Definition:Intercept
Let $\mathcal C$ be a curve embedded in a cartesian plane. An '''intercept''' of $\mathcal C$ is a point where $\mathcal C$ intersects one of the coordinate axes. In particular: === $x$-Intercept === {{:Definition:Intercept/X-Intercept}} === $y$-Intercept === {{:Definition:Intercept/Y-Intercept}}
Definition:Intercept/X-Intercept
An '''$x$-intercept''' of $\mathcal C$ is a point where $\mathcal C$ intersects the $x$-axis.
Definition:Intercept/Y-Intercept
A '''$y$-intercept''' of $\mathcal C$ is a point where $\mathcal C$ intersects the $y$-axis.
Definition:Perpendicular Distance
Let $\mathcal L$ be a straight line. Let $P$ be a point not on $\mathcal L$. Let a perpendicular be dropped from $P$ to $\mathcal L$, intersecting $\mathcal L$ at $A$. The '''perpendicular distance''' from $P$ to $\mathcal L$ is defined as being the length of the line $AP$. :300px
Definition:Lowest Common Multiple/Integers/General Definition
Let $S = \set {a_1, a_2, \ldots, a_n} \subseteq \Z$ such that $\displaystyle \prod_{a \mathop \in S} a = 0$ (that is, all elements of $S$ are non-zero). Then the '''lowest common multiple''' of $S$: :$\lcm \paren S = \lcm \set {a_1, a_2, \ldots, a_n}$ is defined as the smallest $m \in \Z_{>0}$ such that: :$\forall x \in S: x \divides m$
Definition:Sphere/Metric Space
Let $M = \struct{A, d}$ be a metric space or pseudometric space. Let $a \in A$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. The '''$\epsilon$-sphere of $a$ in $M$''' is defined as: :$\map {S_\epsilon} a = \set {x \in A: \map d {x, a} = \epsilon}$
Definition:Sphere/Metric Space/Center
In $S_\epsilon \paren{a}$, the value $a$ is referred to as the '''center''' of the $\epsilon$-sphere.
Definition:Sphere/Metric Space/Radius
In $S_\epsilon \paren{a}$, the value $\epsilon$ is referred to as the '''radius''' of the $\epsilon$-sphere.
Definition:Sphere/Normed Division Ring
Let $\struct{R, \norm{\,\cdot\,}}$ be a normed division ring. Let $a \in R$. Let $\epsilon \in \R_{>0}$ be a strictly positive real number. The '''$\epsilon$-sphere of $a$ in $\struct{R, \norm{\,\cdot\,}}$''' is defined as: :$S_\epsilon \paren{a} = \set {x \in R: \norm{x - a} = \epsilon}$
Definition:Sphere/Normed Division Ring/Radius
In $\map {S_\epsilon} a$, the value $\epsilon$ is referred to as the '''radius''' of the $\epsilon$-sphere.
Definition:Sphere/Normed Division Ring/Center
In $\map {S_\epsilon} a$, the value $a$ is referred to as the '''center''' of the $\epsilon$-sphere.
Definition:Stellation/Polygon/Regular
A '''regular stellated polygon''' is a stellated polygon which is both equilateral and equiangular. That is, in which all the sides are the same length, and all the vertices have the same angle: :300px
Definition:Triskaidecagon
A '''triskaidecagon''' is a polygon with exactly $13$ sides. :300px === Regular Triskaidecagon === {{:Definition:Triskaidecagon/Regular}}
Definition:Topology Induced by Division Ring Norm
Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring whose norm is $\norm {\,\cdot\,}$. The '''topology on $\struct {R, \norm {\,\cdot\,} }$ induced by $\norm {\,\cdot\,}$''' is defined as the topology $\tau$ generated by the basis consisting of the set of all open balls of $\struct {R, \norm {\,\cdot\,} }$. Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$. By the definition of the metric induced by a norm on division ring it follows that the '''topology on the normed division ring $\struct{R, \norm{\,\cdot\,}}$ induced by (the norm) $\norm{\,\cdot\,}$''' is the topology induced by the metric d.
Definition:Complete Residue System
Let $m \in \Z_{\ne 0}$ be a non-zero integer. Let $S := \set {r_1, r_2, \dotsb, r_s}$ be a set of integers with the properties that: :$(1): \quad i \ne j \implies r_i \not \equiv r_j \pmod m$ :$(2): \quad \forall n \in \Z: \exists r_i \in S: n \equiv r_i \pmod m$ Then $S$ is a '''complete residue system modulo $m$'''.
Definition:Deck of Cards
A '''deck of cards''' is a set of pieces of (traditionally) cardboard with a design on one side uniquely identifying the card, and on the other side a single design which is exactly the same for all cards in the '''deck'''. === Card === {{:Definition:Deck of Cards/Card}} === Suit === {{:Definition:Deck of Cards/Suit}}
Definition:Deck of Cards/Card
Each of the elements of a deck of cards is a '''card'''.
Definition:Deck of Cards/Suit
The cards in a deck of cards are partitioned into a number of '''suits'''. The traditional '''deck of cards''' that is usually discussed in the context of probability theory is divided into $4$ '''suits''' of $13$ cards each. There may also be a small number (usually between $1$ and $3$) of extra cards, usually called '''jokers''', which do not belong to any '''suit'''.
Definition:Shuffle
A '''shuffle''' is an operation on a deck of cards which changes the order of the cards in the deck.
Definition:Order of Cards
Let $D$ be a deck of cards. The cards in the deck can be considered to be stacked uniformly, usually all in the same orientation, with a card on the top and a card on the bottom. The '''order''' of the cards is the total ordering $\prec$ defined as: :$a \prec b$  $a$ is somewhere underneath $b$
Definition:Modified Perfect Faro Shuffle
A '''modified perfect Faro shuffle''' is a shuffle performed on a deck of cards $D$ of size $2 n$. Cut $D$ into $2$ piles $D_1$ (the top half) and $D_2$ (the bottom half) of exactly $n$ cards each. The shuffled deck $D'$ is assembled by alternating the cards in $D$: :The cards in positions $2, 4, 6, \ldots, 2 n$ of $D'$ are to be occupied by the cards $1, 2, \ldots, n$ of $D_1$ in the same order :The cards in positions $1, 3, 5, \ldots, 2 n - 1$ of $D'$ are to be occupied by the cards $1, 2, \ldots, n$ of $D_2$ in the same order. Thus a card in position $k$ in $D$ ends up in position $2 k \bmod \paren {2 n + 1}$ in $D'$.
Definition:Fifth Root/Real
Let $x \in \R_{\ge 0}$ be a positive real number. The '''fifth root of $x$''' is the real number defined as: :$x^{\paren {1 / 5} } := \set {y \in \R: y^5 = x}$ where $x^{\paren {1 / 5} }$ is the $5$th root of $x$. The notation: :$y = \sqrt [5] x$ is usually encountered.
Definition:Affine Space/Weyl's Axioms
Let $K$ be a field. Let $\struct{V, +_V, \circ}$ be a vector space over $K$. Let $\EE$ be a set on which a mapping is defined: :$- : \EE \times \EE \to V$ satisfying the following associativity conditions: {{begin-axiom}} {{axiom | n = \text W 1         | q = \forall p \in \EE: \forall v \in V: \exists ! q \in \EE         | m = v = q - p }} {{axiom | n = \text W 2         | q = \forall p, q, r \in \EE         | m = \paren{r - q} +_V \paren{q - p} = r - p }} {{end-axiom}} Then the ordered pair $\tuple {\EE, -}$ is an '''affine space'''.
Definition:Affine Space/Group Action
Let $K$ be a field. Let $\left({V, +_V, \circ}\right)$ be a vector space over $K$. Let $\mathcal E$ be a set. Let $\phi: \mathcal E \times V \to \mathcal E$ be a free and transitive group action of $\struct{V, +_V}$ on $\mathcal E$. Then the ordered pair $\tuple{\mathcal E, \phi}$ is an '''affine space'''.
Definition:Affine Space/Associativity Axioms
Let $K$ be a field. Let $\struct {V, +_V, \circ}$ be a vector space over $K$. Let $\EE$ be a set on which two mappings are defined: :$+ : \EE \times V \to \EE$ :$- : \EE \times \EE \to V$ satisfying the following associativity conditions: {{begin-axiom}} {{axiom | n = \text A 1         | q = \forall p, q \in \EE         | m = p + \paren {q - p} = q }} {{axiom | n = \text A 2         | q = \forall p \in \EE: \forall u, v \in V         | m = \paren {p + u} + v = p + \paren {u +_V v} }} {{axiom | n = \text A 3         | q = \forall p, q \in \EE: \forall u \in V         | m = \paren {p - q} +_V u = \paren {p + u} - q }} {{end-axiom}} Then the ordered triple $\struct {\EE, +, -}$ is an '''affine space'''.
Definition:Tangent Space (Affine Geometry)
Let $\tuple {\EE, +, -}$ be an affine space. Let $V$ be the vector space that is the codomain of $-$. Then $V$ is the '''tangent space''' of $\EE$.
Definition:Vector (Affine Geometry)
Let $\EE$ be an affine space. Let $V$ be the tangent space of $\EE$. Any element $v$ of $V$ is called a '''vector'''.
Definition:Subtraction (Affine Geometry)
Let $\tuple {\EE, +, -}$ be an affine space. Then the mapping $-$ is called '''affine subtraction'''.
Definition:Addition (Affine Geometry)
Let $\tuple {\EE, +, -}$ be an affine space. Then the mapping $+$ is called '''affine addition'''.
Definition:Addition/Sum
Let $a + b$ denote the operation of addition on two objects $a$ and $b$. Then the result $a + b$ is referred to as the '''sum''' of $a$ and $b$.
Definition:Multiplication/Notation
There are several variants of the notation for '''multiplication''': :$n \times m$ This is usually used when numbers are under consideration, for example: $3 \times 5 = 15$. However, it can be used in the context of algebra where extra clarity is needed. :$n m$ This is most common in algebra, but not with numbers, as it is difficult to make it obvious where one number ends and the next number begins. :$\paren n \paren m$ This form can be used for either symbols denoting variables or numbers, for example: $\paren 3 \paren 4 = 12$. :$n \cdot m$ or $n . m$ These have their uses in algebra, but the dot has the danger of being confused with the decimal point when used for numbers. :$n * m$ This notation specifically evolved in the field of computer science, but can occasionally be seen encroaching into mathematics. Its use is not recommended, as it can be confused with other operations that use the same or similar notation, for example convolution.
Definition:Multiplication/Multiplicand
Let $a \times b$ denote the operation of multiplication on two objects. The object $b$ is known as the '''multiplicand of $a$'''. That is, it is the object which is to be multiplied by the multiplier.
Definition:Multiplication/Multiplier
Let $a \times b$ denote the operation of multiplication on two objects. The object $a$ is known as the '''multiplier of $b$'''. That is, it is the object which is to multiply the multiplicand.
Definition:Multiplication/Product
Let $a \times b$ denote the operation of multiplication on two objects $a$ and $b$. Then the result $a \times b$ is referred to as the '''product''' of $a$ and $b$.
Definition:Real Number/Real Number Line/Origin
The point representing the number $0$ (zero) is referred to as the '''origin'''.
Definition:Variable/Real
A '''real variable''' is a symbol which can stand for any one of a set of real numbers.
Definition:Complex Number/Definition 2/Scalar Product
Let $\tuple {x, y}$ be a complex numbers. Let $m \in \R$ be a real number. Then $m \tuple {x, y}$ is defined as: :$m \tuple {x, y} := \tuple {m x, m y}$
Definition:Complex Conjugate Coordinates
Let $P$ be a point in the complex plane. $P$ may be located using '''complex conjugate coordinates''' $\tuple {z, \overline z}$ based on: {{begin-eqn}} {{eqn | l = x       | r = \dfrac {z + \overline z} 2       | c = Sum of Complex Number with Conjugate }} {{eqn | l = y       | r = \dfrac {z - \overline z} {2 i}       | c = Difference of Complex Number with Conjugate }} {{end-eqn}} where $P = \tuple {x, y}$ is expressed in Cartesian coordinates.
Definition:Connected Set (Complex Analysis)/Definition 1
$D$ is '''connected'''  every pair of points in $D$ can be joined by a staircase contour.
Definition:Connected Set (Complex Analysis)/Definition 2
$D$ is '''connected'''  every pair of points in $D$ can be joined by a polygonal path all points of which are in $D$.
Definition:Polygonal Path
Let $\C$ denote the complex plane. A '''polygonal path''' in $\C$ is a contour in $\C$ which is composed of a finite number of straight line segments.
Definition:Valuation Ring Induced by Non-Archimedean Norm
Let $\struct {R, \norm{\,\cdot\,}}$ be a non-Archimedean normed division ring with zero $0_R$. The '''valuation ring induced by the non-Archimedean norm''' $\norm{\,\cdot\,}$ is the set: :$\mathcal O = \set {x \in R: \norm x \le 1}$ That is, the '''valuation ring induced by the non-Archimedean norm''' $\norm{\,\cdot\,}$ is the closed ball ${B_1}^- \paren {0_R}$.
Definition:Valuation Ideal Induced by Non-Archimedean Norm
Let $\struct {R, \norm{\,\cdot\,}}$ be a non-Archimedean normed division ring with zero $0_R$. The '''valuation ideal induced by the non-Archimedean norm''' $\norm{\,\cdot\,}$ is the set: :$\mathcal P = \set{x \in R: \norm{x} \lt 1}$ That is, the '''valuation ideal induced by the non-Archimedean norm''' $\norm{\,\cdot\,}$ is the open ball $B_1 \paren {0_R}$.
Definition:Residue Division Ring Induced by Non-Archimedean Norm
Let $\struct {R, \norm {\,\cdot\,}}$ be a non-Archimedean normed division ring. Let $\OO$ be the valuation ring induced by the non-Archimedean norm $\norm {\,\cdot\,}$. Let $\PP$ be the valuation ideal induced by the non-Archimedean norm $\norm {\,\cdot\,}$. The '''residue division ring induced by the norm''' $\norm {\,\cdot\,}$ is the quotient ring $\OO / \PP$. If $R$ is a field then the quotient ring $\OO / \PP$ is called the '''residue field induced by the norm''' $\norm {\,\cdot\,}$.
Definition:P-adic Integers
Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the $p$-adic numbers $\Q_p$ for some prime $p$. The set of '''$p$-adic integers''', denoted $\Z_p$, is the valuation ring induced by $\norm {\,\cdot\,}_p$, that is: :$\Z_p = \set {x \in \Q_p: \norm x_p \le 1}$
Definition:Coincident Straight Lines
Two straight lines are '''coincident'''  they lie on top of each other. That is, two straight lines $L_1$ and $L_2$ are '''coincident''' : :$L_1 \cap L_2 = L_1 = L_2$.
Definition:Equilibrant
Let $\mathbf F_1, \mathbf F_2, \ldots, \mathbf F_n$ be a set of $n$ forces acting on a particle $B$ at a point $P$ in space. The '''equilibrant''' of $\mathbf F_1, \mathbf F_2, \ldots, \mathbf F_n$ is defined as the force which is needed to prevent $B$ from moving.
Definition:Lemniscate of Bernoulli/Major Axis
The line $P_1 P_2$ is the '''major axis''' of the lemniscate.
Definition:Lemniscate of Bernoulli/Major Semiaxis
Each of the lines $O P_1$ and $O P_2$ is a '''major semiaxis''' of the lemniscate.
Definition:Division Ring/Definition 2
:Every non-zero element of $R$ is a unit.
Definition:Ring of Quaternions
The '''ring of quaternions''' $\struct {\H, +, \times}$ is the set of quaternions under the two operations of quaternion addition and quaternion multiplication.
Definition:Subfield/Field
Let $\struct {F, +, \circ}$ be a field. Let $K$ be a subset of $F$ such that $\struct {K, +, \circ}$ is also a field. Then $\struct {K, +, \circ}$ is a '''subfield''' of $\struct {F, +, \circ}$.
Definition:Subfield/Ring
Let $\struct {R, +, \circ}$ be a ring with unity. Let $K$ be a subset of $R$ such that $\struct {K, +, \circ}$ is a field. Then $\struct {K, +, \circ}$ is a '''subfield''' of $\struct {R, +, \circ}$.
Definition:Subfield/Proper Subfield
Let $\struct {K, +, \circ}$ be a subfield of $\struct {F, +, \circ}$. Then $\struct {K, +, \circ}$ is a '''proper subfield''' of $\struct {F, +, \circ}$  $K \ne F$. That is, $\struct {K, +, \circ}$ is a '''proper subfield''' of $\struct {F, +, \circ}$ : :$(1): \quad \struct {K, +, \circ}$ is a subfield of $\struct {F, +, \circ}$ :$(2): \quad K$ is a proper subset of $F$.
Definition:Periodic Function/Periodic Element
Let $L \in X_{\ne 0}$. Then $L$ is a '''periodic element''' of $f$ : :$\forall x \in X: \map f x = \map f {x + L}$
Definition:Trivial Factorization/Non-Trivial Factorization
A factorization in $\struct {D, +, \circ}$ of the form $x = z \circ y$, where neither $y$ nor $z$ is a unit of $D$, is called a '''non-trivial factorization'''.
Definition:Associate/Integral Domain/Definition 3
$x$ and $y$ are '''associates (in $D$)'''  there exists a unit $u$ of $\struct {D, +, \circ}$ such that: :$y = u \circ x$ and consequently: :$x = u^{-1} \circ y$ That is,  $x$ and $y$ are unit multiples of each other.
Definition:Cauchy 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 '''Cauchy distribution''' if it has probability density function: :$\map {f_X} x = \dfrac 1 {\pi \gamma \paren {1 + \paren {\frac {x - x_0} \gamma}^2} }$ for some $\gamma > 0$.  This is written:  :$X \sim \Cauchy {x_0} \gamma$
Definition:Antiperiodic Function
=== Antiperiodic Real Function === {{:Definition:Antiperiodic Function/Real}} === Antiperiodic Complex Function === {{:Definition:Antiperiodic Function/Complex}}
Definition:Antiperiodic Function/Antiperiodic Element
Let $L \in X_{\ne 0}$. Then $L$ is an '''anti-periodic element''' of $f$ : : $\forall x \in X: - \map f x = \map f {x + L}$
Definition:Antiperiodic Function/Antiperiod
The '''antiperiod''' of $f$ is the smallest value $\cmod L \in \R_{\ne 0}$ such that: : $\forall x \in X: - \map f x = \map f {x + L}$ where $\cmod L$ is the modulus of $L$.
Definition:Skewness
'''Skewness''' is a measure of the asymmetry of a probability distribution about its mean. Let $X$ be a random variable with mean $\mu$ and standard deviation $\sigma$. Then the '''skewness''' of $X$, usually denoted $\gamma_1$, is defined as:  :$\gamma_1 = \expect {\paren {\dfrac {X - \mu} \sigma}^3}$ where $\expect X$ denotes the expectation of $X$.
Definition:Uniform Distribution
=== Discrete Uniform Distribution === {{:Definition:Discrete Uniform Distribution}} === Continuous Uniform Distribution === {{:Definition:Continuous Uniform Distribution}}
Definition:Prime Ideal of Ring/Commutative and Unitary Ring/Definition 1
A '''prime ideal''' of $R$ is a proper ideal $P$ such that: :$\forall a, b \in R : a \circ b \in P \implies a \in P$ or $b \in P$
Definition:Prime Ideal of Ring/Commutative and Unitary Ring/Definition 2
A '''prime ideal''' of $R$ is a proper ideal $P$ of $R$ such that: :$I \circ J \subseteq P \implies I \subseteq P \text { or } J \subseteq P$ for all ideals $I$ and $J$ of $R$.
Definition:Prime Ideal of Ring/Commutative and Unitary Ring
=== Definition 1 === {{:Definition:Prime Ideal of Commutative and Unitary Ring/Definition 1}} === Definition 2 === {{:Definition:Prime Ideal of Commutative and Unitary Ring/Definition 2}} === Definition 3 === {{:Definition:Prime Ideal of Commutative and Unitary Ring/Definition 3}}
Definition:Prime Ideal of Ring/Commutative and Unitary Ring/Definition 3
A '''prime ideal''' of $R$ is a proper ideal $P$ of $R$ such that: :the complement $R \setminus P$ of $P$ in $R$ is closed under the ring product $\circ$.
Definition:Moment (Probability Theory)
Let $X$ be a random variable on some probability space.  Let $a$ be a real number. Then the '''$n$th moment of $X$ about $a$''', usually denoted $\map {\mu_n} a$, is defined as:  :$\map {\mu_n} a = \expect {\paren {X - a}^n}$ where $\expect X$ denotes the expectation of $X$.
Definition:Raw Moment
Let $X$ be a random variable on some probability space.  Then the '''$n$th raw moment of $X$''', usually denoted $\mu'_n$, is defined as:  :$\mu'_n = \expect {X^n}$ where $\expect X$ denotes the expectation of $X$. That is, the $n$th raw moment of $X$ is its $n$th moment about $0$.  The first raw moment of $X$ is the mean of $X$, and is instead usually denoted $\mu$.
Definition:Central Moment
Let $X$ be a random variable on some probability space with mean $\mu$.  Then the '''$n$th central moment of $X$''', usually denoted $\mu_n$, is defined as:  :$\mu_n = \expect {\paren {X - \mu}^n}$ where $\expect X$ denotes the expectation of $X$. That is, the $n$th central moment of $X$ is its $n$th moment about $\mu$.  The second central moment of $X$ is the variance of $X$, and is instead usually denoted $\sigma^2$.
Definition:Norm Axioms
=== Multiplicative Norm Axioms === Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\norm {\, \cdot \,}: R \to \R_{\ge 0}$ be a multiplicative norm on $R$. The '''multiplicative norm axioms''' are the conditions on $\norm {\, \cdot \,}$ which are satisfied for all elements of $R$ in order for $\norm {\, \cdot \,}$ to be a multiplicative norm: {{:Definition:Multiplicative Norm Axioms}} When the concept of '''norm axioms''' is raised without qualification, it is usually the case that '''multiplicative norm axioms''' are under discussion. === Submultiplicative Norm Axioms === Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$. Let $\norm {\, \cdot \,}: R \to \R_{\ge 0}$ be a submultiplicative norm on $R$. The '''submultiplicative norm axioms''' are the conditions on $\norm {\, \cdot \,}$ which are satisfied for all elements of $R$ in order for $\norm {\, \cdot \,}$ to b a submultiplicative norm: {{:Definition:Submultiplicative Norm Axioms}} === Norm Axioms (Vector Space) === Let $\struct {R, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_R$. Let $V$ be a vector space over $R$, with zero $\mathbf 0_V$. Let $\norm {\,\cdot\,}: V \to \R_{\ge 0}$ be a norm on $V$. The '''norm axioms''' are the following conditions on $\norm {\,\cdot\,}$ which define $\norm {\,\cdot\,}$ as being a norm: {{:Definition:Norm Axioms (Vector Space)}}
Definition:Submultiplicative Norm Axioms
{{begin-axiom}} {{axiom | n = \text N 1         | lc= Positive Definiteness:         | q = \forall x \in R         | ml= \norm x = 0         | mo= \iff         | mr= x = 0_R }} {{axiom | n = \text N 2         | lc= Submultiplicativity:         | q = \forall x, y \in R         | ml= \norm {x \circ y}         | mo= \le         | mr= \norm x \times \norm y }} {{axiom | n = \text N 3         | lc= Triangle Inequality:         | q = \forall x, y \in R         | ml= \norm {x + y}         | mo= \le         | mr= \norm x + \norm y }} {{end-axiom}}
Definition:Multiplicative Function on Ring
Let $\struct {R, +, \circ}$ be a ring. Let $f: R \to \R$ be a (real-valued) function on $R$. $f$ is a '''multiplicative function on $R$''' : :$\forall x, y \in R: \map f {x \circ y} = \map f x \times \map f y$
Definition:Standardized Moment
Let $X$ be a random variable on some probability space with standard deviation $\sigma$.  Then the '''$n$th standardized moment of $X$''', usually denoted $\alpha_n$, is defined as:  :$\alpha_n = \dfrac {\mu_n} {\sigma^n}$ where $\mu_n$ is the $n$th central moment of $X$. The third standardized moment of $X$ is the skewness of $X$, and is instead usually denoted $\gamma_1$.  The fourth standardized moment of $X$ is the kurtosis of $X$.
Definition:Cyclotomic Ring
Let $\Z \sqbrk {i \sqrt n}$ be the set $\set {a + i b \sqrt n: a, b \in \Z}$. The algebraic structure $\struct {\Z \sqbrk {i \sqrt n}, +, \times}$ is the '''$n$th cyclotomic ring'''.
Definition:Kurtosis
Let $X$ be a random variable with mean $\mu$ and standard deviation $\sigma$. Then the '''kurtosis''' of $X$, usually denoted $\alpha_4$, is defined as:  :$\alpha_4 = \expect {\paren {\dfrac {X - \mu} \sigma}^4}$ where $\expect X$ denotes the expectation of $X$. That is, the '''kurtosis''' of $X$ is the fourth standardized moment of $X$.
Definition:Constant Term of Polynomial
Let $R$ be a commutative ring with unity. Let $P \in R \sqbrk X$ be a nonzero polynomial over $R$: :$\displaystyle f = \sum_{k \mathop = 0}^n a_k \circ x^k$ where $n$ is the degree of $P$. The '''constant term''' of $P$ is the coefficient of $a_0$.
Definition:Ring of Eisenstein Integers
The '''ring of Eisenstein integers''' $\struct {\Z \sqbrk \omega, +, \times}$ is the algebraic structure formed from: :the set of Eisenstein integers $\Z \sqbrk \omega$, where $\omega = e^{2 \pi i / 3}$ :the operation of complex addition :the operation of complex multiplication.