hcju/proofwikips huggingface dataset:
Definition:Rational Point in Plane
Consider the Cartesian plane $\R^2$. The points $\tuple {x, y} \in \R^2$ such that $x, y \in \Q$ are called the '''rational points (of $\R^2$)'''.
Definition:Linear Polynomial
A '''linear polynomial''' is a polynomial whose degree is $1$.
Definition:Algebraic Number/Degree
Let $\alpha$ be an algebraic number. By definition, $\alpha$ is the root of at least one polynomial $P_n$ with rational coefficients. The '''degree''' of $\alpha$ is the degree of the minimal polynomial $P_n$ whose coefficients are all in $\Q$.
Definition:Gelfond-Schneider Constant
The '''Gelfond-Schneider constant''' is defined as $2$ raised to the power of the square root of $2$: :$2^{\sqrt 2}$ Its decimal expansion begins: :$2^{\sqrt 2} \approx 2 \cdotp 66514 \, 4$ {{OEIS|A007507}} {{NamedforDef|Alexander Osipovich Gelfond|name2 = Theodor Schneider|cat = Gelfond|cat2 = Schneider}}
Definition:Bernoulli Numbers/Sequence
The sequence of '''Bernoulli numbers''' begins: {{begin-eqn}} {{eqn | l = B_0       | r = 1       | c =  }} {{eqn | l = B_1       | ro= -       | r = \dfrac 1 2       | c =  }} {{eqn | l = B_2       | r = \dfrac 1 6       | c =  }} {{eqn | l = B_4       | ro= -       | r = \dfrac 1 {30}       | c =  }} {{eqn | l = B_6       | r = \dfrac 1 {42}       | c =  }} {{eqn | l = B_8       | ro= -       | r = \dfrac 1 {30}       | c =  }} {{eqn | l = B_{10}       | r = \dfrac 5 {66}       | c =  }} {{eqn | l = B_{12}       | ro= -       | r = \dfrac {691} {2730}       | c =  }} {{end-eqn}} The odd index Bernoulli numbers, apart from $B_1$, are all equal to $0$.
Definition:Partial Fractions Expansion
Let $\map R x = \dfrac {\map P x} {\map Q x}$ be a rational function, where $\map P x$ and $\map Q x$ are expressible as polynomial functions. Let $\map Q x$ be expressible as: :$\map Q x = \displaystyle \prod_{k \mathop = 1}^n \map {q_k} x$ where the $\map {q_k} x$ are themselves polynomial functions of degree at least $1$. Let $\map R x$ be expressible as: :$\map R x = \map r x \displaystyle \sum_{k \mathop = 0}^n \dfrac {\map {p_k} x} {\map {q_k} x}$ where: :$\map r x$ is a polynomial function which may or may not be the null polynomial, or be of degree $0$ (that is, a constant) :each of the $\map {p_k} x$ are polynomial functions :the degree of $\map {p_k} x$ is strictly less than the degree of $\map {q_k} x$ for all $k$. Then $\map r x \displaystyle \sum_{k \mathop = 0}^n \dfrac {\map {p_k} x} {\map {q_k} x}$ is a '''partial fractions expansion''' of $\map R x$.
Definition:Hypocycloid
Let a circle $C_1$ roll around the inside of another (larger) circle $C_2$. The locus of a given point on the circumference of $C_1$ is known as a '''hypocycloid'''.
Definition:Epicycloid
Let a circle $C_1$ roll around the outside of another circle $C_2$. The locus of a given point on the circumference of $C_1$ is known as an '''epicycloid'''.
Definition:Astroid
An '''astroid''' is a hypocycloid with $4$ cusps. :400px
Definition:Circular Error of Pendulum
The period of a pendulum $P$ is dependent upon the angle from the vertical from which the bob is released. That period of $P$ is described by Motion of Pendulum as being: :$T = 2 \sqrt {\dfrac a g} \, \map K {\map \sin {\dfrac \alpha 2} }$ where: :$a$ is the length of the rod of $P$ :$\map K k$ is the complete elliptic integral of the first kind. The '''circular error''' of $P$ is the difference between $T$ and its approximate period as described by period Approximate Motion of Pendulum: :$T' = 2 \pi \sqrt {\dfrac a g}$
Definition:Tusi Couple
A '''Tusi couple''' is a hypocycloid with $2$ cusps. :400px
Definition:Hypocycloid/Generator
The circles $C_1$ and $C_2$ can be referred to as the '''generators''' of the hypocycloid. === Stator === {{Definition:Hypocycloid/Generator/Stator}} === Rotor === {{Definition:Hypocycloid/Generator/Rotor}}
Definition:Hypocycloid/Generator/Rotor
The circle $C_1$ can be referred to as the '''rotor''' of the hypocycloid.
Definition:Hypocycloid/Generator/Stator
The circle $C_2$ can be referred to as the '''stator''' of the hypocycloid.
Definition:Epicycloid/Generator
The circles $C_1$ and $C_2$ can be referred to as the '''generators''' of the epicycloid. === Stator === {{Definition:Epicycloid/Generator/Stator}} === Rotor === {{Definition:Epicycloid/Generator/Rotor}}
Definition:Epicycloid/Generator/Stator
The circle $C_2$ can be referred to as the '''stator''' of the epicycloid.
Definition:Epicycloid/Generator/Rotor
The circle $C_1$ can be referred to as the '''rotor''' of the epicycloid.
Definition:Deltoid
A '''deltoid''' is a hypocycloid with $3$ cusps. :400px
Definition:Oscillation/Real Space/Oscillation on Set
Let $A \subseteq X$ be any non-empty subset $A$ of $X$. The '''oscillation of $f$ on (or over) $A$''' is defined as: :$\displaystyle \map {\omega_f} A := \sup_{x, y \mathop \in A} \size {\map f x - \map f y}$ where the supremum is taken in the extended real numbers $\overline \R$.
Definition:Oscillation/Real Space/Oscillation at Point
Let $X$ and $Y$ be real sets. Let $f: X \to Y$ be a real function. <section notitle="true" name="definition"> Let $x \in X$. === Definition 1 === {{:Definition:Oscillation/Real Space/Oscillation at Point/Infimum}} === Definition 2 === {{:Definition:Oscillation/Real Space/Oscillation at Point/Epsilon}} === Definition 3 === {{:Definition:Oscillation/Real Space/Oscillation at Point/Limit}} </section>
Definition:Oscillation/Real Space
Let $X$ and $Y$ be real sets. Let $f: X \to Y$ be a real function. === Oscillation on a Set === {{:Definition:Oscillation/Real Space/Oscillation on Set}} {{transclude:Definition:Oscillation/Real Space/Oscillation at Point   |section  = definition   |increase = 1   |title    = Oscillation at a Point   |header   = 3   |link     = true }}
Definition:Nephroid
A '''nephroid''' is an epicycloid with $2$ cusps. :600px
Definition:Curvature
Let $C$ be a curve defined by a real function which is twice differentiable. The '''curvature''' of a $C$ is the reciprocal of the radius of the osculating circle to $C$ and is often denoted $\kappa$ (Greek '''kappa''').
Definition:Radius of Curvature
The '''radius of curvature''' of a curve $C$ at a point $P$ is defined as the reciprocal of the absolute value of its curvature: :$\rho = \dfrac 1 {\size k}$
Definition:Normal to Curve
Let $C$ be a curve embedded in the plane. The '''normal to $C$''' at a point $P$ is defined as the straight line which lies perpendicular to the tangent at $P$ and in the same plane as $P$.
Definition:Center of Curvature
Let $C$ be a curve embedded in the plane. Let $N$ be the normal to $C$ at a point $P$. Let $\rho$ be the radius of curvature of $C$ at $P$. Let $Q$ be the point on $N$ which is at a distance $\rho$ from $P$ on the concave side of $C$. $Q$ is known as the '''center of curvature''' of $C$ at $P$.
Definition:Curvature/Cartesian Form
Let $C$ be embedded in a cartesian plane. The '''curvature''' $\kappa$ of $C$ at a point $P = \tuple {x, y}$ is given by: :$\kappa = \dfrac {y''} {\paren {1 + y'^2}^{3/2} }$ where: :$y' = \dfrac {\d y} {\d x}$ is the derivative of $y$ {{WRT|Differentiation}} $x$ at $P$ :$y'' = \dfrac {\d^2 y} {\d x^2}$ is the second derivative of $y$ {{WRT|Differentiation}} $x$ at $P$.
Definition:Intrinsic Equation
Let $C$ be a curve. An '''intrinsic equation''' of $C$ is an equation that defines $C$ using a relation between the intrinsic properties of $C$. Thus an '''intrinsic equation''' defines the shape of $C$ without specifying its position relative to an arbitrarily defined coordinate system. === Natural Equation === {{:Definition:Intrinsic Equation/Natural Equation}} === Cesàro Equation === {{:Definition:Intrinsic Equation/Cesàro Equation}} === Whewell Equation === {{:Definition:Intrinsic Equation/Whewell Equation}}
Definition:Intrinsic Property
Let $C$ be a curve. The '''intrinsic properties''' of $C$ are those properties that do not depend on the nature of the embedding of $C$ in a coordinate system.
Definition:Intrinsic Property/Examples
Examples of the intrinsic properties of a curve include: :Arc length $s$ :Curvature $\kappa$ :Turning angle $\psi$ :Torsion $\tau$ :Radius of curvature $\rho$
Definition:Turning Angle
Let $C$ be a curve. The '''turning angle''' $\psi$ of $C$ at a point $P$ is defined by the differential equation: :$\kappa = \dfrac {\d \psi} {\d s}$ where: :$\kappa$ is the curvature at $P$ :$\dfrac {\d \psi} {\d s}$ is the derivative of $\psi$ {{WRT|Differentiation}} the arc length $s$. That is, the '''turning angle''' is a measure of how sharply a curve changes direction. The '''turning angle''' is an intrinsic property of $C$.
Definition:Intrinsic Equation/Whewell Equation
A '''Whewell equation''' is an intrinsic equation for a curve $C$ such that $C$ is expressed in terms of its intrinsic properties: :Arc length $s$ :Turning angle $\psi$
Definition:Curvature/Whewell Form
The '''curvature''' $\kappa$ of $C$ at a point $P$ can be expressed in the form of a Whewell equation as: :$\kappa = \dfrac {\d \psi} {\d s}$ where: :$\psi$ is the turning angle of $C$ :$s$ is the arc length of $C$.
Definition:Curvature/Parametric Form
Let $C$ be embedded in a cartesian plane and defined by the parametric equations: :$\begin{cases} x = \map x t \\ y = \map y t \end{cases}$ The '''curvature''' $\kappa$ of $C$ at a point $P = \tuple {x, y}$ is given by: :$\kappa = \dfrac {x' y'' - y' x''} {\tuple {x'^2 + y'^2}^{3/2} }$ where: :$x' = \dfrac {\d x} {\d t}$ is the derivative of $x$ {{WRT|Differentiation}} $t$ at $P$ :$y' = \dfrac {\d y} {\d t}$ is the derivative of $y$ {{WRT|Differentiation}} $t$ at $P$ :$x''$ and $y''$ are the second derivatives of $x$ and $y$ {{WRT|Differentiation}} $t$ at $P$.
Definition:Astronomical Unit/Approximate Values
The '''astronomical unit''' is approximately $150 \, 000 \, 000 \, \mathrm {k m}$, or $93$ million (international) miles.
Definition:Astronomical Unit
The '''astronomical unit''' is a derived unit of length. It is defined as being $149 \, 597 \, 870 \, 700$ metres. The '''astronomical unit''' is the standard unit of measurement used by astronomers when discussing distances within the solar system. It is derived as the approximate mean distance from the Earth to the Sun. === Approximate Values === {{:Definition:Astronomical Unit/Approximate Values}} === Symbol === {{:Definition:Astronomical Unit/Symbol}}
Definition:Division Algebra/Definition 1
$\left({A_F, \oplus}\right)$ is a '''division algebra''' : : $\forall a, b \in A_F, b \ne \mathbf 0_A: \exists_1 x \in A_F, y \in A_F: a = b \oplus x, a = y \oplus b$
Definition:Division Algebra/Definition 2
$A$ is a division algebra  it has no zero divisors: :$\forall a, b \in A_F: a \oplus b = \mathbf 0_A \implies a = \mathbf 0_A \lor b = \mathbf 0_A$
Definition:Conjugate Quaternion/Matrix Form
Let $\mathbf x$ be a quaternion defined in matrix form as: :$\mathbf x = \begin{bmatrix} a + bi & c + di \\ -c + di & a - bi \end{bmatrix}$ The conjugate quaternion of $\mathbf x$ is defined as: :$\overline {\mathbf x} = \begin{bmatrix} a - bi & -c - di \\ c - di & a + bi \end{bmatrix}$
Definition:Conjugate Quaternion/Ordered Pair
Let $\mathbf x$ be a quaternion defined as an ordered pair $\left({a, b}\right)$  of complex numbers. The conjugate quaternion of $\mathbf x$ is defined as: :$\overline {\mathbf x} = \overline {\left({a, b}\right)} = \left({\overline a, -b}\right)$
Definition:Field Norm of Quaternion
Let $\mathbf x = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$ be a quaternion. Let $\overline {\mathbf x}$ be the conjugate of $\mathbf x$. The '''norm''' of $\mathbf x$ is the real number defined as: :$n \left({\mathbf x}\right) := \left\vert{\mathbf x \overline {\mathbf x} }\right\vert = \left\vert{\overline {\mathbf x} \mathbf x }\right\vert = a^2 + b^2 + c^2 + d^2$
Definition:Positive Definite (Ring)
Let $\struct {R, +, \times}$ be a ring whose zero is denoted $0_R$. Let $f: R \to \R$ be a (real-valued) function on $R$. Then $f$ is '''positive definite''' : :$\forall x \in R: \begin {cases} \map f x = 0 & : x = 0_R \\ \map f x > 0 & : x \ne 0_R \end {cases}$
Definition:Nonassociative Algebra
Let $\left({A_R, \oplus}\right)$ be an algebra over a ring. Then $\left({A_R, \oplus}\right)$ is a nonassociative algebra  it is not necessarily the case that $\oplus$ is associative.
Definition:Lie Algebra
Let $\struct {A_R, \oplus}$ be an algebra over a ring. Then $\struct {A_R, \oplus}$ is a Lie algebra : :$\forall a \in A_R: a^2 = 0$ :$\forall a, b, c \in A_R: a \paren {b c} + b \paren {c a} + c \paren {a b} = 0$
Definition:Lie Group
A '''Lie group''' is a group which is also a smooth manifold.
Definition:Cayley Algebra
The '''Cayley algebra''' is the nonassociative division algebra formed by the Octonions under octonion multiplication.
Definition:Normal Family
Let $X = \left({M_1, d_1}\right)$ and $Y = \left({M_2, d_2}\right)$ be complete metric spaces. Let $\mathcal F = \left \langle{f_i}\right \rangle_{i \mathop \in I}$ be a family of continuous mappings $f_i: X \to Y$. Then $\mathcal F$ is a '''normal family''' : : every sequence of mappings in $\mathcal F$ contains a subsequence which converges uniformly on compact subsets of $X$ to a continuous function $f: X \to Y$.
Definition:Locally Bounded/Mapping
Let $M = \left({A, d}\right)$ be a metric space. Let $f$ be a mapping defined on $M$. Then $f$ is said to be '''locally bounded''' : :for all $x \in A$, there is some neighbourhood $N$ of $x$ such that $f \left[{N}\right]$ is bounded.
Definition:Uniformly Bounded
Let $X$ be a  set and let $Y = \left({A, d}\right)$ be a metric space. Let $\mathcal F = \left \langle{f_i}\right \rangle_{i \mathop \in I}$ be a family of mappings $f_i: X \to Y$. Then $\mathcal F$ is said to be '''uniformly bounded''' if every mapping $f \in \mathcal F$ can be bounded by the same constant. That is, if there exists some $M \in \R$ such that: :$\forall x, y \in X, i \in I : d \left({f_i \left({x}\right), f_i \left({y}\right)}\right) \le M$
Definition:Locally Bounded/Family of Mappings
Let $M = \left({A, d}\right)$ be a metric space. Let $\mathcal F = \left \langle{f_i}\right \rangle_{i \mathop \in I}$ be a family of mappings defined on $M$. Then $\mathcal F$ is said to be '''locally bounded''' : : for all $x \in A$, there is some neighbourhood $N$ of $x$ such that $\mathcal F$ is uniformly bounded on $N$.
Definition:Locally Bounded
=== Locally Bounded Mapping === {{:Definition:Locally Bounded/Mapping}} === Locally Bounded Family of Mappings === {{:Definition:Locally Bounded/Family of Mappings}}
Definition:Axiom/Also known as
An '''axiom''' is also known as a '''postulate'''. Among ancient Greek philosophers, the term '''axiom''' was used for a general truth that was common to everybody (see Euclid's "common notions"), while  '''postulate''' had a specific application to the subject under discussion. For most authors, the distinction is no longer used, and the terms are generally used interchangeably. This is the position of . However, some believe there is a difference significant enough to matter: :''... we shall use "postulate" instead of "axiom" hereafter, as "axiom" has a pernicious historical association of "self-evident, necessary truth", which "postulate" does not have; a postulate is an arbitrary assumption laid down by the mathematician himself and not by God Almighty.'' :::: -- {{BookReference|Men of Mathematics|1937|Eric Temple Bell}}: Chapter $\text{II}$: Modern Minds in Ancient Bodies
Definition:Bounded Sequence/Complex
Let $\sequence {z_n}$ be a complex sequence. Then $\sequence {z_n}$ is '''bounded''' : :$\exists M \in \R$ such that $\forall i \in \N: \cmod {z_i} \le M$ where $\cmod {z_i}$ denotes the complex modulus of $z_i$.
Definition:Bounded Sequence/Complex/Unbounded
$\sequence {z_n}$ is '''unbounded'''  it is not bounded.
Definition:Mechanical Curve
A '''mechanical curve''' is a curve which cannot be constructed using a compass and straightedge. Note that while it is not possible to construct such a curve in this manner, it may well be possible to construct as many points on it as required.
Definition:Conic Section/Intersection with Cone/Degenerate Hyperbola
Let $\phi < \theta$, that is: so as to make $K$ a hyperbola. However, let $D$ pass through the apex of $C$. Then $K$ degenerates into a pair of intersecting straight lines.
Definition:Geometric Curve
A '''geometric curve''' is a curve which can be constructed using a compass and straightedge.
Definition:Number Theory/Also known as
Older sources refer to '''number theory''' variously as: :'''theory of numbers''' :'''the higher arithmetic''' :'''arithmetic'''. {{AuthorRef|Carl Friedrich Gauss}} himself used the term '''arithmetic''', after the manner of the ancient Greeks. '''Number theory''' can also be seen described by the fanciful name '''the Queen of Mathematics''' for various whimsical reasons.
Definition:Ore Graph
Let $G = \struct {V, E}$ be an undirected simple graph. Then $G$ is an '''Ore graph''' : :the sum of the degrees of every pair of non-adjacent vertices is greater than or equal to the order of $G$. That is, : :$\forall u, v \in V: \set {u, v} \notin E \implies \map {\deg_G} u + \map {\deg_G} v \ge \card V$ {{NamedforDef|Øystein Ore|cat = Ore}}
Keith.U/Whatever/Definition:Exponential/Real
For all definitions of the '''real exponential function''': : The domain of $\exp$ is $\R$ : The codomain of $\exp$ is $\R_{>0}$ For $x \in \R$, the real number $\exp x$ is called the '''exponential of $x$'''. === As a Sum of a Series === {{:Definition:Exponential Function/Real/Sum of Series}} === As a Limit of a Sequence === {{:Definition:Exponential Function/Real/Limit of Sequence}} === As an Extension of the Rational Exponential === {{:Definition:Exponential Function/Real/Extension of Rational Exponential}} === As the Inverse to the Natural Logarithm === {{:Definition:Exponential Function/Real/Inverse of Natural Logarithm}} === As the Solution of a Differential Equation === {{:Definition:Exponential Function/Real/Differential Equation}}
Definition:Exponential Function/Real
For all definitions of the '''real exponential function''': : The domain of $\exp$ is $\R$ : The codomain of $\exp$ is $\R_{>0}$ For $x \in \R$, the real number $\exp x$ is called the '''exponential of $x$'''. === As a Sum of a Series === {{:Definition:Exponential Function/Real/Sum of Series}} === As a Limit of a Sequence === {{:Definition:Exponential Function/Real/Limit of Sequence}} === As an Extension of the Rational Exponential === {{:Definition:Exponential Function/Real/Extension of Rational Exponential}} === As the Inverse to the Natural Logarithm === {{:Definition:Exponential Function/Real/Inverse of Natural Logarithm}} === As the Solution of a Differential Equation === {{:Definition:Exponential Function/Real/Differential Equation}}
Definition:Boundary Value Problem
A '''boundary value problem''' is a differential equation for which a particular solution is to be found, such that the particular solution must satisfy certain boundary conditions.
Definition:Boundary Condition
Let $\Phi$ be a differential equation to which a particular solution is to be found. A '''boundary condition''' is an equation relating particular values of the dependent and independent variables which the particular solution to $\Phi$ must satisfy. It is usual for such '''boundary conditions''' to correspond to the physical extremities of bodies, aspects of whose internal nature is being modelled by means of $\Phi$.
Definition:Subdivision (Graph Theory)
Let $G = \struct {V, E}$ be a graph. === Edge Subdivision === {{:Definition:Subdivision (Graph Theory)/Edge}} === Graph Subdivision === {{:Definition:Subdivision (Graph Theory)/Graph}}
Definition:Isoperimetrical Problem
An '''isoperimetrical problem''' is a mathematical problem that asks what figure has the greatest area of all those figures which have a perimeter of a given length.
Definition:Subdivision (Graph Theory)/Edge
The '''edge subdivision operation''' for an edge $\set {u, v} \in E$ is the deletion of $\set {u, v}$ from $G$ and the addition of two edges $\set {u, w}$ and $\set {w, v}$ along with the new vertex $w$.  This operation generates a new graph $H$: :$H = \struct {V \cup \set w, \paren {E \setminus \set {u, v} } \cup \set {\set {u, w}, \set {w, v} } }$
Definition:Subdivision (Graph Theory)/Graph
A graph which has been derived from $G$ by a sequence of edge subdivision operations is called a '''subdivision of $G$'''.
Definition:Algorism/Algorist
During the middle ages, an '''algorist''' was an arithmetician who used algorism, as opposed to an  abacist who calculated using an abacus. By the time of {{AuthorRef|Leonhard Paul Euler}}, the term had evolved to mean a mathematician who devised algorithms for more complicated operations than those of the classical algorithms.
Definition:Abacism/Abacist
An '''abacist''' is an arithmetician who uses an abacus to do arithmetic, as opposed to an  algorist who calculates using algorism.
Definition:Riesel Number
Let $k$ be an odd positive integer. Then $k$ is a '''Riesel number''' : :For all positive integers $n$, the number $k \, 2^n - 1$ is composite.
Definition:Real Interval/Unit Interval/Open
The open interval between $0$ and $1$ is referred to as the '''open unit interval'''. :$\left({0 \,.\,.\, 1}\right) = \left\{ {x \in \R: 0 < x < 1}\right\}$
Definition:Real Interval/Unit Interval/Closed
The closed interval from $0$ to $1$ is denoted $\mathbb I$ (or a variant) by some authors: :$\mathbb I := \closedint 0 1 = \set {x \in \R: 0 \le x \le 1}$ This is often referred to as the '''closed unit interval'''.
Definition:Power (Algebra)/Real Number/Definition 2
Let $f : \Q \to \R$ be the real-valued function defined as: :$f \left({ q }\right) = x^q$ where $a^q$ denotes $a$ to the power of $q$. Then we define $x^r$ as the unique continuous extension of $f$ to $\R$.
Definition:Power (Algebra)/Real Number/Definition 1
We define $x^r$ as: :$x^r := \map \exp {r \ln x}$ where $\exp$ denotes the exponential function.
Definition:Nonnumerical Analysis
'''Nonnumerical analysis''' is a branch of discrete mathematics that studies subjects emerging from the evolution of computer science, and includes such topics as: :information processing :analysis of algorithms and so on.
Definition:Replacement Operation
The '''replacement operation''' on two variables $a$ and $b$ is denoted: :$a \gets b$ and is interpreted as: :the value of variable $a$ is to be replaced by the current value of variable $b$, while leaving the value of variable $b$ the same.
Definition:Exchange Operation
The '''exchange operation''' on two variables $a$ and $b$ is denoted: :$a \leftrightarrow b$ and is interpreted as: :the value of variable $a$ is to be replaced by the current value of variable $b$ and at the same time: :the value of variable $b$ is to be replaced by the current value of variable $a$.
Definition:Computational Method/Computational Sequence
Each $x \in I$ defines a '''computational sequence''' $x_0, x_1, x_2, \ldots$ as follows: : $x_0 = x$ : $\forall k \ge 0: x_{k+1} = f \left({x_k}\right)$
Definition:Algorithm/Step
An algorithm consists of a finite set of '''steps''', uniquely identified by means of a label, conventionally numeric. A '''step''' of an algorithm consists of: :an '''operation''' :an '''instruction''' as to what the algorithm is to do next, which will be one of the following: ::$(1): \quad$ By default: to move onto the next '''step''' in the sequence ::$(2): \quad$ Based on the result of a '''condition''', the specific '''step''' to perform next ::$(3): \quad$ To '''terminate'''.
Definition:Flow Chart
A '''flow chart''' is a graphical depiction of an algorithm in which the steps are depicted in the form of boxes connected together by arrows. Conventionally, the shape of the box representing a step is dependent upon the type of operation encapsulated within the step: :Rectangular for an action :A different shape, conventionally a diamond, for a condition. On , the preferred shape for condition boxes is rectangular with rounded corners.  This is to maximise ease and neatness of presentation: configuring a description inside a diamond shaped boxes in order for it to be aesthetically pleasing can be challenging and tedious. Also on , it is part of the accepted style to implement the start and end points of the algorithm using a box of a particular style, in this case with a double border.