hcju/proofwikips huggingface dataset:
Definition:Principal Ultrafilter
Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. Let $\FF \subset \powerset S$ be an ultrafilter on $S$ with a cluster point. Then $\FF$ is a '''principal ultrafilter on $S$'''.
Definition:Neighborhood Filter
Let $T = \left({S, \tau}\right)$ be a topological space. === Set === {{:Definition:Neighborhood Filter/Set}} === Point === {{:Definition:Neighborhood Filter/Point}}
Definition:Group Automorphism
Let $\struct {G, \circ}$ be a group. Let $\phi: G \to G$ be a (group) isomorphism from $G$ to itself. Then $\phi$ is a group automorphism.
Definition:Scheme of Abbreviation
When translating from a series of sentences in natural language into a collection of propositional formulas in symbolic logic, it is necessary to replace each simple statement with a statement label. As stated in the definition of statement label, it is imperative that each sentence be replaced with a different label. A '''scheme of abbreviation''' is the list of simple statements, together with their associated statement labels, for reference and unambiguous key in order to translate any conclusion reached back into natural language.
Definition:Negation Normal Form
A propositional formula $P$ is in '''negation normal form''' ('''NNF''') : * The only logical connectives connecting substatements of $P$ are Not, And and Or, that is, elements of the set $\left\{{\neg, \land, \lor}\right\}$; * The Not sign $\neg$ appears only in front of atomic statements. That is $P$ is in '''negation normal form''' iff it consists of literals, conjunctions and disjunctions.
Definition:Finite Group
A '''finite group''' is a group of finite order. That is, a group $\struct {G, \circ}$ is a '''finite group'''  its underlying set $G$ is finite.
Definition:Cayley Table
A '''Cayley table''' is a technique for describing an algebraic structure (usually a finite group) by putting all the products in a square array: :$\begin {array} {c|cccc} \circ & a & b & c & d \\ \hline a & a & a & b & a \\ b & b & c & a & d \\ c & d & e & f & a \\ d & c & d & a & b \\ \end {array}$
Definition:Latin Square
Let $n \in \Z_{>0}$ be some given (strictly) positive integer $n$. A '''Latin square''' of order $n$ is a square array of size $n \times n$ containing $n$ different symbols, such that every row and column contains '''exactly one''' of each symbol. That is, each row and column is a permutation of the same $n$ symbols.
Definition:Infinite Group
A group which is not finite is an '''infinite group'''.
Definition:General Linear Group
Let $K$ be a field. The set of all invertible order-$n$ square matrices over $K$ is a group under (conventional) matrix multiplication. This group is called the '''general linear group (of degree $n$)''' and is denoted $\GL {n, K}$, or $\GL n$ if the field is implicit. The field itself is usually $\R$, $\Q$ or $\C$, but can be ''any'' field.
Definition:Noetherian Ring
=== Definition 1 === {{Definition:Noetherian Ring/Definition 1}} === Definition 2 === {{Definition:Noetherian Ring/Definition 2}} === Definition 3 === {{Definition:Noetherian Ring/Definition 3}} === Definition 4 === {{Definition:Noetherian Ring/Definition 4}}
Definition:Product (Abstract Algebra)
=== General Operation === {{:Definition:Operation/Binary Operation/Product}} === Group Product === {{:Definition:Product (Abstract Algebra)/Group}} === Ring Product === {{:Definition:Product (Abstract Algebra)/Ring}} === Field Product === {{:Definition:Product (Abstract Algebra)/Field}}
Definition:Product Inverse
The inverse of $x \in U_R$ by $\circ$ is called the '''(ring) product inverse of $x$'''. The usual means of denoting the product inverse of an element $x$ is by $x^{-1}$. Thus it is distinguished from the additive inverse of $x$, that is, the (ring) negative of $x$, which is usually denoted $-x$.
Definition:Affine Algebraic Variety
Let $k$ be a field. Then a subset $X \subseteq k^n$ is an '''affine algebraic variety''' if the following hold: * $X$ is an affine algebraic set  * $X$ is irreducible
Definition:Zariski Topology
=== On an Affine Space === {{:Definition:Zariski Topology/Affine Space}} === On the spectrum of a ring === {{:Definition:Zariski Topology/Spectrum of Ring}}
Definition:Radical of Integer
The '''radical of an integer''' $n \in \Z$ is the product of the individual prime factors of $n$. The radicals of the first few integers are given here: {| |- ! align="right" | $n$ !! ! align="right" | Decomposition !! ! align="right" | $\map {\operatorname {rad} } n$ |- | align="right" | $1$ || | align="right" | $1$ || | align="right" | $1$ |- | align="right" | $2$ || | align="right" | $2$ || | align="right" | $2$ |- | align="right" | $3$ || | align="right" | $3$ || | align="right" | $3$ |- | align="right" | $4$ || | align="right" | $2^2$ || | align="right" | $2$ |- | align="right" | $5$ || | align="right" | $5$ || | align="right" | $5$ |- | align="right" | $6$ || | align="right" | $2 \times 3$ || | align="right" | $6$ |- | align="right" | $7$ || | align="right" | $7$ || | align="right" | $7$ |- | align="right" | $8$ || | align="right" | $2^3$ || | align="right" | $2$ |- | align="right" | $9$ || | align="right" | $3^2$ || | align="right" | $3$ |- | align="right" | $10$ || | align="right" | $2 \times 5$ || | align="right" | $10$ |- | align="right" | $11$ || | align="right" | $11$ || | align="right" | $11$ |- | align="right" | $12$ || | align="right" | $2^2 \times 3$ || | align="right" | $6$ |- | align="right" | $13$ || | align="right" | $13$ || | align="right" | $13$ |- | align="right" | $14$ || | align="right" | $2 \times 7$ || | align="right" | $14$ |- | align="right" | $15$ || | align="right" | $3 \times 5$ || | align="right" | $15$ |- | align="right" | $16$ || | align="right" | $2^4$ || | align="right" | $2$ |} {{OEIS|A007947}} The radical of $n$ can alternatively be described as the largest square-free integer which divides $n$.
Definition:Generator of Subgroup
Let $\struct {G, \circ}$ be a group. Let $S \subseteq G$. Let $H$ be the subgroup generated by $S$. Then '''$S$ is a generator of $H$''', denoted $H = \gen S$,  $H$ is the subgroup generated by $S$.
Definition:Join of Subgroups
Let $\struct {G, \circ}$ be a group. Let $A$ and $B$ be subgroups of $G$. The '''join''' of $A$ and $B$ is written and defined as: :$A \vee B := \gen {A \cup B}$ where $\gen {A \cup B}$ is the subgroup generated by $A \cup B$. By the definition of subgroup generator, this can alternatively be written: :$\displaystyle A \vee B := \bigcap \set {T: T \text { is a subgroup of } G: A \cup B \subseteq T}$ === General Definition === {{:Definition:Join of Subgroups/General Definition}}
Definition:Argand Diagram
An '''Argand diagram''' is a graphical representation of a set of complex numbers on the complex plane: :400px
Definition:Topological Manifold/Smooth Manifold
Let $M$ be a second-countable locally Euclidean space of dimension $d$.  Let $\mathscr F$ be a smooth differentiable structure on $M$. Then $\left({M, \mathscr F}\right)$ is called a '''smooth manifold of dimension $d$'''.
Definition:Kampyle of Eudoxus
:300px {{refactor|Separate pages for Cartesian and Polar equations}} The '''Kampyle of Eudoxus''' has the Cartesian equation: :$x^4 = x^2 + y^2$ where the point $x = y = 0$ is specifically excluded. In polar coordinates, it is described by the equation: :$r = \sec^2 \theta$  {{NamedforDef|Eudoxus of Cnidus|cat = Eudoxus}}
Definition:Cube Number
A '''cube number''' (or just '''cube''') is a number which can be expressed as the third power of an integer.
Definition:One
The immediate successor element of zero in the set of natural numbers $\N$ is called '''one''' and has the symbol $1$. === Naturally Ordered Semigroup === {{:Definition:Unit (One)/Naturally Ordered Semigroup}}
Definition:Ordered Tuple as Ordered Set
The rigorous definition of an ordered tuple is as a finite sequence whose domain is $\N^*_n$. However, it is possible to treat an ordered tuple as an extension of the concept of an ordered pair. === Ordered Triple === {{:Definition:Ordered Tuple as Ordered Set/Ordered Triple}} === Ordered Quadruple === {{:Definition:Ordered Tuple as Ordered Set/Ordered Quadruple}} === Ordered Tuple === {{:Definition:Ordered Tuple as Ordered Set/Ordered Tuple}} === Ordered Singleton === {{:Definition:Ordered Tuple as Ordered Set/Ordered Singleton}}
Definition:Zero Element
An element $z \in S$ is called a '''two-sided zero element''' (or simply '''zero element''' or '''zero''')  it is both a '''left zero''' and a '''right zero''': :$\forall x \in S: x \circ z = z = z \circ x$
Definition:Invertible Operation
The operation $\circ$ is '''invertible''' : :$\forall a, b \in S: \exists r, s \in S: a \circ r = b = s \circ a$
Definition:Mathematical System
A '''mathematical system''' is a set $\SS = \struct {E, O, A}$ where: :$E$ is a non-empty set of elements :$O$ is a set of relations and operations on the elements of $E$ :$A$ is a set of axioms concerning the elements of $E$ and $O$. === Abstract System === A '''mathematical system''' $\SS = \struct {E, O, A}$ is classed as '''abstract''' if the elements of $E$ and $O$ are defined only by their properties as specified in $A$. === Concrete System === A '''mathematical system''' $\SS = \struct {E, O, A}$ is classed as '''concrete''' if the elements of $E$ and $O$ are understood as objects independently of their existence in $\SS$ itself. The distinction between '''abstract''' and '''concrete''' is of questionable value from a modern standpoint, as it is a moot point, for example, as to whether the natural numbers exist independently of Peano's axioms or are specifically '''defined''' by them. === Algebraic System === A '''mathematical system''' $\SS = \struct {E, O, A}$ is classed as '''algebraic''' if it has many of the properties of the set of integers. This is usually because such a system is itself an abstraction of certain properties of the integers. The axioms are usually not considered as separate entities from the operations, as their nature is implicit in the operations themselves. Specifically, an algebraic system can be defined as follows: {{:Definition:Algebraic System}}
Definition:Scalar Field
Let $\struct {G, +_G, \circ}_K$ be a vector space, where: :$\struct {K, +_K, \times_K}$ is a field :$\struct {G, +_G}$ is an abelian group $\struct {G, +_G}$ :$\circ: K \times G \to G$ is a binary operation. Then the field $\struct {K, +_K, \times_K}$ is called the '''scalar field''' of $\struct {G, +_G, \circ}_K$.
Definition:Bilinear Mapping
Let $\left({R, +_R, \times_R}\right)$ be a commutative ring. Let $\left({A_1, +_1, \circ_1}\right)_R, \left({A_2, +_2, \circ_2}\right)_R, \left({A_3, +_3, \circ_3}\right)_R$ be $R$-modules. Let $\oplus: A_1 \times A_2 \to A_3$ be a binary operator with the property that: $\forall \left({a_1, a_2}\right) \in A_1 \times A_2$: : $a_1 \mapsto a_1 \oplus a_2$ is a linear transformation from $A_1$ to $A_3$ : $a_2 \mapsto a_1 \oplus a_2$ is a linear transformation from $A_2$ to $A_3$ Then $\oplus$ is a '''bilinear mapping'''. That is, $\forall a, b \in R, \forall x, y \in A_2, z \in A_3$: : $\left({\left({a \circ_1 x}\right) +_1 \left({y \circ_1 b}\right)}\right) \oplus z = \left({a \circ_3 \left({x \oplus z}\right)}\right) +_3 \left({\left({y \oplus z}\right) \circ_3 b}\right)$ and for all $z \in A_1, x,y \in A_2$: : $z \oplus \left({\left({a \circ_2 x}\right) +_2 \left({y \circ_2 b}\right)}\right) = \left({a \circ_3 \left({z \oplus x}\right)}\right) +_3 \left({\left({z \oplus y}\right) \circ_3 b}\right)$ Equivalently, this can be expressed: : $\left({x +_1 y}\right) \oplus z = \left({x \oplus z}\right) +_3 \left({y \oplus z}\right)$ : $z \oplus \left({x +_2 y}\right) = \left({z \oplus x}\right) +_3 \left({z \oplus y}\right)$ : $\left({a \circ_1 x}\right) \oplus z = a \circ_3 \left({x \oplus z}\right)$ : $z \oplus \left({y \circ_2 b}\right) = \left({z \oplus y}\right) \circ_3 b$ If $\left({A, +, \circ}\right)_R = A_1 = A_2 = A_3$, the notation simplifies considerably: : $\left({\left({a \circ x}\right) + \left({b \circ y}\right)}\right) \oplus z = \left({a \circ \left({x \oplus z}\right)}\right) + \left({b \circ \left({y \oplus z}\right)}\right)$ : $z \oplus \left({\left({a \circ x}\right) + \left({y \circ b}\right)}\right) = \left({a \circ \left({z \oplus x}\right)}\right) + \left({\left({z \oplus y}\right) \circ b}\right)$ or equivalently, more easily digested: : $\left({x + y}\right) \oplus z = \left({x \oplus z}\right) + \left({y \oplus z}\right)$ : $z \oplus \left({x + y}\right) = \left({z \oplus x}\right) + \left({z \oplus y}\right)$ : $\left({a \circ x}\right) \oplus z = a \circ \left({x \oplus z}\right)$ : $z \oplus \left({y \circ b}\right) = \left({z \oplus y}\right) \circ b$ === Non-Commutative Ring === {{:Definition:Bilinear Mapping/Non-Commutative Ring}}
Definition:Algebra over Field
Let $F$ be a field. An '''algebra over $F$''' is an ordered pair $\struct {A, *}$ where: :$A$ is a vector space over $F$ :$* : A^2 \to A$ is a bilinear mapping That is, it is an algebra $\struct {A, *}$ over the ring $F$ where: :$F$ is a field :the $F$-module $A$ is a vector space.
Definition:Algebra over Ring
Let $R$ be a commutative ring. An '''algebra over $R$''' is an ordered pair $\left({A, *}\right)$ where: :$A$ is an $R$-module :$*: A^2 \to A$ is an $R$-bilinear mapping
Definition:Second Order Ordinary Differential Equation
A '''second order ordinary differential equation''' is an ordinary differential equation in which any derivatives with respect to the independent variable have order no greater than $2$.
Definition:Physical Universe
The '''physical universe''', or usually just '''universe''', is commonly defined as, and understood to be, '''{{WP|Universe|the totality of everything that exists}}'''. === Real-World === Used to describe a phenomenon or object that has a genuine existence in the physical universe.
Definition:Body
In the context of physics and applied mathematics, a '''body''' is an object (possibly idealized) which is imagined to have a material existence in the physical universe. A '''body''' can be considered as an aggregate of particles.
Definition:Physical Law
A '''physical law''' is a statement about the physical universe whose truth value stems from observations.
Definition:Force
A '''force''' is an influence which causes a body to undergo a change in velocity. '''Force''' is a vector quantity.
Definition:Velocity
The '''velocity''' $\mathbf v$ of a body $M$ is defined as the first derivative of the displacement $\mathbf s$ of $M$ from a given point of reference {{WRT|Differentiation}} time $t$: :$\mathbf v = \dfrac {\d \mathbf s} {\d t}$
Definition:Frame of Reference
A '''frame of reference''' is a coordinate system which is more or less arbitrarily or conveniently disposed in a model of the physical universe. === Point of Reference === {{:Definition:Frame of Reference/Point of Reference}}
Definition:Mathematical Model
A '''mathematical model''' is an equation, or a system of equations, whose purpose is to provide an approximation to the behavior of a real-world phenomenon.
Definition:Displacement
The '''(physical) displacement''' of a body is a measure of its position relative to a given point of reference in a particular frame of reference. '''Displacement''' is a vector quantity, so it specifies a magnitude and direction from the point of reference.
Definition:Scalar Quantity
A '''scalar quantity''' is a real-world concept that needs for its model a mathematical object which contains only one (usually numeric) component.
Definition:Magnitude
The '''magnitude''' (or '''size''') of a quantity (either vector or scalar) is a measure of how big it is. It is usually encountered explicitly in the context of vectors: If $\mathbf v$ is the vector quantity in question, then its '''magnitude''' is denoted: :$\size {\mathbf v}$ or :$v$
Definition:Direction
The '''direction''' of a vector quantity is a measure of which way it is pointing, relative to a particular frame of reference.
Definition:Speed
The '''speed''' of a body is a measure of the magnitude of its velocity, taking no account of its direction. It is, therefore, a scalar quantity.
Definition:Acceleration
The '''acceleration''' $\mathbf a$ of a body $M$ is defined as the first derivative of the velocity $\mathbf v$ of $M$ relative to a given point of reference {{WRT|Differentiation}} time $t$: :$\mathbf a = \dfrac {\d \mathbf v} {\d t}$
Definition:Linear Momentum
The '''linear momentum''' of a body is its mass multiplied by its velocity. :$\mathbf p = m \mathbf v$
Definition:Mass
The '''mass''' of a body is a measure of how much matter it contains. '''Mass''' is equivalent to inertia. '''Mass''' also determines the degree to which a body creates or is affected by a gravitational field. It is a scalar quantity.
Definition:Second Order Partial Differential Equation
A '''second order partial differential equation''' is a partial differential equation in which any derivatives with respect to the independent variable have order no greater than $2$.
Definition:Proportion
Two real variables $x$ and $y$ are '''proportional'''  one is a constant multiple of the other: :$\forall x, y \in \R: x \propto y \iff \exists k \in \R, k \ne 0: x = k y$
Definition:Aggregate
An '''aggregate''' is an archaic word for an '''infinite set''', as used by {{AuthorRef|Georg Cantor}}. It is used in current parlance to mean a general accumulation of objects, but the concept is vague.
Definition:Ideal (Physics)
An '''ideal''' (or '''idealized''') object is one in which certain attributes are approximated to zero or infinity.
Definition:Particle
A '''particle''' is a representation of an object in the physical university which is idealized as having no magnitude. That is, it is modelled as being a mass concentrated at a single point.
Definition:One-Parameter Family of Curves
Consider the implicit function $\map f {x, y, c} = 0$ in the $\tuple {x, y}$-plane where $c$ is a constant. For each value of $c$, we have that $\map f {x, y, c} = 0$ defines a relation between $x$ and $y$ which can be graphed in the cartesian plane. Thus, each value of $c$ defines a particular curve. The complete set of all these curve for each value of $c$ is called a '''one-parameter family of curves'''. === Parameter === {{:Definition:One-Parameter Family of Curves/Parameter}}
Definition:Orthogonal Trajectories
Let $\map f {x, y, c}$ define a one-parameter family of curves $F$. Let $\map g {x, y, c}$ also define a one-parameter family of curves $G$, with the property that: :Every curve in $F$ is orthogonal to every curve in $G$. Then $F$ is a '''family of (reciprocal) orthogonal trajectories''' of $G$, and contrariwise.
Definition:Orthogonal (Analytic Geometry)
Two curves are '''orthogonal''' if they intersect at right angles. The term perpendicular can also be used, but the latter term is usual when the intersecting lines are straight.
Definition:Quadrature
'''Quadrature''' is an archaic term meaning '''the process of finding area'''.
Definition:Lune
A '''lune''' is a geometric figure formed by the boundary of two intersecting circles: :300px The area coloured red is a '''lune'''.
Definition:Physical Quantity
A '''physical quantity''' is a physical property which can be quantified.
Definition:Rest Mass
The '''rest mass''' of a body is its '''mass''' when it is '''at rest'''.
Definition:Physical Property
A '''physical property''' is any measurable property whose describes the state of a physical system.
Definition:Measurable Property
A '''measurable property''' is a property of a physical system which can be compared against a standard unit of measurement to provide a numerical value defining the quantity of that property.
Definition:Physical System
A '''physical system''' is a portion of the physical universe which has been chosen for investigation for a particular purpose.
Definition:Quantitative Property
A '''quantitative property''' is a property of a physical system which exists in a range of values corresponding with a number, such that the system can be described as having a particular number of a unit of measurement of that property.
Definition:Unit of Measurement
A '''unit of measurement''' is a specified magnitude of a given physical quantity, defined by convention. It is used as a standard for measurement of that physical quantity. Any other value of the physical quantity can be expressed as a multiple of that '''unit of measurement'''.
Definition:Inertia
'''Inertia''' is the tendency of a body to maintain the same velocity in the absence of an external force, in accordance with Newton's First Law of Motion. Equivalently put, '''inertia''' is the resistance of a body to a change in its motion. '''Inertia''' is equivalent to mass.
Definition:Friction
'''Friction''' is a force whose tendency is to reduce the velocity of a body relative to its surroundings. It is caused by interaction of the matter in the body with that of the environment in which it is traveling. '''Friction''' is (at a fundamental level) an electromagnetic force. That is, bits of the medium in or on which the body is traveling get in the way (on a molecular level) of bits of the body. The less friction, the less the reduction in velocity. If there were no '''friction''', then motion would continue for ever.
Definition:Force of Gravity
The '''force of gravity''' or '''gravitational force''' is the force on a body as a result of Newton's Law of Universal Gravitation. When used in an unqualified sense, it is usual for this to mean the force on a body at the surface of the Earth. From Gravity at Earth's Surface, this is approximately $9.8 \ \mathrm N \ \mathrm{kg}^{-1}$ The force of gravity varies across the earth's surface, and therefore it makes little sense to use it as a standard. However, the {{WP|General_Conference_on_Weights_and_Measures|CGPM}} adopted a standard acceleration of gravity of $9.806 \, 65  \ \mathrm N \ \mathrm{kg}^{-1}$ in 1901.
Definition:Dimension (Measurement)
Every physical quantity has a '''dimension''' associated with it. No attempt is made here to provide an abstract definition of this term. Instead, it will be defined by example.
Definition:Rate
The '''rate''' of a physical process is its (first) derivative with respect to time. Loosely speaking, it means '''how fast something progresses''', with a wider scope than change in physical displacement. === Dimension === {{:Definition:Rate/Dimension}}
Definition:Physical Process
A '''physical process''' is a physical system whose importance is in the way it changes over time.
Definition:Periodic Process
A '''periodic process''' is a physical process whose state can be modeled by a periodic function. === Example === The canonical example of such a process is the motion of a pendulum.
Definition:Time
A true definition of what '''time''' actually ''is'' has baffled philosophers from the dawn of, er, time. Therefore it will be left as an exercise for the reader. It is usually treated as a scalar quantity.
Definition:Time Period
The '''time period''' of a periodic process is the period of the periodic function which models that process.
Definition:Fundamental Unit (Physics)
A '''fundamental unit''' a unit of measurement for a measurable physical property from which every other unit for that quantity can be derived. The '''fundamental unit''' for a particular measurable quantity is chosen by convention. They are the '''units''' of the fundamental dimensions of physics.
Definition:SI Units
The '''SI Units''' are the elements of the [http://en.wikipedia.org/wiki/International_System_of_Units International System of Units].
Definition:Time/Unit/Second
The '''second''' is the SI base unit of time, and also therefore of the MKS system. It is also the base unit of time for the FPS and CGS systems. The '''second''' is defined as: :the duration of $9 \ 192 \ 631 \ 770$ periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium $133$ atom at rest at $0 \ \mathrm K$.