hcju/proofwikips huggingface dataset:
Definition:Galois Connection
Let $\struct {S, \preceq}$ and $\struct {T, \precsim}$ be ordered sets. Let $g: S \to T$, $d: T \to S$ be mappings. Then $\struct {g, d}$ is a '''Galois connection''' : :$g$ and $d$ are increasing mappings and ::$\forall s \in S, t \in T: t \precsim \map g s \iff \map d t \preceq s$ $g$ is '''upper adjoint''' and $d$ is '''lower adjoint''' of a Galois connection. {{definition wanted|monotone, antitone, closure operator, closed}} {{NamedforDef|Évariste Galois|cat = Galois}}
Definition:Taylor Series/Remainder
Consider the Taylor series expansion $\map T {\map f \xi}$ of $f$ about the point $\xi$: :$\displaystyle \sum_{n \mathop = 0}^\infty \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$ Let $\map {T_n} {\map f \xi}$ be the Taylor polynomial: :$\displaystyle \sum_{n \mathop = 0}^n \frac {\paren {x - \xi}^n} {n!} \map {f^{\paren n} } \xi$ for some $n \in \N$. The difference: :$\displaystyle \map {R_n} x = \map f x - \map {T_n} {\map f \xi} = \int_\xi^x \map {f^{\paren {n + 1} } } t \dfrac {\paren {x - t}^n} {n!} \rd t$ is known as the '''remainder of $\map T {\map f \xi}$''' at $x$.
Definition:Taylor Series/Remainder/Lagrange Form
The '''Lagrange form''' of the remainder $R_n$ is given by: :$R_n = \dfrac {\map {f^{\paren {n + 1} } } {x^*} } {\paren {n + 1}!} \paren {x - \xi}^{n + 1}$ where $x^* \in \openint \xi x$.
Definition:Taylor Series/Remainder/Cauchy Form
The '''Cauchy form''' of the remainder $R_n$ is given by: :$R_n = \dfrac {\paren {x - \eta}^n} {n!} \paren {x - \xi} \map {f^{\paren {n + 1} } } \eta$ where $\eta \in \closedint \xi x$.
Definition:Pentagonal Number/Examples
The first few pentagonal numbers are as follows: :700px === Sequence of Pentagonal Numbers === {{:Definition:Pentagonal Number/Sequence}}
Definition:Polygonal Number/Degenerate Case
Consider the polygonal number $P \left({2, n}\right)$ when $k = 2$. In this case, the polygon degenerates into a straight line, and the recurrence formula becomes: :$P \left({2, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({2, n-1}\right) + 0 \times \left({n-1}\right) + 1 & : n > 0 \end{cases}$ Hence: :$P \left({2, n}\right) = P \left({2, n-1}\right) + 1$ and the sequence goes: :$0, 1, 2, 3, \ldots$ which is of course the natural numbers.
Definition:Triangular Number/Definition 2
:$\displaystyle T_n = \sum_{i \mathop = 1}^n i = 1 + 2 + \cdots + \left({n-1}\right) + n$
Definition:Triangular Number/Definition 1
:$T_n = \begin{cases} 0 & : n = 0 \\ n + T_{n-1} & : n > 0 \end{cases}$
Definition:Triangular Number/Definition 3
:$\forall n \in \N: T_n = P \left({3, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({3, n - 1}\right) + \left({n - 1}\right) + 1 & : n > 0 \end{cases}$ where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.
Definition:Polygonal Number/Examples
=== Triangular Numbers === When $k = 3$, the recurrence relation is: {{:Definition:Triangular Number/Definition 3}} See Triangular Number. === Square Numbers === When $k = 4$, the recurrence relation is: {{:Definition:Square Number/Definition 4}} See Square Number. Square numbers are of course better known as: :$S_n = n^2$ === Pentagonal Numbers === When $k = 5$, the recurrence relation is: {{:Definition:Pentagonal Number/Definition 3}} See Pentagonal Number. === Hexagonal Numbers === When $k = 6$, the recurrence relation is: {{:Definition:Hexagonal Number/Definition 3}} See Hexagonal Number. === Heptagonal Numbers === When $k = 7$, the recurrence relation is: {{:Definition:Heptagonal Number/Definition 3}} See Heptagonal Number. === Octagonal Numbers === When $k = 8$, the recurrence relation is: {{:Definition:Octagonal Number/Definition 3}} See Octagonal Number.
Definition:Pentagonal Number/Definition 1
:$P_n = \begin{cases} 0 & : n = 0 \\ P_{n-1} + 3 n - 2 & : n > 0 \end{cases}$
Definition:Pentagonal Number/Definition 2
:$\displaystyle P_n = \sum_{i \mathop = 1}^n \left({3 i - 2}\right) = 1 + 4 + \cdots + \left({3 \left({n - 1}\right) - 2}\right) + \left({3 n - 2}\right)$
Definition:Pentagonal Number/Definition 3
:$\forall n \in \N: P_n = \map P {5, n} = \begin {cases} 0 & : n = 0 \\ \map P {5, n - 1} + 3 \paren {n - 1} + 1 & : n > 0 \end {cases}$ where $\map P {k, n}$ denotes the $k$-gonal numbers.
Definition:Hexagonal Number/Definition 1
:$H_n = \begin{cases} 0 & : n = 0 \\ H_{n-1} + 4 \left({n-1}\right) + 1 & : n > 0 \end{cases}$
Definition:Hexagonal Number/Definition 2
:$\displaystyle H_n = \sum_{i \mathop = 1}^n \left({4 \left({i - 1}\right) + 1}\right) = 1 + 5 + \cdots + \left({4 \left({n-2}\right) + 1}\right) + \left({4 \left({n - 1}\right) + 1}\right)$
Definition:Hexagonal Number/Definition 3
:$\forall n \in \N: H_n = P \left({6, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({6, n - 1}\right) + 4 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$ where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.
Definition:Square Number/Definition 2
:$S_n = \begin {cases} 0 & : n = 0 \\ S_{n - 1} + 2 n - 1 & : n > 0 \end {cases}$
Definition:Square Number/Definition 1
An integer $n$ is classified as a '''square number''' : :$\exists m \in \Z: n = m^2$ where $m^2$ denotes the integer square function. ==== Euclid's Definition ====  :''{{:Definition:Euclid's Definitions - Book VII/18 - Square Number}}'' {{EuclidDefRefNocat|VII|18|Square Number}}
Definition:Square Number/Definition 3
:$\displaystyle S_n = \sum_{i \mathop = 1}^n \paren {2 i - 1} = 1 + 3 + 5 + \cdots + \paren {2 n - 1}$
Definition:Square Number/Definition 4
:$\forall n \in \N: S_n = \map P {4, n} = \begin{cases} 0 & : n = 0 \\ \map P {4, n - 1} + 2 \paren {n - 1} + 1 & : n > 0 \end{cases}$ where $\map P {k, n}$ denotes the $k$-gonal numbers.
Definition:Triangular Number/Sequence
The sequence of triangular numbers, for $n \in \Z_{\ge 0}$, begins: :$0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, \ldots$
Definition:Pentagonal Number/Sequence
The sequence of pentagonal numbers, for $n \in \Z_{\ge 0}$, begins: :$0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, \ldots$
Definition:Hexagonal Number/Sequence
The sequence of hexagonal numbers, for $n \in \Z_{\ge 0}$, begins: :$0, 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, \ldots$
Definition:Odd Integer/Definition 1
An integer $n \in \Z$ is '''odd'''  it is not divisible by $2$. That is,  it is not even.
Definition:Odd Integer/Definition 2
An integer $n \in \Z$ is '''odd''' : :$\exists m \in \Z: n = 2 m + 1$
Definition:Convex Polyhedron
Let $P$ be a polyhedron. $P$ is a '''convex polyhedron''' : :For all points $A$ and $B$ located inside $P$, the line $AB$ is also inside $P$.
Definition:Neusis Construction/Historical Note
Many of the constructions performed by {{AuthorRef|Archimedes of Syracuse|Archimedes}} were '''neusis constructions'''. Some sources suggest that {{AuthorRef|Oenopides of Chios}} (c. $440$ BCE) was the first to rank '''compass and straightedge constructions''' above '''neusis constructions''' in importance. It may have been {{AuthorRef|Hippocrates of Chios}} (c. $430$ BCE) who publicised the principle to use a '''compass and straightedge construction''' in preference to a '''neusis construction''' whenever possible.  uses '''compass and straightedge constructions''' exclusively. Under {{AuthorRef|Plato}}, '''neusis''' was considered a last resort when all other methods of construction had failed. By the time of {{AuthorRef|Pappus of Alexandria}}, to use a '''neusis construction''' when other methods would work was considered a serious error.
Definition:Center of Gravity
Let $B$ be a body. The '''center of gravity''' of $B$ is that point through which the resultant of the system of parallel forces formed by the weights of all the particles constituting $B$ passes for all positions of $B$.
Definition:Archimedean Spiral/Archimedes' Definition
:''If a straight line of which one extremity remains fixed be made to revolve at a uniform rate in a plane until it returns to the position from which it started, and if, at the same time as the straight line revolves, a point moves at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane.''
Definition:Information Theory
'''Information theory''' is that branch of mathematics concerned with the flow of information.
Definition:Special Function
'''Special functions''' are particular functions which have more or less established names and notations due to their importance in, for example,  analysis, functional analysis or physics. There is no general formal definition, but the list of mathematical functions contains functions which are commonly accepted as special.
Definition:Stochastic Process
A '''stochastic process''' is a collection of random variables representing the evolution of some real-world system over time.
Definition:Non-Linear System
A '''non-linear system''' is a system of differential equations which are non-linear.
Definition:Catastrophe Theory
'''Catastrophe theory''' is a branch of bifurcation theory in the study of dynamical systems. It is also a particular special case of more general singularity theory in geometry.
Definition:Chaos Theory
'''Chaos theory''' is the branch of non-linear systems that studies the behavior and condition of dynamical systems that are highly sensitive to initial conditions.
Definition:Atomic Physics
'''Atomic physics''' is the branch of physics concerned with the structure and behaviour of the atom.
Definition:Wave Mechanics
'''Wave mechanics''' is the branch of physics concerned with the physical behaviour of waves.
Definition:Oscillation/Oscillation on Set
Let $A \subseteq X$ be any non-empty subset $A$ of $X$. The '''oscillation of $f$ on (or over) $A$ with respect to $d$''', denoted $\omega_f \left({A; d}\right)$, is defined as the diameter of $f \left({A}\right)$: :$\displaystyle \omega_f \left({A; d}\right) := \operatorname{diam} \left({f \left({A}\right)}\right) = \sup_{x,y \mathop \in A} d \left({f \left({x}\right), f \left({y}\right)}\right)$ where the supremum is taken in the extended real numbers $\overline \R$. The metric $d$ is often suppressed from the notation if it is clear from context, in which case one would simply write $\omega_f \left({A}\right)$. Similarly, one would speak of the '''oscillation of $f$ on $A$''' in this case.
Definition:Oscillation/Oscillation at Point
Let $x \in X$. Let $X$ be a topological space. Denote with $\mathcal N_x$ the set of neighborhoods of $x$. The '''oscillation of $f$ at $x$ with respect to $d$''', denoted by $\omega_f \left({x; d}\right)$, is defined as: :$\displaystyle \omega_f \left({x; d}\right) := \inf_{U \mathop \in \mathcal N_x} \omega_f \left({U; d}\right)$ where $\omega_f \left({U; d}\right)$ denotes the oscillation of $f$ on $U$. The metric $d$ is often suppressed from the notation if it is clear from context, in which case one would simply write $\omega_f \left({x}\right)$. Similarly, one would speak of the '''oscillation of $f$ at $x$''' in this case.
Definition:Point of Inflection
Let $f$ be a real function which is differentiable on an interval $\Bbb I \subseteq \R$. Let $\xi \in \Bbb I$. === Definition 1 === {{:Definition:Point of Inflection/Definition 1}} === Definition 2 === {{:Definition:Point of Inflection/Definition 2}}
Definition:Point of Inflection/Definition 1
$f$ has a '''point of inflection at $\xi$'''  $\xi$ is a point on $f$ at which $f$ changes from being concave to convex, or vice versa.
Definition:Point of Inflection/Definition 2
$f$ has a '''point of inflection at $\xi$'''  the derivative $f'$ of $f$ has either a local maximum or a local minimum at $\xi$.
Definition:Ordering on Mappings
Let $S$ be a set. Let $\left({T, \preceq}\right)$ be an ordered set. Let $f, g: S \to T$ be mappings. Then '''ordering on mappings''' $f$ and $g$ denoted $f \preceq g$ is defined by :$\forall s \in S: f\left({s}\right) \preceq g\left({s}\right)$
Definition:Euler-Lagrange Equation
The '''Euler–Lagrange equation''' is an equation satisfied by a function $\mathbf q$ of a real argument $t$, which is a stationary point of the functional: :$\ds \map S {\mathbf q} = \int_a^b \map L {t, \map {\mathbf q} t, \map {\mathbf q'} t} \rd t$ where: :$\mathbf q$ is the function to be found: :$\mathbf q: \closedint a b \subset \R \to X : t \mapsto x = \map {\mathbf q} t$ such that: ::$\mathbf q$ is differentiable ::$\map {\mathbf q} a = \mathbf x_a$ ::$\map {\mathbf q} b = \mathbf x_b$ :$\mathbf q'$ is the derivative of $\mathbf q$: ::$\mathbf q': \closedint a b \to T_{\map {\mathbf q} t} X: t \mapsto v = \map {\mathbf q'} t$ :$T_{\map {\mathbf q} t} X$ denotes the tangent space to $X$ at the point $\map {\mathbf q} t$ :$L$ is a real-valued function with continuous first partial derivatives: ::$L: \closedint a b \times T X \to \R: \tuple {t, x, v} \mapsto \map L {t, x, v}$ where: :$T X$ is the tangent bundle of $X$ defined by: ::$T X = \bigcup_{x \mathop \in X} \set x \times T_x X$ The ''Euler–Lagrange equation'', then, is given by: :$\displaystyle \map {L_x} {t, \map {\mathbf q} t, \map {\mathbf q'} t} - \dfrac \d {\d t} \map {L_v} {t, \map {\mathbf q} t, \map {\mathbf q'} t} = 0$ where: :$L_x$ and $L_v$ denote the partial derivatives of $L$ with respect to the second and third arguments respectively.
Definition:Oscillation/Real Space/Oscillation at Point/Infimum
Let $\mathcal N_x$ be the set of open subset neighborhoods of $x$. The '''oscillation of $f$ at $x$''' is defined as: :$\displaystyle \omega_f \left({x}\right) := \inf_{U \mathop \in \mathcal N_x} \omega_f \left({U \cap X}\right)$ where $\omega_f \left({U \cap X}\right)$ denotes the oscillation of $f$ on $U \cap X$.
Definition:Oscillation/Real Space/Oscillation at Point/Limit
The '''oscillation of $f$ at $x$''' is defined as: :$\displaystyle \omega_f \left({x}\right) := \lim_{h \to 0^+} \omega_f \left({\left({x - h \,.\,.\, x + h}\right) \cap X}\right)$ where $\omega_f \left({\left({x - h \,.\,.\, x + h}\right) \cap X}\right)$ denotes the oscillation of $f$ on $\left({x - h \,.\,.\, x + h}\right) \cap X$.
Definition:Language of Predicate Logic
In order to define $\LL_1$, it is necessary to specify: * An alphabet $\AA$ * A collation system with the unique readability property for $\AA$ * A formal grammar (which determines the WFFs of $\LL_1$)
Definition:Abel's Integral Equation
'''Abel's integral equation''' is an integral equation whose purpose is to solve Abel's mechanical problem, which finds how long it will take a bead to slide down a wire. The purpose of '''Abel's integral equation''' is to find the shape of the curve into which the wire is bent in order to yield that result: Let $T \left({y}\right)$ be a function which specifies the total time of descent for a given starting height. :$\displaystyle T \left({y_0}\right) = \int_{y \mathop = y_0}^{y \mathop = 0} \, \mathrm d t = \frac 1 {\sqrt {2 g} } \int_0^{y_0} \frac 1 {\sqrt {y_0 - y} } \frac {\mathrm d s} {\mathrm d y} \, \mathrm d y$ where: :$y$ is the height of the bead at time $t$ :$y_0$ is the height from which the bead is released :$g$ is Acceleration Due to Gravity :$s \left({y}\right)$ is the distance along the curve as a function of height.
Definition:Abelian Integral
An '''Abelian integral''' is a complex Riemann integral of the form :$\displaystyle \int_{z_0}^z R \left({x, w}\right) \mathrm d x$ where $R \left({x, w}\right)$ is an arbitrary rational function of the two variables $x$ and $w$. These variables are related by the equation: :$F \left({x, w}\right) = 0$  where $F \left({x, w}\right)$ is an irreducible polynomial in $w$: :$F \left ({x, w}\right) \equiv \phi_n \left({x}\right) w^n + \cdots + \phi_1 \left({x}\right) w + \phi_0 \left({x}\right)$ whose coefficients $\phi_j \left({x}\right), j = 0, 1, \ldots, n$ are rational functions of $x$.  {{NamedforDef|Niels Henrik Abel|cat = Abel}}
Definition:Abelian Function
An '''Abelian function''' is an inverse function of an Abelian integral. {{NamedforDef|Niels Henrik Abel|cat = Abel}}
Definition:Dirichlet Function
A '''Dirichlet function''' $D: \R \to \R$ is a real function defined as: :$\forall x \in \R: \map D x = \begin {cases} c & : x \in \Q \\ d & : x \notin \Q \end {cases}$ for $c, d \in \R$ such that $c \ne d$. The canonical example of this has $c = 1$ and $d = 0$. {{NamedforDef|Johann Peter Gustav Lejeune Dirichlet|cat = Dirichlet}}
Definition:Liouville's Constant
'''Liouville's constant''' is the real number defined as: {{begin-eqn}} {{eqn | l = \sum_{n \mathop \ge 1} \dfrac 1 {10^{n!} }       | r = \frac 1 {10^1} + \frac 1 {10^2} + \frac 1 {10^6} +  \frac 1 {10^{24} } + \cdots       | c =  }} {{eqn | r = 0 \cdotp 11000 \, 10000 \, 00000 \, 00000 \, 00010 \, 00 \ldots       | c =  }} {{end-eqn}} {{OEIS|A012245}}
Definition:Riemann Surface/Elliptic/Definition 2
A '''Riemann surface''' is '''elliptic'''  it admits a metric of constant positive curvature.
Definition:Riemann Surface/Elliptic
{{:Definition:Riemann Surface/Elliptic/Definition 1}}
Definition:Riemann Surface/Parabolic
{{:Definition:Riemann Surface/Parabolic/Definition 1}}
Definition:Riemannian Geometry (Mathematical Branch)
'''Riemannian geometry''' is the branch of differential geometry that studies Riemannian manifolds.
Definition:Riemannian Manifold
A '''Riemannian manifold''' is a smooth manifold on the real space $\R^n$ upon which a Riemannian metric has been imposed. === Dimension === {{:Definition:Riemannian Manifold/Dimension}} {{NamedforDef|Bernhard Riemann|cat = Riemann}}
Definition:Riemannian Metric
Consider a smooth manifold $\mathcal M$ on the real space $\R^n$. A '''Riemannian metric''' on $\mathcal M$ is a metric $\mathrm d s$ between nearby points $\left({x_1, x_2, \ldots, x_n}\right)$ and $\left({x_1 + \mathrm d x_1, x_2 + \mathrm d x_2, \ldots, x_n + \mathrm d x_n}\right)$ by means of the quadratic differential form: :$\displaystyle \mathrm d s^2 = \sum_{i, j \mathop = 1}^n g_{i j} \, \mathrm d x_i \, \mathrm d x_j$ where each $g_{i j}$ is a suitable real-valued function of $x_1, \ldots, x_n$. Different instances of $g_{i j}$ define different Riemannian geometries on the manifold under discussion. A manifold with such a '''Riemannian metric''' applied is known as a '''Riemannian manifold'''. {{NamedforDef|Bernhard Riemann|cat = Riemann}}
Definition:Riemann-Christoffel Tensor
A '''Riemann-Christoffel tensor''' is a tensor field which expresses the curvature of a Riemannian manifold. The '''Riemann-Christoffel tensor''' is given in terms of the Levi-Civita connection $\nabla$ by: :$R \left({u, v}\right) w = \nabla_u \nabla_v w - \nabla_v \nabla_u w - \nabla_{\left[{u, v}\right]} w$ where $\left[{u, v}\right]$ is the Lie bracket of vector fields. It measures the extent to which the metric tensor is not locally isometric to that of Euclidean space.
Definition:Riemannian Manifold/Dimension
The '''dimension''' of the '''Riemannian manifold''' on $\R^n$ is $n$.
Definition:Riemannian Geometry
A '''Riemannian geometry''' is a Riemannian manifold $\mathcal M$ together with the Riemannian metric that has been applied to $M$.
Definition:Riemann Zeta Function/Critical Line
Let $s = \sigma + i t$. The line defined by the equation $\sigma = \dfrac 1 2$ is known as the '''critical line'''. Hence the popular form of the statement of the Riemann Hypothesis: :''All the nontrivial zeroes of the Riemann zeta function lie on the '''critical line'''.''
Definition:Weierstrass Function
A '''Weierstrass function''' is a real function defined on a closed real interval $I$ which is: :$(1):\quad$ continuous on $I$ :$(2):\quad$ nowhere differentiable on $I$.
Definition:Entire Function/Transcendental
Let $f$ be an '''entire function''' that has an essential singularity at $\infty$. Then $f$ is a '''transcendental entire function'''.
Definition:Dedekind Cut/Definition 1
A '''Dedekind cut''' of $\struct {S, \preceq}$ is a non-empty proper subset $L \subsetneq S$ such that: :$(1): \quad \forall x \in L: \forall y \in S: y \prec x \implies y \in L$ ($L$ is a lower set in $S$) :$(2): \quad \forall x \in L: \exists y \in L: x \prec y$
Definition:Dedekind Cut/Definition 2
A '''Dedekind cut''' of $\struct {S, \preceq}$ is an ordered pair $\tuple {L, R}$ such that: :$(1): \quad \set {L, R}$ is a partition of $S$. :$(2): \quad L$ does not have a greatest element. :$(3): \quad \forall x \in L: \forall y \in R: x \prec y$.
Definition:Cartesian Product of Ordered Sets
Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be ordered sets. '''Cartesian product of (ordered sets)''' $\left({S_1, \preceq_1}\right)$ and $\left({S_2, \preceq_2}\right)$ is a pair $\left({S_1 \times S_2, \preceq}\right)$ where :${\preceq} \subseteq \left({S_1 \times S_2}\right) \times \left({S_1 \times S_2}\right)$ such that :$\forall s_1, t_1 \in S_1, s_2, t_2 \in S_2: \left({s_1,s_2}\right) \preceq \left({t_1,t_2}\right) \iff s_1 \preceq_1 t_1 \land s_2 \preceq_2 t_2$
Definition:Prime Number/Definition 7
A '''prime number''' $p$ is an integer greater than $1$ which cannot be written in the form: :$p = a b$ where $a$ and $b$ are both positive integers less than $p$.
Definition:Perfect Number/Definition 1
A '''perfect number''' is a (strictly) positive integer equal to its aliquot sum.
Definition:Perfect Number/Definition 2
A '''perfect number''' $n$ is a (strictly) positive integer such that: : $\sigma \left({n}\right) = 2 n$ where $\sigma: \Z_{>0} \to \Z_{>0}$ is the sigma function.
Definition:Momentum
=== Linear Momentum === {{:Definition:Linear Momentum}} === Angular Momentum === {{:Definition:Angular Momentum}}
Definition:Linear Momentum/Relativistic Model
A more accurate model for the linear momentum of a body is given by: :$\mathbf p = \gamma m \mathbf v$ where $\gamma$ is the Lorentz Factor: :$\gamma = \dfrac c {\sqrt {c^2 - v^2} } = \dfrac 1 {\sqrt {1 - v^2 / c^2} }$ where: : $c$ is the speed of light in vacuum : $v$ is the magnitude of $\mathbf v$: $v = \size {\mathbf v}$ It is clear $\gamma \approx 1$ (and thus that $\mathbf p \approx m \mathbf v$) for values of $v$ much less than $c$.
Definition:Thrust
The '''thrust''' of a rocket is defined as the product of the exhaust velocity and the rate at which its fuel is being consumed: :$\mathbf t = \mathbf b \paren {\dfrac {\d m} {\d t} }$ where: :$\mathbf b$ is the exhaust velocity :$m$ is the mass of the rocket as a function of time $t$.
Definition:Meet-Continuous Lattice
Let $\left({S, \preceq}\right)$ be a meet semilattice. Then $\left({S, \preceq}\right)$ is '''meet-continuous'''  :$\left({S, \preceq}\right)$ is up-complete and ::(MC): for every an element $x \in S$ and a directed subset $D$ of $S$: $x \wedge \sup D = \sup \left\{ {x \wedge d: d \in D}\right\}$
Definition:Twin Primes/Sequence
The sequence of twin primes begins: :$3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73, 101, 103, 107, 109, \ldots$
Definition:Prime-Counting Function/Approximations
A table of some of the values of the prime-counting ($\pi$) function compared with $\dfrac x {\ln x}$ and the Eulerian logarithmic integral $\displaystyle \map \Li x = \int_2^x \frac {\d t} {\map \ln t}$: :{| border="1" |- ! align="right" style = "padding: 2px 10px" | $n$  ! align="right" style = "padding: 2px 10px" | $\map \pi n$  ! align="right" style = "padding: 2px 10px" | $\dfrac x {\ln x}$ ! align="right" style = "padding: 2px 10px" | $\map \Li x$ |- | align="right" style = "padding: 2px 10px" | $1 \, 000$ | align="right" style = "padding: 2px 10px" | $168$ | align="right" style = "padding: 2px 10px" | $145$ | align="right" style = "padding: 2px 10px" | $178$ |- | align="right" style = "padding: 2px 10px" | $10 \, 000$ | align="right" style = "padding: 2px 10px" | $1 \, 229$ | align="right" style = "padding: 2px 10px" | $1 \, 068$ | align="right" style = "padding: 2px 10px" | $1 \, 246$ |- | align="right" style = "padding: 2px 10px" | $100 \, 000$ | align="right" style = "padding: 2px 10px" | $9 \, 596$ | align="right" style = "padding: 2px 10px" | $8 \, 686$ | align="right" style = "padding: 2px 10px" | $9 \, 630$ |- | align="right" style = "padding: 2px 10px" | $1 \, 000 \, 000$ | align="right" style = "padding: 2px 10px" | $78 \, 498$ | align="right" style = "padding: 2px 10px" | $72 \, 382$ | align="right" style = "padding: 2px 10px" | $78 \, 628$ |- | align="right" style = "padding: 2px 10px" | $10 \, 000 \, 000$ | align="right" style = "padding: 2px 10px" | $664 \, 579$ | align="right" style = "padding: 2px 10px" | $620 \, 421$ | align="right" style = "padding: 2px 10px" | $664 \, 918$ |}