hcju/proofwikips huggingface dataset:
Definition:Nash Equilibrium
Let a strategic game $G$ be modelled by: :$G = \stratgame N {A_i} {\succsim_i}$ A '''Nash equilibrium''' of $G$ is a profile $a^* \in A$ of moves which has the property that: :$\forall i \in N: \forall a_i \in A_i: \tuple {a^*_{-i}, a^*_i} \succsim_i \tuple {a^*_{-i}, a_i}$ Thus, for $a^*$ to be a '''Nash equilibrium''', no player $i$ has a move yielding a preferable outcome to that when $a^*_i$ is chosen, given that every other player $j$ has chosen his own equilibrium move. That is, no player can profitably deviate, if no other player also deviates. {{NamedforDef|John Forbes Nash|cat = Nash}}
Definition:Best-Response Function
Let a strategic game $G$ be modelled by: :$G = \left\langle{N, \left\langle{A_i}\right\rangle, \left\langle{\succsim_i}\right\rangle}\right\rangle$ Let $a^*$ be a Nash equilibrium of $G$: :$\forall i \in N: \forall a_i \in A_i: \left({a^*_{-i}, a^*_i}\right) \succsim_i \left({a^*_{-i}, a_i}\right)$ For any $a_{-1} \in A_{-i}$, let $B_i \left({a_{-i} }\right)$ be the set of player $i$'s best moves, defined as: :$B_i \left({a_{-i} }\right) = \left\{ {a_i \in A_i: \forall a'_i \in A_i: \left({a_{-i}, a_i}\right) \precsim_i  \left({a_{-i}, a'_i}\right)}\right\}$ Then $B_{-i}$ is known as the '''best-response function''' of player $i$.
Definition:Bach or Stravinsky?
There are two players: $\text A$lexis and $\text B$everley. They wish to go out together to a musical concert, but $\text A$ prefers Bach and $\text B$ prefers Stravinsky. The key points are: :$\text A$lexis and $\text B$everley wish to coordinate their behaviour but: :they have conflicting interests.
Definition:Coordination Game
There are two players: $\text A$lexis and $\text B$everley. They wish to go out together to a musical concert to experience either the music of Mozart or Mahler. Unaccountably, both $\text A$ and $\text B$ prefer Mozart.  (It takes all sorts to make a world.) The key points are: :$\text A$lexis and $\text B$everley wish to coordinate their behaviour but: :they have common interests.
Definition:Strictly Competitive Game
A '''strictly competitive game''' is a game in which the interests of each player are diametrically opposed.
Definition:Sealed-Bid Auction
A '''sealed-bid auction''' is a game in the following format: An object $F$ is to be assigned to a player in a set $\left\{ {1, 2, \ldots, n}\right\}$ in exchange for a payment. The valuation of $F$ by player $i$ is $v_i$, and: :$v_1 > v_2 > \cdots > v_n > 0$ Each player simultaneously submits a non-negative number, called a '''bid'''. $F$ is then given to the player with the lowest index among those who submit the highest bid, in exchange for a payment. === First Price Auction === {{:Definition:Sealed-Bid Auction/First Price}} === Second Price Auction === {{:Definition:Sealed-Bid Auction/Second Price}}
Definition:Sealed-Bid Auction/First Price
A '''first price auction''' is a sealed-bid auction in which the payment made by the winner is the price which is bid by that player.
Definition:Sealed-Bid Auction/Second Price
A '''second price auction''' is a sealed-bid auction in which the payment made by the winner is the highest bid made by the players who did ''not'' win. So if only one player submits the highest bid, the price paid is the second highest bid.
Definition:Weakly Dominant Move
Let $G$ be a game. Let $a$ be a move available to player $P$. Then $a$ is '''weakly dominant'''  :for any move of an opposing player, the payoff of $a$ is not less than the payoff of any other move available to $P$.
Definition:Intelligent Player
Let $G$ be a game. Let $P$ be a player in $G$. Then $P$ is described as '''intelligent'''  $P$ knows everything about $G$ that there is to know, and can use that knowledge to make informed decisions about what move to make. In game theory, all players are assumed to be '''intelligent'''.
Definition:Decision Theory
'''Decision theory''' is the study of the reasoning underlying an agent's choices.
Definition:Bayesian Decision Theory
'''Bayesian decision theory''' is a branch of decision theory which is informed by Bayesian probability. It is a statistical system that tries to quantify the tradeoff between various decisions, making use of probabilities and costs.
Definition:Subjective Probability Distribution
A '''subjective probability distribution''' is a probability distribution based on the beliefs of a rational decision-maker about all relevant unknown factors concerning a game. As new information becomes available, the '''subjective probabilities''' can then revised according to Bayes' Formula.
Definition:Lottery/Probability Model
In a '''probability model''', a lottery is a probability distribution over a set of prizes.
Definition:Lottery/State-Variable Model
In a '''state-variable model''', a lottery is defined as a mapping from a set of possible states into a set of prizes.
Definition:Objective Unknown
An '''objective unknown''' is an event which has a well-defined objective probability of occurring. Two '''objective unknowns''' with the same probability are equivalent in the field of decision theory. An '''objective unknown''' is appropriately modelled by means of a probability model.
Definition:Subjective Unknown
An '''subjective unknown''' is an event which does not have a well-defined objective probability of occurring. An '''subjective unknown''' is appropriately modelled by means of a state-variable model. === Examples === {{:Definition:Subjective Unknown/Examples}}
Definition:Prize
Let $G$ be a game. A prize in $G$ is a payoff whose utility value is strictly positive. The term is usually seen in the context of a lottery.
Definition:State
Let $G$ be a game whose outcome is determined by the realization of a random variable $X$. Each of the possible values that can be taken by $X$ is known as a '''state''' of $G$.
Definition:True State
Let $G$ be a game whose outcome is determined by the realization of a random variable $X$. The particular state of $G$ which $X$ actually takes is known as the '''true state (of the world)'''.
Definition:Conditional Preference
Let $G$ be a lottery. Let $P$ be a player of $G$. Let $X$ denote the set of prizes which $P$ may receive. Let $\Omega$ denote the set of possible states of $G$. Let $\Xi$ be the event space of $G$. Let $L$ be the set of all plays of $G$. Let $f$ and $g$ be two lotteries in $G$. Let $S \subseteq \Xi$ be an event. A '''conditional preference''' is a preference relation $\succsim_S$ such that: :$f \succsim_S g$  $f$ would be at least as desirable to $P$ as $g$, if $P$ was aware that the true state of the world was $S$. That is, $f \succsim_S g$  $P$ prefers $f$ to $g$ and he knows only that $S$ has occurred. The notation $a \sim_S b$ is defined as: :$a \sim_S b$  $a \succsim_S b$ and $b \succsim_S a$ The notation $a \succ_S b$ is defined as: :$a \succ_S b$  $a \succsim_S b$ and $a \not \sim_S a$ When no conditioning event $S$ is mentioned, the notation $a \succsim_\Omega b$, $a \succ_\Omega b$ and $a \sim_\Omega b$ can be used, which mean the same as $a \succsim b$, $a \succ b$ and $a \sim b$. {{handwaving|Myerson is as lax as all the other game theory writers when it comes to defining rigorous concepts. I am going to have to abandon this field of study until I really understand exactly what the underlying mathematical objects are.}}
Definition:Biquadrate
A '''biquadrate number''' (or just '''biquadrate''') is a number which can be expressed as the fourth power of an integer. The sequence of positive '''biquadrates''' begins: :$1, 16, 81, 256, 625, 1297, 2401, 4096, 6561, 10000, \ldots$ {{OEIS|A000583}}
Definition:Egyptian Fraction
An '''Egyptian fraction''' is a fraction either: :whose numerator is $1$ or :which is $\dfrac 2 3$
Definition:Primorial/Prime
Let $p_n$ be the $n$th prime number. Then the '''$n$th primorial''' $p_n \#$ is defined as: :$\displaystyle p_n \# := \prod_{k \mathop = 1}^n p_k$ That is, $p_n \#$ is the product of the first $n$ primes.
Definition:Primorial/Positive Integer
Let $n$ be a positive integer. Then: : $\displaystyle n\# := \prod_{i \mathop = 1}^{\pi \left({n}\right)} p_i = p_{\pi \left({n}\right)}\#$ where $\pi \left({n}\right)$ is the prime counting function. That is, $n\#$ is defined as the product of all primes less than or equal to $n$. Thus: :$n\# = \begin{cases} 1 & : n \le 1 \\ n \left({\left({n - 1}\right)\#}\right) & : n \mbox { prime} \\ \left({n - 1}\right)\# & : n \mbox { composite} \end{cases}$
Definition:Repunit
Let $b \in \Z_{>1}$ be an integer greater than $1$. Let a (positive) integer $n$, greater than $b$ be expressed in base $b$. $n$ is a '''repunit base $b$'''  $n$ is a repdigit number base $b$ whose digits are $1$. That is, $n$ is a '''repunit base $b$'''  all of the digits of $n$ are $1$. When $b$ is the usual base $10$, $n$ is merely referred to as a '''repunit'''.
Definition:Unit Fraction
A '''unit fraction''' is a fraction whose numerator is $1$.
Definition:Myriad
'''Myriad''' is an archaic word for the number $10 \ 000$ (ten thousand).
Definition:Thousand
A '''thousand''' is $1000$: $10$ to the power of $3$: :$1000 = 10^3 = 10 \times 10 \times 10$
Definition:Almost All/Set Theory/Countable
Let $S$ be a countably infinite set. Let $P: S \to \left\{ {\text{true}, \text{false} }\right\}$ be a property of $S$ such that: :$\left\{ {s \in S: \neg P \left({s}\right)}\right\}$ is finite. Then $P$ holds for '''almost all''' of the elements of $S$.
Definition:Champernowne Constant
The '''Champernowne constant''' is the real number whose decimal expansion is formed by concatenating the positive integers in ascending order: :$C_{10} = 0 \cdotp 12345 \, 67891 \, 01112 \, 13141 \, 51617 \, 18192 \, 02122 \ldots$ {{OEIS|A033307}}
Definition:Absolutely Normal Real Number
A real number $r$ is '''absolutely normal''' if it is normal with respect to ''every'' number base $b$. That is,  its basis expansion in every number base $b$ is such that: :no finite sequence of digits of $r$ of length $n$ occurs more frequently than any other such finite sequence of length $n$.
Definition:Anomalous Cancellation
Let $r = \dfrac a b$ be a fraction where $a$ and $b$ are integers expressed in conventional decimal notation. '''Anomalous cancellation''' is a phenomenon whereby deleting (that is, cancelling) common digits from the numerator $a$ and the denominator $b$ of $r$, the value of $r$ the fraction does not change.
Definition:Newton-Mercator Series
Let $\ln x$ denote the natural logarithm function. Then: {{begin-eqn}} {{eqn | l = \map \ln {1 + x}       | r = x - \dfrac {x^2} 2 + \dfrac {x^3} 3 - \dfrac {x^4} 4 + \cdots       | c =  }} {{eqn | r = \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^{n + 1} } n x^n       | c =  }} {{end-eqn}} The series converges to the natural logarithm (shifted by $1$) for $-1 < x \le 1$. This is known as the '''Newton-Mercator series'''.
Definition:Filter on Set/Filtered Set
Let $\FF$ be a filter on $S$. Then $S$ is said to be '''filtered by $\FF$''', or just a '''filtered set'''.
Definition:Filter on Set/Trivial Filter
A filter $\FF$ on $S$ by definition specifically does ''not'' include the empty set $\O$. If a filter $\FF$ ''were'' to include $\O$, then from Empty Set is Subset of All Sets it would follow that ''every'' subset of $S$ would have to be in $\FF$, and so $\FF = \powerset S$. Such a "filter" is called '''the trivial filter''' on $S$.
Definition:Filter on Set/Definition 1
A '''filter on $S$''' (or '''filter of $S$''') is a set $\FF \subset \powerset S$ which satisfies the following conditions: {{begin-axiom}} {{axiom | n = \text F 1         | m = S \in \FF }} {{axiom | n = \text F 2         | m = \O \notin \FF }} {{axiom | n = \text F 3         | m = U, V \in \FF \implies U \cap V \in \FF }} {{axiom | n = \text F 4         | m = \forall U \in \FF: U \subseteq V \subseteq S \implies V \in \FF }} {{end-axiom}}
Definition:Filter on Set/Definition 2
A '''filter on $S$''' (or '''filter of $S$''') is a set $\FF \subset \powerset S$ which satisfies the following conditions: {{begin-axiom}} {{axiom | n = \text F 1         | m = S \in \FF }} {{axiom | n = \text F 2         | m = \O \notin \FF }} {{axiom | n = \text F 3         | m = \forall n \in \N: U_1, \ldots, U_n \in \FF \implies \bigcap_{i \mathop = 1}^n U_i \in \FF }} {{axiom | n = \text F 4         | m = \forall U \in \FF: U \subseteq V \subseteq S \implies V \in \FF }} {{end-axiom}}
Definition:Filter/Proper Filter
Let $\mathcal F$ be a filter on $\left({S, \preccurlyeq}\right)$. Then $\mathcal F$ is a '''proper filter''' on $S$  $\mathcal F \ne S$. That is,  $\mathcal F$ is a proper subset of $S$.
Definition:Closed under Mapping/Arbitrary Product
Let $\phi: X^I \to T$ be a mapping or a partial mapping, taking $I$-indexed families as arguments. Denote with $\Dom \phi$ the domain of $\phi$ (if $\phi$ is a mapping, this is simply $X^I$). A set $S$ is '''closed under $\phi$''' : :$\forall \family {s_i}_{i \mathop \in I} \in S^I \cap \Dom \phi: \map \phi {\family {s_i}_{i \mathop \in I} } \in S$ Phrased in terms of image of a mapping, this translates to: :$\map \phi {S^I \cap \Dom \phi} \subseteq S$ Thus, in words, $S$ is '''closed under $\phi$''', : :Whenever $\phi$ is defined for an $I$-indexed family from $S$, it maps that indexed family into $S$ again.
Definition:Filter Basis/Equivalent Filter Bases
Two '''filter bases''' are '''equivalent'''  they both generate the same filter.
Definition:Unit Cube
A '''unit cube''' is a cube each of whose edges is one unit long.
Definition:Cube Root/Real
Let $x \in \R_{\ge 0}$ be a positive real number. The '''cube roots of $x$''' is the real number defined as: :$x^{\paren {1 / 3} } := \set {y \in \R: y^3 = x}$ where $x^{\paren {1 / 3} }$ is the $3$rd root of $x$. The notation: :$y = \sqrt [3] x$ is usually encountered.
Definition:Archytas Curve
The '''Archytas curve''' is a curve formed as the intersection of a cone, a cylinder and a horn torus.  {{NamedforDef|Archytas of Tarentum|cat = Archytas}}
Definition:Conchoid of Nicomedes
The '''conchoid of Nicomedes''' is the plane curve defined in Cartesian coordinates as: :$\paren {x - a}^2 \paren {x^2 + y^2} = b^2 x^2$ or in polar coordinates as: :$r = b + a \sec \theta$ for some real constants $a \in \R$, $b \in \R_{> 0}$. :600px The above diagram illustrates the '''conchoid of Nicomedes''' for $b = 1$ and various values of $a$ from $0$ to $3$.
Definition:Cissoid of Diocles
Let $C$ be a a circle of radius $a$ with a distinguished point $O$ on its circumference. Let $L$ be the tangent to $C$ at the other end of the diameter of $C$ through $O$. Let $R$ be a point on the circumference of $C$. Let $OR$ be produced to meet $L$ at $S$. Let $P$ be the point on $OS$ such that $OP$ = $RS$. The '''cissoid of Diocles''' is the locus of points $P$ as $R$ travels around the circumference of $C$.
Definition:Pell Numbers
The '''Pell numbers''' are a sequence $\sequence {P_n}$ which is formally defined by the recurrence relation: :$P_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ 2 P_{n - 1} + P_{n - 2} & : \text {otherwise}\end{cases}$ The sequence of '''Pell numbers''' begins: :$0, 1, 2, 5, 12, 29, 70, 169, 408, 985, \ldots$
Definition:Pell-Lucas Numbers
The '''Pell-Lucas numbers''' are a sequence $\left \langle {Q_n}\right \rangle$ which is formally defined by the recurrence relation: :$P_n = \begin{cases} 2 & : n = 0 \\ 2 & : n = 1 \\ 2 P_{n -1} + P_{n - 2} & : \text {otherwise}\end{cases}$ The sequence of '''Pell-Lucas numbers''' begins: :$2, 2, 6, 14, 34, 82, 198, 478, 1154, \ldots$ {{OEIS|A002203}}
Definition:Best Rational Approximation
Let $x \in \R$ be an (irrational) real number. The rational number $a = \dfrac p q$ is a '''best rational approximation''' to $x$ : :$(1): \quad a$ is in canonical form, that is $p$ is coprime to $q$: $p \perp q$ :$(2): \quad \left\vert{x - \dfrac p q}\right\vert = \min \left\{ {\left\vert{x - \dfrac {p'} {q'} }\right\vert: q' \le q}\right\}$ That is: :$\left\vert{x - \dfrac p q}\right\vert$ is smaller than for any $\dfrac {p'} {q'}$ where $q' \le q$ where $\left\vert{x}\right\vert$ denotes the absolute value of $x$. === Sequence === {{definition wanted|sorted by increasing denominator}}
Definition:Prime Filter (Order Theory)
Let $\left({S, \preceq}\right)$ be an ordered set. Let $F$ be a filter in $\left({S, \preceq}\right)$. $F$ is a '''prime filter''' : :$\forall x, y \in S: \left({x \vee y \in F \implies x \in F \lor y \in F}\right)$
Definition:Quadratic Irrational/Reduced
An irrational root $\alpha$ of a quadratic equation with integer coefficients is a '''reduced quadratic irrational'''  :$(1): \quad \alpha > 1$ :$(2): \quad$ its conjugate $\tilde{\alpha}$ satisfies: ::::$-1 < \tilde{\alpha} < 0$
Definition:Power Tower Function
The '''power tower function''' is the real function $f: \R \to \R$ defined on the closed interval $\closedint {e^{-e} } {e^{1/e} }$ as: :$\map f x = x^{x^{x^{x^{x^{\cdot^{\cdot^\cdot} } } } } }$ where the tower of powers tends to infinity.
Definition:Pentagram
A '''pentagram''' is a geometric figure formed by the line segments connecting alternate vertices of a regular pentagon: :400px
Definition:Golden Rectangle
A '''golden rectangle''' is a rectangle whose sides are in extreme and mean ratio: :500px Thus, in the above rectangle: :$AB = AD$ :$AB : BC = AC : AB$
Definition:Beatty Sequence
Let $x$ be an irrational number. The '''Beatty sequence on $x$''' is the integer sequence $\BB_x$ defined as: :$\BB_x := \sequence{\floor{n x} }_{n \mathop \in \Z_{\ge 0} }$
Definition:Beatty Sequence/Complementary
Let $\mathcal B_x$ be the '''Beatty sequence''' on $x$. The '''complementary Beatty sequence on $x$''' is the integer sequence formed by the integers which are missing from $\mathcal B_x$.
Definition:Lower Wythoff Sequence
The '''lower Wythoff sequence''' is the Beatty sequence on the golden section $\phi$. It starts: :$0, 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, \ldots$
Definition:Ultrafilter (Order Theory)
Let $O = \left({S, \preceq}\right)$ be an ordered set. Let $F$ be a filter in $O$. Then $F$ is '''ultrafilter''' (on $O$)  :$F$ is proper subset of $S$ and :for all filter $G$ in $O$: $\left({F \subseteq G \implies F = G \lor G = S}\right)$
Definition:Upper Wythoff Sequence
{{:Definition:Upper Wythoff Sequence/Definition 2}} The '''upper Wythoff sequence''' starts: :$0, 2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, \ldots$
Definition:Upper Wythoff Sequence/Definition 1
The '''upper Wythoff sequence''' is the complementary Beatty sequence on the golden section $\phi$.
Definition:Upper Wythoff Sequence/Definition 2
The '''upper Wythoff sequence''' is the Beatty sequence on the square $\phi^2$ of the golden section $\phi$.
Definition:Wythoff Pair
A '''Wythoff pair''' is an ordered pair of integers of the form: :$\tuple {\floor {k \phi}, \floor {k \phi^2} }$ where: :$\phi$ denotes the golden section: $\phi = 1 \cdotp 618 \ldots$ :$\floor x$ denotes the floor of $x$ :$k$ signifies a positive integer: $k \in \Z_{\ge 0}$. Thus the coordinates of a '''Wythoff pair''' are corresponding terms of the lower and upper Wythoff sequences. {{NamedforDef|Willem Abraham Wythoff|cat = Wythoff}}
Definition:Wythoff Sequence
=== Lower Wythoff Sequence === {{:Definition:Lower Wythoff Sequence}} === Upper Wythoff Sequence === {{:Definition:Upper Wythoff Sequence}}
Definition:Finite Infima Set
Let $P = \left({S, \preceq}\right)$ be an ordered set. Let $X$ be a subset of $S$. Then '''finite infima set''' of $X$, denoted $\operatorname{fininfs}\left({X}\right)$, is defined by :$\left\{ {\inf A: A \in \mathit{Fin}\left({X}\right) \land A}\right.$ admits an infimum$\left.{}\right\}$ where $\mathit{Fin}\left({X}\right)$ denotes the set of all finite subsets of $X$.
Definition:Wythoff's Game
'''Wythoff's game''' is a two-person game whose rules are as follows: :$(1): \quad$ The game starts with two piles of counters. :$(2): \quad$ Each player takes it in turns to remove either: ::::A number of counters from one pile ::::A number of counters from both pile such that the number of counters removed from each pile must be equal. :$(3): \quad$ The person who removes the last counter (or counters) is the winner. {{NamedforDef|Willem Abraham Wythoff|cat = Wythoff}}
Definition:Tribonacci Constant
The '''Tribonacci constant''' $\eta$ is the one real root of the cubic: :$x^3 - x^2 - x - 1 = 0$ Its decimal expansion starts: :$\eta = 1 \cdotp 83928 \, 67552 \, 1416 \ldots$ {{OEIS|A058265}}
Definition:Tribonacci Sequence/General
A '''general Tribonacci sequence''' is a sequence $\left \langle {u_n}\right \rangle$ which is formally defined recursively as: :$u_n = \begin{cases} a & : n = 0 \\ b & : n = 1 \\ c & : n = 2 \\ u_{n - 1} + u_{n - 2} + u_{n - 3} & : n > 2 \end{cases}$ where $a, b, c \in \Z$ are constants.
Definition:Tribonacci Sequence
The '''Tribonacci sequence''' is a sequence $\left \langle {u_n}\right \rangle$ which is formally defined recursively as: :$u_n = \begin{cases} 0 & : n = 0 \\ 0 & : n = 1 \\ 1 & : n = 2 \\ u_{n - 1} + u_{n - 2} + u_{n - 3} & : n > 2 \end{cases}$
Definition:Brun's Constant
'''Brun's constant''' is the sum of the series consisting of the reciprocals of the twin primes: :$B_2 := \paren {\dfrac 1 3 + \dfrac 1 5} + \paren {\dfrac 1 5 + \dfrac 1 7} + \paren {\dfrac 1 {11} + \dfrac 1 {13} } + \paren {\dfrac 1 {17} + \dfrac 1 {19} } + \paren {\dfrac 1 {29} + \dfrac 1 {31} } + \cdots$ Its approxmiate decimal expansion is: :$B_2 \approx 1 \cdotp 90216 \, 05831 \, 04 \ldots$ {{OEIS|A065421}} Estimates of its value are occasionally refined as further work is done to establish its nature.
Definition:Bilateral Symmetry
Let $F$ be a geometric figure. Let there exists a reflection $R$ in the such that $\map R F$ is a symmetry. Then $F$ has '''bilateral symmetry'''. === Axis of Symmetry === {{:Definition:Bilateral Symmetry/Axis}}
Definition:Metre/Square Metre
The '''square metre''' is the SI unit of area. The symbol for the '''square metre''' is $\mathrm m^2$.
Definition:Centimetre/Square Centimetre
The '''square centimetre''' is the CGS unit of area. The symbol for the '''square centimetre''' is $\mathrm {cm}^2$ or (the informal and ugly) $\mathrm {sq}. \ \mathrm {cm}$.
Definition:SI Units/Derived
The units derived from the SI base units include the following: {{begin-axiom}} {{axiom | lc= Square metre:         | m = \mathrm m^2 }} {{axiom | lc= Cubic metre:         | m = \mathrm m^3 }} {{axiom | lc= Newton:         | m = \mathrm {kg} \, \mathrm m \, \mathrm s^{-2} }} {{axiom | lc= Coulomb:         | m = \mathrm A \, \mathrm s }} {{end-axiom}}
Definition:FPS/Base Units
{| class="wikitable" style="margin:1em auto 1em auto" |+ FPS base units |- !Name ! Unit symbol ! Dimension ! Symbol |- ! foot | $\mathrm {ft}$ | Length | $l$ |- ! pound | $\mathrm {lb}$ | Mass | $m$ |- ! second | $\mathrm s$ | Time | $t$ |- |} === Foot === {{:Definition:FPS/Foot}} === Pound === {{:Definition:FPS/Pound}} === Second === {{:Definition:FPS/Second}}
Definition:FPS/Pound
The '''pound''' is the FPS unit of measurement of mass, which has a number of different standards. The most common standard is for it to be defined in kilograms to be exactly $0.453 \, 592 \, 37 \ \mathrm{kg}$.
Definition:Imperial/Volume
The imperial units of volume are based on a binary system, in which each unit is a factor of $2$ larger than the next smaller unit. === Fluid Ounce === {{Definition:Imperial/Volume/Fluid Ounce}} === Gill === {{Definition:Imperial/Volume/Gill}} === Chopin === {{Definition:Imperial/Volume/Chopin}} === Pint === {{Definition:Imperial/Volume/Pint}} === Quart === {{Definition:Imperial/Volume/Quart}} === Pottle === {{Definition:Imperial/Volume/Pottle}} === Gallon === {{Definition:Imperial/Volume/Gallon}} === Peck === {{Definition:Imperial/Volume/Peck}} === Demi-Bushel === {{Definition:Imperial/Volume/Demi-Bushel}} === Bushel === {{Definition:Imperial/Volume/Bushel}} === Kilderkin === {{Definition:Imperial/Volume/Kilderkin}} === Barrel === {{Definition:Imperial/Volume/Barrel}} === Hogshead === {{Definition:Imperial/Volume/Hogshead}} === Pipe === {{Definition:Imperial/Volume/Pipe}} === Tun === {{Definition:Imperial/Volume/Tun}}
Definition:Imperial/Volume/Tun
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''tun''' }} {{eqn | r = 2       | c = pipes }} {{eqn | r = 1 \cdotp 16379 \, 904       | c = cubic metres }} {{eqn | r = 1 \, 163 \cdotp 79904       | c = litres }} {{end-eqn}}