Definition:Imperial/Volume/Barrel

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''barrel''' }} {{eqn | r = 2       | c = kilderkins }} {{eqn | r = 0 \cdotp 14547 \, 488       | c = cubic metres }} {{eqn | r = 145 \cdotp 47488       | c = litres }} {{end-eqn}}


Definition:Litre

The '''litre''' is an SI unit of volume. It is defined as: {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''litre''' }} {{eqn | r = 1000       | c = cubic centimetres }} {{eqn | r = 10^{-3}       | c = cubic metres }} {{end-eqn}}


Definition:Imperial/Volume/Kilderkin

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''kilderkin''' }} {{eqn | r = 2       | c = bushels }} {{eqn | r = 0 \cdotp 07273 \, 744       | c = cubic metres }} {{eqn | r = 72 \cdotp 73744       | c = litres }} {{end-eqn}}


Definition:Imperial/Volume/Bushel

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''bushel''' }} {{eqn | r = 2       | c = demi-bushels }} {{eqn | r = 0 \cdotp 03636 \, 872       | c = cubic metres }} {{eqn | r = 36 \cdotp 36872       | c = litres }} {{end-eqn}}


Definition:Imperial/Volume/Demi-Bushel

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''demi-bushel''' }} {{eqn | r = 2       | c = pecks }} {{eqn | r = 18 \cdotp 18436       | c = litres }} {{end-eqn}}


Definition:Imperial/Volume/Peck

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''peck''' }} {{eqn | r = 2       | c = (imperial) gallons }} {{eqn | r = 9 \cdotp 09218       | c = litres }} {{end-eqn}}


Definition:Imperial/Volume/Gallon

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''gallon''' }} {{eqn | r = 2       | c = pottles }} {{eqn | r = 4 \cdotp 54609       | c = litres }} {{eqn | r = 4 \, 546 \cdotp 09       | c = millilitres }} {{end-eqn}}


Definition:Imperial/Volume/Pottle

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''pottle''' }} {{eqn | r = 2       | c = quarts }} {{eqn | r = 2 \cdotp 273045       | c = litres }} {{eqn | r = 2 \, 273 \cdotp 045       | c = millilitres }} {{end-eqn}}


Definition:Imperial/Volume/Quart

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''quart''' }} {{eqn | r = 2       | c = pints }} {{eqn | r = 1 \cdotp 13652 25       | c = litres }} {{eqn | r = 1 \, 136 \cdotp 5225       | c = millilitres }} {{end-eqn}}


Definition:Imperial/Volume/Pint

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''pint''' }} {{eqn | r = 2       | c = chopins }} {{eqn | r = 0 \cdotp 5682 \, 6125       | c = litres }} {{eqn | r = 568 \cdotp 26125       | c = millilitres }} {{end-eqn}}


Definition:Imperial/Volume/Chopin

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''chopin''' }} {{eqn | r = 2       | c = gills }} {{eqn | r = 0 \cdotp 28413 \, 0625       | c = litres }} {{eqn | r = 284 \cdotp 13062 \, 5       | c = millilitres }} {{end-eqn}}


Definition:Imperial/Volume/Gill

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''gill''' }} {{eqn | r = 5       | c = fluid ounces }} {{eqn | r = 0 \cdotp 14206 \, 53125       | c = litres }} {{eqn | r = 142 \cdotp 06531 \, 25       | c = millilitres }} {{end-eqn}}


Definition:Imperial/Volume/Fluid Ounce

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''fluid ounce''' }} {{eqn | r = \dfrac 1 {20}       | c = pint }} {{eqn | r = 28 \cdotp 41306 \, 25       | c = millilitres }} {{end-eqn}}


Definition:Wallis's Number

'''Wallis's number''' is the real root of the cubic: :$x^3 - 2 x - 5 = 0$ Its approximate value is: :$2 \cdotp 09455 \, 1 \ldots$ {{OEIS|A007493}}


Definition:Stirling's Constant

'''Stirling's constant''' is a name given to the square root of $2$ times $\pi$ (pi). It is approximated by the decimal expansion: :$\sqrt {2 \pi} \approx 2 \cdotp 50662 \, 82746 \, 31000 \, 5 \ldots$ {{OEIS|A019727}} {{NamedforDef|James Stirling|cat = Stirling}}


Definition:Gelfond's Constant

'''Gelfond's constant''' is the number obtained by raising Euler's number $e$ to the power of $\pi$ (pi): :$e^\pi$


Definition:Time/Fourth Dimension

In the Theory of Relativity, it is convenient to consider time as another dimension in addition to, and treated similarly to, the $3$ dimensions of ordinary space.


Definition:Quadratrix of Hippias/Definition 1

The '''quadratrix of Hippias''' is the plane curve defined in Cartesian coordinates as: :$y = x \cot \left({\dfrac {\pi x }{2 a} }\right)$ for some real constant $a \in \R$. :600px The above diagram illustrates the '''quadratrix of Hippias'''.


Definition:Quadratrix of Hippias

=== Definition 1 === {{:Definition:Quadratrix of Hippias/Definition 1}} === Definition 2 === {{:Definition:Quadratrix of Hippias/Definition 2}}


Definition:Orthocenter

Let $\triangle ABC$ be a triangle. The '''orthocenter''' of $\triangle ABC$ is where its three altitudes meet. :500px


Definition:Medial Triangle

Let $\triangle ABC$ be a triangle. The '''medial triangle''' of $\triangle ABC$ is the triangle formed by joining the midpoint of each of the sides of $\triangle ABC$.


Definition:Euler Line

Let $\triangle ABC$ be a triangle which is not equilateral. The '''Euler line''' of $\triangle ABC$ is the straight line through the orthocenter, centroid and circumcenter of $\triangle ABC$. :700px In the above, the line $OGK$ is the '''Euler line''' of $\triangle ABC$.


Definition:Tessellation

A '''tessellation''' is a division of the plane into plane geometric figures, referred to as tiles. The division is made in such a way that the tiles are fitted together with no gaps and no overlaps.


Definition:Tile

A '''tile''' is a plane geometric figure which forms part of a tessellation. Category:Definitions/Tessellations fpbdcx8a60ckfnpjfhai3x6htecvlwa


Definition:Regular Tessellation

A '''regular tessellation''' is a tessellation such that: :Its tiles consist of congruent regular polygons :No vertex of one tile may lie against the side of another tile.


Definition:Magic Square

A '''magic square''' is an arrangement of $n^2$ numbers into an $n \times n$ square array such that: :the sum of the elements of each row :the sum of the elements in each column :the sum of the elements along each diagonal are the same. When the elements are not specified, it is conventional for them to be the first $n^2$ (strictly) positive integers.


Definition:Magic Square/Order

An $n \times n$ magic square is called an '''order $n$ magic square'''.


Definition:Coloring/Why Colors?

It is clear that the nature of the actual elements of a coloring $C$ is irrelevant. They are traditionally referred to as '''colors''' because this subfield of graph theory arose from considerations of the coloring of the faces of planar graphs such that adjacent faces have different '''colors'''. This was the origin of the famous Four Color Theorem.


Definition:Coloring/Vertex Coloring

A '''vertex $k$-coloring''' of a simple graph $G = \left({V, E}\right)$ is defined as an assignment of one element from a set $C$ of $k$ colors to each vertex in $V$. That is, a '''vertex $k$-coloring''' of the graph $G = \left({V, E}\right)$ is a mapping $c: V \to \left\{{1, 2, \ldots k}\right\}$. A graph with such a coloring is called a '''vertex-colored graph'''.


Definition:Continued Square Root

A '''continued square root''' is an expression of the form: :$a_0 + \sqrt {a_1 + \sqrt {a_2 + \sqrt {a_3 + \sqrt {\cdots} } } }$ where $\left\langle{a_n}\right\rangle$ is an arbitrary sequence of numbers.


Definition:Perfect Power

A '''perfect power''' is an integer which can be expressed in the form: :$a^k$ where both $a$ and $k$ are integers with $a \ge 1$ and $k \ge 2$. === Sequence === {{:Definition:Perfect Power/Sequence}}


Definition:Perfect Power/Sequence

The sequence of perfect powers begins: :$1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, \ldots$


Definition:Zu Chongzhi Fraction

The '''Zu Chongzhi fraction''' is an exceptionally accurate approximation to $\pi$ (pi): :$\pi \approx \dfrac {355} {113}$ whose decimal expansion is: :$\pi \approx 3 \cdotp 14159 \, 292$ {{OEIS|A068079}}


Definition:Squarely Irrational Number

A number is '''squarely irrational''' if both the number itself and its square is irrational. Thus, the set of the '''squarely irrational numbers''' is $\left\{{x \in \R \setminus \Q : x^2 \in \R \setminus \Q}\right\}$.


Definition:Rationally Expressible Number

A number is '''rationally expressible''' if its square is rational. Thus, the set of the '''rationally expressible numbers''' is $\set {x \in \R : x^2 \in \Q}$.


Definition:Square Root/Historical Note

It is suggested by some sources that the symbol $\surd$ (a stylised '''r''' for '''radix''') for the '''square root''' may have originated with {{AuthorRef|René Descartes}}, but there is evidence that it may have been around a lot earlier than that.


Definition:Philosophical Element

The Pythagoreans postulated that the Universe was composed of $4$ '''elements''': :Earth :Water :Air :Fire. Subsequently, the alchemical tradition proposed the existence of a fifth '''element''': quintessence, or aether. === Earth === {{:Definition:Philosophical Element/Earth}} === Water === {{:Definition:Philosophical Element/Water}} === Air === {{:Definition:Philosophical Element/Air}} === Fire === {{:Definition:Philosophical Element/Fire}} === Quintessence === {{:Definition:Philosophical Element/Quintessence}}


Definition:Philosophical Element/Earth

'''Earth''' was one of the original four elements that were postulated by the Pythagoreans to compose the Universe. All the '''earthy''' matter of the Universe was supposed to collect at its natural place, which was the centre of the Universe, that is, Earth itself.


Definition:Philosophical Element/Air

'''Air''' was one of the original four elements that were postulated by the Pythagoreans to compose the Universe. It had its natural place around the rim of the sphere of water.


Definition:Philosophical Element/Fire

'''Fire''' was one of the original four elements that were postulated by the Pythagoreans to compose the Universe. It had its natural place outside the sphere of air.


Definition:Philosophical Element/Water

'''Water''' was one of the original four elements that were postulated by the Pythagoreans to compose the Universe. It had its natural place about the rim of the surface of Earth.


Definition:Linearly Irrational Number

A number is '''linearly irrational''' if both the number itself is irrational but its square is rational. Thus, the set of the '''linearly irrational numbers''' is $\left\{{x \in \R \setminus \Q : x^2 \in \Q}\right\}$.


Definition:Exponential Order

Let $f: \R \to \F$ be a function, where $\F \in \set {\R, \C}$. Let $f$ be continuous on the real interval $\hointr 0 \to$, except possibly for some finite number of discontinuities of the first kind in every finite subinterval of $\hointr 0 \to$. {{explain|Establish whether it is "finite subinterval" that is needed here, or what we have already defined as "Definition:Finite Subdivision".}} Then $f$ is said to be of '''exponential order''', denoted $f \in \mathcal E$, {{iff}} it is of exponential order $a$ for some $a > 0$.


Definition:Philosophical Element/Quintessence

'''Quintessence''' was the fifth element postulated by the Pythagoreans, and was supposed to be the material out of which the heavens were made. It was postuiated in order to explain how the heavenly bodies were in constant motion, and never came to rest like everything else that could be seen in the universe.


Definition:Pythagoreans/Quadrivium

The '''quadrivium''' was the medieval name of the required course of study of the Pythagoreans, which had been adopted by the educational establishments in Europe. The required bodies of knowledge were divided into '''discrete''' and '''continuous''': Thus: :'''Arithmetic''': study of the absolute discrete :'''Music''': study of the relative discrete :'''Geometry''': study of the '''stable''' continuous :'''Astronomy''': study of the '''moving''' continuous.


Definition:Pythagoreans/Trivium

The '''trivium''' was the medieval name of the supplementary course of study of the Pythagoreans, adopted by the educational establishments in Europe. These supplementary bodies of knowledge were: :'''Grammar''' :'''Rhetoric''' :'''Logic'''.


Definition:Pythagoreans/Quadrivium/Linguistic Note

The word '''quadrivium''' is a Latin word meaning '''the four ways''', or '''place where four roads meet''' (that is, literally: '''crossroads''').


Definition:Pythagoreans/Trivium/Linguistic Note

The word '''trivium''' is a Latin word meaning '''the three ways''', or '''place where three roads meet'''. As the '''trivium''' was a preliminary course of study to the quadrivium, and thereby simpler, from this root the word '''trivial''' evolved, which has a meaning in accordance with '''insignificant''', '''self-evident''' and '''commonplace''', and the like.


Definition:Dual Polyhedron

Let $P$ be a polyhedron. The '''dual polyhedron''' $D$ of $P$ is the polyhedron which can be constructed as follows: :$(1): \quad$ The vertices of $D$ are the centroids of the faces of $P$. :$(2): \quad$ For each edge of $P$ which is adjacent to two faces $F_1$ and $F_2$ of $P$, an edge of $D$ is constructed which is adjacent to the vertices of $D$ forming the centroids of $F_1$ and $F_2$.


Definition:Brown Numbers

'''Brown numbers''' are pairs of integers $\left({m, n}\right)$ such that: :$n! + 1 = m^2$ There are only $3$ known such pairs: :$\left({5, 4}\right)$ :$\left({11, 5}\right)$ :$\left({71, 7}\right)$


Definition:Feigenbaum Constants

The '''Feigenbaum constants''' are a pair of real constants which arise in bifurcation theory. === First Feigenbaum Constant === {{:Definition:Feigenbaum Constants/First}} === Second Feigenbaum Constant === {{:Definition:Feigenbaum Constants/Second}} {{NamedforDef|Mitchell Jay Feigenbaum|cat = Feigenbaum}} Category:Definitions/Bifurcation Theory 0as984mc1pfs2fgfhoyhgnds31c9x4p


Definition:Feigenbaum Constants/First

The '''first Feigenbaum constant''' is the limiting ratio of each bifurcation interval to the next between every period doubling of a one-parameter mapping: :$x_{i + 1} = f \left({x_i}\right)$ where $f \left({x}\right)$ is a function parameterized by the bifurcation parameter $a$. It is given by the limit: :$\displaystyle \delta = \lim_{n \mathop \to \infty} \dfrac{a_ {n - 1} - a_{n - 2} } {a_n - a_{n - 1} } = 4 \cdotp 66920 \, 16091 \, 02990 \, 67185 \, 32038 \, 20466 \, 20161 \, 72 \ldots$ where $a_n$ are discrete values of $a$ at the $n$th period doubling. Its precise value appears to be a topic of research, as various resources quote it differently from the above, including the {{AuthorRef|David Wells}} $1997$ source work {{BookLink|Curious and Interesting Numbers|David Wells|ed = 2nd|edpage = Second Edition}}.


Definition:Feigenbaum Constants/Second

The '''secondFeigenbaum constant''' $\alpha$ is the ratio between the width of a tine and the width of one of its two subtines (except the tine closest to the fold). A negative sign is applied to $\alpha$ when the ratio between the lower subtine and the width of the tine is measured. Its approximate value is given by: :$\alpha \approx 2 \cdotp 50290 \, 78750 \, 95892 \, 82228 \, 39028 \, 73218 \, 21578 \cdots$


Definition:Pythagorean Triangle

A '''Pythagorean triangle''' is a right triangle whose sides all have lengths which are integers.


Definition:Automorphic Number

An '''automorphic number''' is a positive integer all of whose powers end in that number.


Definition:Absolutely Convergent Integral

Let $I = \displaystyle \int_S \map f x \rd x$ be an improper integral, where $S$ is an unbounded closed interval. $I$ is '''absolutely convergent''' {{iff}} $\displaystyle \int_S \size {\map f x} \rd x$ converges.


Definition:Untouchable Number

An '''untouchable number''' is a (strictly) positive integer that is not the aliquot sum of any integer.


Definition:Polytope

A '''polytope''' is an $n$-dimensional generalization of a polygon and polyhedron, for $n \in \Z_{>0}$.


Definition:Simplex

A '''simplex''' is an $n$-dimensional generalization of a triangle and tetrahedron, for $n \in \Z_{>0}$. A $k$-simplex is a $k$-dimensional polytope which is the convex hull of its $k + 1$ vertices.


Definition:Pentatope

A '''pentatope''' is a $4$-simplex: :400px


Definition:Triangle (Geometry)/Right-Angled/Legs

In a right-angled triangle, the two sides which are not the hypotenuse are referred to as its '''legs'''.


Definition:Fibonacci Number/Negative

The definition of '''Fibonacci numbers''' for negative integers is an extension of the definition for positive integers: :$F_n = \begin{cases} 0 & : n = 0 \\ 1 & : n = 1 \\ F_{n + 2} - F_{n + 1} & : n < 0 \end{cases}$ for all $n \in \Z$.


Definition:Extension of Sequence

As a sequence is, by definition, also a mapping, the definition of an '''extension of a sequence''' is the same as that for an extension of a mapping: Let: : $\left \langle {a_k} \right \rangle_{k \mathop \in A}$ be a sequence on $A$, where $A \subseteq \N$. : $\left \langle {b_k} \right \rangle_{k \mathop \in B}$ be a sequence on $B$, where $B \subseteq \N$. : $A \subseteq B$ : $\forall k \in A: b_k = a_k$. Then $\left \langle {b_k} \right \rangle_{k \mathop \in B}$ '''extends''' or '''is an extension of''' $\left \langle {a_k} \right \rangle_{k \mathop \in A}$.


Definition:Extension of Sequence/Negative Integers

A sequence on $\N$ can be extended to the negative integers. Let $\left \langle {a_k} \right \rangle_{k \mathop \in \N}$ and $\left \langle {b_k} \right \rangle_{k \mathop \in \N}$ be sequences on $\N$. Let $a_0 = b_0$. Let $c_k$ be defined as: :$\forall k \in \Z: c_k = \begin{cases} a_k & : k \ge 0 \\ b_{-k} : & k \le 0 \end{cases}$ Then $\left \langle {c_k} \right \rangle_{k \mathop \in \Z}$ '''extends''' (or '''is an extension of''') $\left \langle {a_k} \right \rangle_{k \mathop \in \N}$ to the negative integers.


Definition:Ansatz

An '''Ansatz''' is a guessed solution, or general form of a solution, to a mathematical problem. A mathematician making the '''Ansatz''' will then attempt to prove that the '''Ansatz''' is indeed a solution to the original problem. If he or she succeeds, the '''Ansatz''' is justified retroactively.


Definition:Finite Suprema Set

Let $P = \left({S, \preceq}\right)$ be an ordered set. Let $X$ be a subset of $S$. Then '''finite suprma set''' of $X$, denoted $\operatorname{finsups}\left({X}\right)$, is defined by :$\left\{ {\sup A: A \in \mathit{Fin}\left({X}\right) \land A}\right.$ admits a supremum$\left.{}\right\}$ where $\mathit{Fin}\left({X}\right)$ denotes the set of all finite subsets of $X$.


Definition:Contour Integral/Complex/Closed

Let $C$ be a closed contour in $\C$. Then the symbol $\displaystyle \oint$ is used for the '''contour integral''' on $C$. The definition remains the same: :$\displaystyle \oint_C f \left({z}\right) \rd z := \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right) \rd t$


Definition:Goldbach Decomposition

Let $n \in \Z_{>0}$ be a positive integer such that $n$ is even. A '''Goldbach decomposition''' for $n$ is a pair of primes $p_1, p_2$ such that: :$p_1 + p_2 = n$


Definition:Cauchy Principal Value

The '''Cauchy principal value''' is an extension of the concept of an improper integral when the latter might not exist. === Real Integral === {{:Definition:Cauchy Principal Value/Real Integral}} === Complex Integral === {{:Definition:Cauchy Principal Value/Complex Integral}} === Contour Integral === {{:Definition:Cauchy Principal Value/Contour Integral}}


Definition:Tesseract

A '''tesseract''' is the $4$-dimensional regular polytope corresponding to the cube: :500px {{stub}}


Definition:Steiner Tree

Let $G$ be a configuration of points in space. A '''Steiner tree''' is a network of lines joining the points of $G$ such that the total length of those lines is a minimum. {{NamedforDef|Jakob Steiner|cat = Steiner}}


Definition:Fermat Quotient

Let $a$ be a positive integer. Let $p$ be an odd prime. The '''Fermat quotient of $a$ with respect to $p$''' is defined as: :$\map {q_p} a = \dfrac {a^{p - 1} - 1} p$


Definition:Rational Diagonal

The number $7$ (seven) was referred to by the ancient Greeks as the '''rational diagonal''' of the $5 \times 5$ square. This was on account of the fact that: :$5^2 + 5^2 = 50 \approx 7^2 = 49$


Definition:Dissection

A '''dissection''' is a partition of a geometric figure into two or more components.


Definition:Frieze

A '''frieze''' is a plane geometrical figure which has translational symmetry in one line only.


Definition:Pseudoprime (Order Theory)

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be an up-complete lattice. Let $p \in S$. Then $p$ is '''pseudoprime''' {{iff}} :there exists a prime ideal $P$ in $L$: $p = \sup P$


Definition:Cyclotomic Polynomial

Let $n \ge 1$ be a natural number. The '''$n$th cyclotomic polynomial''' is the polynomial :$\displaystyle \Phi_n \paren x = \prod_\zeta \paren {x - \zeta}$ where the product runs over all primitive complex $n$th roots of unity, that is, those whose order is $n$.


Definition:Fano Plane

The '''Fano plane''' is the finite projective plane of order $2$. It can be denoted: :$\operatorname {PG} \left({2, 2}\right)$ where: :$\operatorname {PG}$ stands for '''projective geometry''' :the first parameter $2$ specifies the dimension :the second parameter $2$ specifies the order. $\operatorname {PG} \left({2, 2}\right)$ has the following properties: : It has $7$ points : It has $7$ lines : There are $3$ points on each line : There are $3$ lines through each point. :400px


Definition:Separable Polynomial

=== Definition 1 === {{:Definition:Separable Polynomial/Definition 1}} === Definition 2 === {{:Definition:Separable Polynomial/Definition 2}} === Definition 3 === {{:Definition:Separable Polynomial/Definition 3}}


Definition:Multiplicity (Polynomial)

Let $R$ be a commutative ring with unity. Let $P \in R \left[{X}\right]$ be a nonzero polynomial. Let $a \in R$ be a root of $P$. The '''multiplicity''' of $a$ in $P$ is the largest positive integer $n$ such that $\left({X - a}\right)^n$ divides $P \left({X}\right)$ in $R \left[{X}\right]$. {{refactor|Extract the following into a separate page, transcluded (perhaps an "examples" page)}} A '''double root''' is a root of multiplicity at least $2$. Category:Definitions/Polynomial Theory 744jj09nfz55cql8hm6ls1w0j0ryn7o


Definition:Knot Product

The '''product''' of two knots is informally defined as what you get when you tie those two knots together on the same string. {{stub|Yes I know this is sketchy and inadequate -- a lot more background is needed before this can be fleshed out and made rigorous.}}


Definition:Prime Knot

A '''prime knot''' is a knot which is not the product of two or more other knots.


Definition:Monoid Ring

Let $R$ be a ring with unity. Let $\left({G, *}\right)$ be a monoid. Let $R^{\left({G}\right)}$ be the free $R$-module on $G$. Let $\left\{ {e_g: g \in G}\right\}$ be its canonical basis. {{explain|Before the above notation can be properly understood, the precise nature of the canonical basis needs to be expanded so as to make the operations completely explicit.}} By Multilinear Mapping from Free Modules is Determined by Bases, there exists a unique bilinear map: :$\circ: R^{\left({G}\right)} \times R^{\left({G}\right)} \to R^{\left({G}\right)}$ which satisfies: :$e_g \circ e_h = e_{g \mathop * h}$ Then $R \left[{G}\right] = \left({R^{\left({G}\right)}, +, \circ}\right)$ is called the '''monoid ring of $G$ over $R$'''.


Definition:Free Module on Set

Let $R$ be a ring. {{explain|For this concept to be more easily understood, it is suggested that the ring be defined using its full specification, i.e. complete with operators, $\left({R, +, \circ}\right)$ and so on, and similarly that a symbol be used to make the module scalar product equally explicit.}} Let $I$ be an indexing set. The '''free $R$-module on $I$''' is the direct sum of $R$ as a module over itself: :$\displaystyle R^{\left({I}\right)} := \bigoplus_{i \mathop \in I} R$ of the family $I \to \{R\}$ to the singleton $\{R\}$. === Canonical Basis === {{Definition:Free Module on Set/Canonical Basis}} === Canonical Mapping === {{Definition:Free Module on Set/Canonical Mapping}}


Definition:Multiplicative Relation

Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $\mathcal R$ be a relation on $S$. Then $\mathcal R$ is '''multiplicative (relation)''' {{iff}} :$\forall a, x, y \in S: \left({\left({a, x}\right), \left({a, y}\right) \in \mathcal R \implies \left({a, x \wedge y}\right) \in \mathcal R}\right)$


Let $R, S$ be commutative rings with unity. Let $\kappa : R \to S$ be a unital ring homomorphism. === Single Indeterminate === {{:Definition:Polynomial Evaluation Homomorphism/Single Indeterminate}} === Multiple Indeterminates === {{:Definition:Polynomial Evaluation Homomorphism/Multiple Indeterminates}}


Definition:Free Module on Set/Canonical Basis

Let $R$ be a ring with unity.


Definition:Homomorphism Ring between Modules

Let $R$ be a ring. Let $M, N$ be $R$-modules. The '''homomorphism ring between $M$ and $N$''' is defined as the set of all $R$-module homomorphisms from $M$ to $N$, denoted: :$\map {\operatorname {Hom}_R} {M, N}$ or just: :$\map {\operatorname {Hom} } {M, N}$ if the specific ring $R$ in question is clear from the context.


Definition:Group Ring

Let $R$ be a ring. Let $\struct {G, +}$ be a group. The '''group ring of $G$ over $R$''' is the monoid ring of $G$ over $R$.


Definition:Power (Algebra)/Complex Number/Principal Branch

The principal branch of a complex number raised to a complex power is defined as: :$z^k = e^{k \Ln z}$ where $\Ln z$ is the principal branch of the natural logarithm.


Definition:Power (Algebra)/Complex Number/Principal Branch/Positive Real Base

Let $t > 0$ be a real number and let $k$ be a complex number. The principal branch of a positive real number raised to a complex power is defined as: :$t^k = e^{k \ln t}$ where $\ln$ is the natural logarithm of a positive real number.


Let $\struct {G, *}$ be an abelian group with identity $e$. Let $\struct {\Z, +, \times}$ be the ring of integers. === Definition 1 === {{:Definition:Z-Module Associated with Abelian Group/Definition 1}} === Definition 2 === {{:Definition:Z-Module Associated with Abelian Group/Definition 2}}


Definition:Octagonal Number

'''Octagonal numbers''' are those denumerating a collection of objects which can be arranged in the form of a regular octagon.


Definition:Octagonal Number/Definition 1

:$O_n = \begin{cases} 0 & : n = 0 \\ O_{n - 1} + 6 n - 5 & : n > 0 \end{cases}$


Definition:Octagonal Number/Definition 2

:$\displaystyle O_n = \sum_{i \mathop = 1}^n \left({6 i - 5}\right) = 1 + 7 + \cdots + \left({6 \left({n - 1}\right) - 5}\right) + \left({6 n - 5}\right)$


Definition:Octagonal Number/Definition 3

:$\forall n \in \N: O_n = P \left({8, n}\right) = \begin{cases} 0 & : n = 0 \\ P \left({8, n - 1}\right) + 6 \left({n - 1}\right) + 1 & : n > 0 \end{cases}$ where $P \left({k, n}\right)$ denotes the $k$-gonal numbers.


Definition:Octagonal Number/Sequence

The sequence of octagonal numbers, for $n \in \Z_{\ge 0}$, begins: :$0, 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, \ldots$


Definition:Ring of Integers

The '''ring of integers''' $\struct {\Z, +, \times}$ is the set of integers under the two operations of integer addition and integer multiplication.


Definition:Perfect Magic Cube

A '''perfect magic cube''' is an arrangement of the first $n^3$ (strictly) positive integers into an $n \times n \times n$ cubic array such that: :the sum of the entries in each row in each of the $3$ dimensions :the sum of the entries along the main diagonal of each plane :the sum of the entries along the space diagonals are the same.


Definition:Magic Cube/Order

An $n \times n \times n$ magic cube is called an '''order $n$ magic cube'''.


Definition:Magic Cube

A '''magic cube''' is an arrangement of the first $n^3$ (strictly) positive integers into an $n \times n \times n$ cubic array such that: :the sum of the entries in each row in each of the $3$ dimensions :the sum of the entries along the space diagonals are the same. It is not guaranteed that the entries along each of the main diagonals of each plane also sum to the same constant.


Definition:Deltahedron

A '''deltahedron''' is a polyhedron each of whose faces are equilateral triangles.


Definition:Bipyramid

A '''bipyramid''' is a polyhedron formed by taking two pyramids with congruent bases and placing those bases together: :400px


Definition:Triangular Bipyramid

A '''triangular bipyramid''' is a polyhedron formed by taking two regular tetrahedra and placing their bases together: :300px The '''triangular bipyramid''' is an example of a deltahedron.


Definition:Pentagonal Bipyramid

A '''pentagonal bipyramid''' is a polyhedron formed by taking two pyramids whose bases are regular pentagons and whose other faces are equilateral triangles, and placing their bases together: :300px The '''pentagonal bipyramid''' is an example of a deltahedron.


Definition:Lucky Number

Start with the list of (strictly) positive integers: :$1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, \ldots$ Remove every $2$nd number from $2$ onwards (that is, all the even integers): :$1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, \ldots$ The $2$nd term is $3$. Starting from the $3$rd element in this list (which is $5$), remove every $3$rd element from what is left: :$1, 3, 7, 9, 13, 15, 19, 21, 25, \ldots$ The $3$rd number is now $7$. Starting from the $7$th element in this list (which is $19$), remove every $7$th element from what is left: :$1, 3, 7, 9, 13, 15, 21, 25, \ldots$ The numbers remaining are the '''lucky numbers'''.


Definition:Lucky Number/Sequence

The sequence of lucky numbers begins: :$1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, \ldots$


Definition:Kaprekar Number

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Suppose that $n^2$, when expressed in number base $b$, can be split into two parts that add up to $n$. Then $n$ is a '''Kaprekar number''' for base $b$.


Definition:Kaprekar Number/Sequence

The sequence of Kaprekar numbers begins: :$1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, \ldots$


Definition:Subfactorial

Let $n \in \Z_{\ge0}$ be a (strictly) positive integer. The '''subfactorial''' of $n$ is defined and denoted as: {{begin-eqn}} {{eqn | l = !n       | o = :=       | r = n! \sum_{k \mathop = 0}^n \frac {\left({-1}\right)^k} {k!}       | c =  }} {{eqn | r = n! \left({1 - \dfrac 1 {1!} + \dfrac 1 {2!} - \dfrac 1 {3!} + \cdots + \dfrac {\left({-1}\right)^n} {n!} }\right)       | c =  }} {{end-eqn}} It arises as the number of derangements of $n$ distinct objects.


Definition:Compact Subset of Lattice

Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. '''Compact subset''' of $S$, denoted $K\left({L}\right)$, equals to the set of all compact elements of $S$.


Definition:Compact Closure

Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $x \in S$. Then '''compact closure''' of $x$, denoted $x^{\mathrm{compact}}$, is defined by :$x^{\mathrm{compact}} := \left\{ {y \in S: y \preceq x \land y}\right.$ is compact$\left.{}\right\}$


Definition:Axiom of K-Approximation

Let $\left({S, \preceq}\right)$ be an ordered set. '''Axiom of K-approximation''' says  :$\forall x \in S: x = \sup \left({x^{\mathrm{compact} } }\right)$ where $x^{\mathrm{compact} }$ denotes the compact closure of $x$.


Definition:Algebraic Ordered Set

Let $\left({S, \preceq}\right)$ be an ordered set. Then $\left({S, \preceq}\right)$ is '''algebraic''' {{iff}} :(for all elements $x$ of $S$: $x^{\mathrm{compact} }$ is directed) and :$\left({S, \preceq}\right)$ is up-complete and satisfies axiom of K-approximation where $x^{\mathrm{compact} }$ denotes the compact closure of $x$.


Definition:Asymmetric Relation/Definition 1

$\RR$ is '''asymmetric''' {{iff}}: :$\tuple {x, y} \in \RR \implies \tuple {y, x} \notin \RR$


Definition:Asymmetric Relation/Definition 2

$\RR$ is '''asymmetric''' {{iff}} it and its inverse are disjoint: :$\RR \cap \RR^{-1} = \O$


Definition:Idempotence/Relation

Let $\RR \subseteq S \times S$ be a relation on $S$. Then $\RR$ is '''idempotent''' {{iff}}: :$\RR \circ \RR = \RR$ where $\circ$ denotes composition of relations.


Definition:Idempotence/Algebraic Structure

Let $\left({S, \circ}\right)$ be an algebraic structure. Let $\circ$ be an idempotent operation on $S$. Then $\left({S, \circ}\right)$ is an '''idempotent algebraic structure'''.


Definition:Meet Closed

Let $L = \left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $X$ be a subset of $S$. Then $X$ is '''meet closed''' {{iff}} :$\forall x, y \in X: x \wedge y \in X$


Definition:Arithmetic Ordered Set

Let $L = \left({S, \vee, \preceq}\right)$ be a bounded below join semilattice. $L$ is '''arithmetic''' {{iff}} :$L$ is algebraic and $K\left({L}\right)$ is meet closed where $K\left({L}\right)$ denotes the compact subset of $L$.


Definition:Endomorphism Ring of Abelian Group

Let $\struct {G, +}$ be an abelian group. Let $\map {\mathrm {End} } G$ be the set of endomorphisms of $G$. The '''endomorphism ring of $G$''' is the algebraic structure: :$\struct {\map {\mathrm {End} } G, +, \circ}$ where: :$\circ$ denotes composition :$+$ denotes pointwise addition.


Definition:Support of Permutation

Let $S$ be a set. Let $f$ be a permutation on $S$. The '''support''' of $f$ is the subset of moved elements: :$\operatorname {supp} \left({f}\right) = \left\{ {x \in S: f \left({x}\right) \ne  x}\right\}$


Definition:Support of Element of Direct Product

Let $\left({S_i, \circ_i}\right)_{i \mathop \in I}$ be a family of algebraic structures with identity. Let $\displaystyle S = \prod_{i \mathop \in I} S_i$ be their direct product. Let $e_i$ be an identity of $S_i$ for all $i \in I$. Let $m = \left({m_i}\right)_{i \mathop \in I} \in S$. The '''support''' of $m$ is defined as: :$\operatorname {supp} \left\{ {i \in I: m_i \ne e_i}\right\}$ === Finite Support === The element is said to have '''finite support''' {{iff}} its support is a finite set.


Definition:Liouville Function

Let $n \in \Z_{>0}$, that is, a strictly positive integer. The '''Liouville function''' is the function $\lambda: \Z_{>0} \to \Z_{>0}$ defined as: :$\lambda \left({n}\right) = \left({-1}\right)^{\Omega \left({n}\right)}$ where $\Omega \left({n}\right)$ is the number of prime factors of $n$, counted with multiplicity.


Definition:Module of All Mappings

Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {M, +_M, \circ}_R$ be an $R$-module. Let $S$ be a set. Let $M^S$ be the set of all mappings from $S$ to $M$. Let: :$+$ be the operation induced on $M^S$ by $+_M$ :$\forall \lambda \in R: \forall f \in M^S: \forall x \in S: \map {\paren {\lambda \circ f} } x = \lambda \circ \paren {\map f x}$ Then $\struct {M^S, +, \circ}_R$ is the '''module of all mappings from $S$ to $M$'''.


Definition:Module Direct Product/Finite Case

Let $R$ be a ring. Let $\left({M_1, +_1, \circ_1}\right)_R, \left({M_2, +_2, \circ_2}\right)_R, \ldots, \left({M_n, +_n, \circ_n}\right)_R$ be $R$-modules. Let: :$\displaystyle M = \prod_{k \mathop = 1}^n M_k$ be the cartesian product of $M_1$ to $M_n$. Let: : $+$ be the operation induced on $M$ by the operations $+_1, +_2, \ldots, +_n$ on $M_1, M_2, \ldots, M_n$ : $\circ$ be defined as $\lambda \circ \left({x_1, x_2, \ldots, x_n}\right) = \left({\lambda \circ_1 x_1, \lambda \circ_2 x_2, \ldots, \lambda \circ_n x_n}\right)$ In Finite Direct Product of Modules is Module, it is shown that $\left\langle{M, +, \circ}\right\rangle$ is an $R$-module. The module $\left({M, +, \circ}\right)_R$ is called the '''(external) direct product''' of $M_1$ to $M_n$.


Definition:Module Direct Product/General Case

Let $R$ be a ring. Let $\left\{ \left\langle M_i,+_i,\circ_i\right\rangle\right\}_{i \in I}$ be a family of $R$-modules. {{explain|Expand the notation of the module to specify the operations of the ring.}} Let: :$M = \displaystyle \prod_{i \mathop \in I} M_i$ be the cartesian product of these modules. The operation $+$ induced on $M$ by $(+_i)_{i\in I}$ is the operation defined by: :$\left\langle{a_i}\right\rangle_{i \mathop \in I} + \left\langle{b_i}\right\rangle_{i \mathop \in I} = \left\langle{a_i +_i b_i}\right\rangle_{i \mathop \in I}$ That is, the additive group of the module $M$ is the direct product of the groups $\left\{ \left(M_i,+_i\right)\right\}_{i \in I}$. The $R$-action $\circ$ induced on $M$ by $(\circ_i)_{i\in I}$ is the operation defined by: :$r \circ \left\langle{m_i}\right\rangle_{i \mathop \in I} = \left\langle{r \circ_i m_i}\right\rangle_{i \mathop \in I}$ In Direct Product of Modules is Module, it is shown that $\left\langle{M, +, \circ}\right\rangle$ is an $R$-module. The module $\left\langle{M, +, \circ}\right\rangle$  is called the '''(external) direct product''' of $\left\{ \left\langle M_i,+_i,\circ_i\right\rangle\right\}_{i \in I}$.


Definition:Direct Product of Vector Spaces/Finite Case

Let $K$ be a field. Let $V_1, V_2, \ldots, V_n$ be $K$-vector spaces. Let: :$\displaystyle V = \prod_{k \mathop = 1}^n V_k$ be their cartesian product. Let: :$+$ be the operation induced on $V$ by the operations $+_1, +_2, \ldots, +_n$ on $V_1, V_2, \ldots, V_n$ :$\circ$ be defined as $\lambda \circ \tuple {x_1, x_2, \ldots, x_n} := \tuple {\lambda \circ x_1, \lambda \circ x_2, \ldots, \lambda \circ x_n}$ $\struct {V, +, \circ}_K$ is called the '''direct product of $V_1, \ldots, V_n$'''.


Let $K$ be a field. Let $V, W$ be $K$-vector spaces. The '''direct product of $V$ and $W$''' is their module direct product. === Finite Case === {{:Definition:Direct Product of Vector Spaces/Finite Case}} === General Case === {{:Definition:Direct Product of Vector Spaces/General Case}}


Definition:Direct Product of Vector Spaces/General Case

Let $K$ be a field. Let $\family {V_i, +_i, \circ_i}_{i \mathop \in I}$ be a family of $K$-vector spaces. The '''(external) direct product''' of $\family {V_i, +_i, \circ_i}_{i \mathop \in I}$ is their module direct product.


Definition:Module of Homomorphisms Between Modules

Let $R$ be a commutative ring. Let $M$ and $N$ be $R$-modules. Let $\map {\operatorname {Hom}_{R - \operatorname {mod} } } {M, N}$ denote the set of $R$-module homomorphisms from $M$ to $N$. Let: :$+$ be the operation on $\map {\operatorname {Hom}_{R - \operatorname {mod} } } {M, N}$ defined by $f + g: m \mapsto \map f m + \map g m$ :$\circ$ be defined as $\lambda \circ f: m \mapsto \lambda \map f m$ Then $\struct {\map {\operatorname {Hom}_{R - \operatorname {mod} } } {M, N}, + , \circ}$ is called the '''module of homomorphisms''' between $M$ and $N$.


Definition:Dual Module

Let $R$ be a commutative ring. Let $M$ be an $R$-module. Then the '''dual module''' of $M$ is the module of homomorphisms $\operatorname{Hom}_{R\text{-mod}}(M,R)$. That is, the elements of the dual module are the linear forms on $M$. Category:Definitions/Abstract Algebra Category:Definitions/Module Theory ro54nx7d4h6z3no66m8rlngyeot2ku6


Definition:Order Complete Set/Definition 1

$\struct {S, \preceq}$ is '''order complete''' {{iff}}: :Each non-empty subset $H \subseteq S$ which has an upper bound admits a supremum.


Definition:Order Complete Set/Definition 2

$\struct {S, \preceq}$ is '''order complete''' {{iff}}: :Each non-empty subset $H \subseteq S$ which has a lower bound admits an infimum.


Definition:Order Complete Set

Let $\left({S, \preceq}\right)$ be an ordered set.


Definition:Normal Subgroup/Also known as

It is usual to describe a '''normal subgroup of $G$''' as '''normal in $G$'''. Some sources refer to a '''normal subgroup''' as an '''invariant subgroup''' or a '''self-conjugate subgroup'''. This arises from Definition 6: {{:Definition:Normal Subgroup/Definition 6}} which is another way of stating that $N$ is '''normal''' {{iff}} $N$ stays the same under all inner automorphisms of $G$. Some sources use '''distinguished subgroup'''.


Definition:Kepler-Poinsot Polyhedron

A '''Kepler-Poinsot polyhedron''' is any of the four regular star polyhedra.


Definition:Altitude of Triangle/Foot

The point at which $h_a$ meets $BC$ is the '''foot of the altitude $h_a$'''.


Definition:Feuerbach Circle

Let $\triangle ABC$ be a triangle. The '''Feuerbach circle''' of $\triangle ABC$ is the circle which passes through each of the $9$ points: : the feet of the altitudes of $\triangle ABC$ : the midpoints of the sides of $\triangle ABC$ : the midpoints of the lines from the vertices of $\triangle ABC$ to the orthocenter of $\triangle ABC$. :500px


Definition:Cosine Integral Function

The '''cosine integral function''' is the real function $\Ci: \R_{>0} \to \R$ defined as: :$\map \Ci x = \displaystyle \int_{t \mathop = x}^{t \mathop \to +\infty} \frac {\cos t} t \rd t$


Definition:Magic Square/Magic Constant

The '''magic constant''' of a magic square is the number that each of the rows and columns adds up to.


Definition:Field Extension/Complex

Let $\left({F, +, \times}\right)$ be a subfield of $\left({\Bbb C, +, \times}\right)$, the field of complex numbers. Let $X_1, X_2, \ldots, X_n$ be complex numbers, in or not in $F$. Then $F \left({X_1, X_2, \ldots, X_n}\right)$ is the smallest field extension over $F$ containing $X_1, X_2, \ldots, X_n$.


Definition:Euclid Prime

A '''Euclid prime''' is a natural number which is both a Euclid number and a prime number.


Definition:Lambert W Function/Principal Branch/Real Valued

The principal branch of the Lambert W function is the real function $W_0: \hointr {-\dfrac 1 e} \to \to \hointr {-1} \to$ such that: :$x = \map {W_0} x e^{\map {W_0} x}$


Definition:Lambert W Function

Let $e^z: \C \to \C$ be the complex exponential function. Then $\map W z$ is defined to be the multifunction that satisfies:  :$\map W z e^{\map W z} = z$


Definition:Omega Constant

The '''omega constant''' $\Omega$ is the constant as the value of the principal branch of the Lambert W function at $1$: :$\Omega \, e^ \Omega = 1$ That is, it is the root of the equation: :$x e^x = 1$ where $e$ denotes Euler's number.


Definition:Euclid Prime/Sequence

The sequence of '''Euclid primes''' begins: :$2, 3, 7, 31, 211, 2311, 200 \, 560 \, 490 \, 131, \ldots$


Definition:Euclid Number/Sequence

The sequence of Euclid numbers begins as follows: {{begin-eqn}} {{eqn | ll= E_0       | l = = p_0\# + 1       | r = 1 + 1       | rr= = 2       | c =  }} {{eqn | ll= E_1       | l = = p_1\# + 1       | r = 2 + 1       | rr= = 3       | c =  }} {{eqn | ll= E_2       | l = = p_2\# + 1       | r = 2 \times 3 + 1       | rr= = 7       | c =  }} {{eqn | ll= E_3       | l = = p_3\# + 1       | r = 2 \times 3 \times 5 + 1       | rr= = 31       | c =  }} {{eqn | ll= E_4       | l = = p_4\# + 1       | r = 2 \times 3 \times 5 \times 7 + 1       | rr= = 211       | c =  }} {{eqn | ll= E_5       | l = = p_5\# + 1       | r = 2 \times 3 \times 5 \times 7 \times 11 + 1       | rr= = 2311       | c =  }} {{eqn | ll= E_6       | l = = p_6\# + 1       | r = 2 \times 3 \times 5 \times 7 \times 11 \times 13 + 1       | rr= = 30031       | c =  }} {{end-eqn}}


Definition:Compact Element

Let $\left({S, \preceq}\right)$ be an ordered set. Let $x \in S$. Then $x$ is '''compact (element)''' {{iff}} $x \ll x$ where $\ll$ denotes the way below relation.


Definition:Lambert W Function/Lower Branch

The '''lower branch'''  of the Lambert W function is the real function $W_{-1}: \left[{-\dfrac 1 e \,.\,.\, 0}\right) \to \left({\gets \,.\,.\, -1}\right]$ such that: :$x = W_{-1} \left({x}\right) e^{W_{-1} \left({x}\right)}$


Definition:Foiaș Constant

=== First Foiaș Constant === {{:Definition:Foiaș Constant/First}} === Second Foiaș Constant === {{:Definition:Foiaș Constant/Second}}


Definition:Foiaș Constant/First/Decimal Expansion

The decimal expansion of the first Foiaș Constant starts: :$x_{\infty} = 2 \cdotp 29316 \, 62874 \, 11861 \, 03150 \, 80282 \, 91250 \, 80586 \, 43722 \, 57290 \, 32712 \, 12485 \, 37 \ldots$


Definition:Foiaș Constant/First

Let: :$x_{n+1} = \left({1 + \dfrac 1 {x_n} }\right)^{x_n}$ for $n = 1, 2, 3, \ldots$ The '''first Foiaș constant''' is the limit of $x_n$ as $n \to \infty$.


Definition:Foiaș Constant/Second/Decimal Expansion

The decimal expansion of the second Foiaș Constant starts: :$\alpha = 1 \cdotp 18745 \, 23511 \, 26501 \ldots$


Definition:Foiaș Constant/Second

Let $x_1 \in \R_{>0}$ be a (strictly) positive real number. Let: :$x_{n + 1} = \left({1 + \dfrac 1 {x_n} }\right)^n$ for $n = 1, 2, 3, \ldots$ The '''second Foiaș constant''' is defined as the unique real number $\alpha$ such that if $x_1 = \alpha$ then the sequence $\left\langle{x_{n + 1} }\right\rangle$ diverges to infinity.


Definition:Pandigital Fraction

A '''pandigital fraction''' is a fraction where the digits $1$ to $9$ appear exactly once in the decimal expansions of the numerator and the denominator of the fraction combined.


Definition:Tetrahedral Number

'''Tetrahedral numbers''' are those denumerating a collection of objects which can be arranged in the form of a regular tetrahedron.


Definition:Polyhedral Number/Examples

=== Tetrahedral Number === {{:Definition:Tetrahedral Number}}


Definition:Noncototient

A '''noncototient''' is a positive integer $n$ such that: :$\nexists m \in \Z_{>0}: m - \phi \left({m}\right) = n$ where $\phi \left({m}\right)$ denotes the Euler $\phi$ function. That is, a '''noncototient''' is a positive integer which is not the cototient of any positive integer.


Definition:Noncototient/Sequence

The sequence of '''noncototients''' begins: :$10, 26, 34, 50, 52, 58, 86, 100, 116, 122, 130, 134, 146, 154, 170, 172, 186, 202, \ldots$


Definition:Cototient

Let $n \in \Z_{>0}$ be a (strictly) positive integer. The '''cototient''' of $n$ is defined as: :$n - \phi \left({n}\right)$ where $\phi \left({n}\right)$ denotes the Euler $\phi$ function.


Definition:Latin Square/Order

Let $\mathbf L$ be an $n \times n$ Latin square. The '''order''' of $\mathbf L$ is $n$.


Definition:Latin Square/Row

Let $\mathbf L$ be a Latin square. The '''rows''' of $\mathbf L$ are the lines of elements reading '''across''' the page.


Definition:Latin Square/Column

Let $\mathbf L$ be a Latin square. The '''columns''' of $\mathbf L$ are the lines of elements reading '''down''' the page.


Definition:Latin Square/Element

Let $\mathbf L$ be a Latin square of order $n$. The individual $n \times n$ symbols that go to form $\mathbf L$ are known as the '''elements''' of $\mathbf L$. The element at row $i$ and column $j$ is called '''element $\left({i, j}\right)$ of $\mathbf L$''', and can be written $a_{i j}$, or $a_{i, j}$ if $i$ and $j$ are of more than one character. If the indices are still more complicated coefficients and further clarity is required, then the form $a \left({i, j}\right)$ can be used. Note that the first subscript determines the row, and the second the column, of the Latin square where the element is positioned.


Definition:Orthogonal Latin Squares

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $\mathbf A$ and $\mathbf B$ be Latin squares of order $n$. Let $\left[{a}\right]_{i j}$ and $\left[{b}\right]_{i j}$ be the elements of $\mathbf A$ and $\mathbf B$ respectively whose indices are $i$ and $j$. Then $\mathbf A$ and $\mathbf B$ are '''orthogonal''' {{iff}}: :for all $i$ and $j$, the ordered pairs $\left({\left[{a}\right]_{i j}, \left[{b}\right]_{i j}}\right)$ are distinct. That is, {{iff}} $\left({\left[{a}\right]_{i j}, \left[{b}\right]_{i j}}\right)$ appears no more than once for every ordered pair $\left({i, j}\right)$. === Mutually Orthogonal === {{:Definition:Orthogonal Latin Squares/Mutually Orthogonal}}


Definition:Orthogonal Latin Squares/Mutually Orthogonal

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $S$ be a set of Latin squares of order $n$. Then the $S$ is '''mutually orthogonal''' {{iff}} every element of $S$ is orthogonal to every other element of $S$.


Definition:Multiplicative Persistence

Let $n \in \N$ be a natural number. Let $n$ be expressed in decimal notation. Multiply the digits of $n$ together. Repeat with the answer, and again until a single digit remains. The number of steps it takes to reach $1$ digit is called the '''multiplicative persistence''' of $n$.


Definition:Octave

An '''octave''' is a musical interval whose components have frequencies in the ratio $2 : 1$.


Definition:Music

'''Music''' is the branch of study which creates aesthetically pleasing combinations of sounds.


Definition:Fifth (Music)

A '''fifth''' is a musical interval whose components have frequencies in the ratio $3 : 2$.


Definition:Fourth (Music)

A '''fourth''' is a musical interval whose components have frequencies in the ratio $4 : 3$.


Definition:Tone (Music)

A '''tone''' is a musical interval whose components have frequencies in the ratio $9 : 8$.


Definition:Well-Tempered Scale

A '''well-tempered scale''' is an octave which has been divided into $12$ equal intervals called semitones.


Definition:Semitone

A '''semitone''' is a musical interval whose components have frequencies in the ratio $2^{1/12} : 1$.


Definition:Imperial/Length

The imperial units of length are as follows: === Mil === {{Definition:Imperial/Length/Mil}} === Inch === {{Definition:Imperial/Length/Inch}} === Foot === {{Definition:Imperial/Length/Foot}} === Yard === {{Definition:Imperial/Length/Yard}} === Rod, Pole or Perch === {{Definition:Imperial/Length/Rod, Pole or Perch}} === Chain === {{Definition:Imperial/Length/Chain}} === Furlong === {{Definition:Imperial/Length/Furlong}} === International Mile === {{Definition:Imperial/Length/International Mile}} === League === {{Definition:Imperial/Length/League}}


Definition:Imperial/Length/Inch

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''inch''' }} {{eqn | r = 2 \cdotp 54       | c = centimetres }} {{eqn | r = 0 \cdotp 0254       | c = metres }} {{end-eqn}} This definition is exact, and is how the imperial unit of length is defined.


Definition:Imperial/Length/Foot

The '''foot''' is the FPS base unit of length. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''foot''' }} {{eqn | r = 12       | c = inches }} {{eqn | r = 30.48       | c = centimetres }} {{eqn | r = 0 \cdotp 3048       | c = metres }} {{end-eqn}}


Definition:Imperial/Length/Yard

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''yard''' }} {{eqn | r = 3       | c = feet }} {{eqn | r = 36       | c = inches }} {{eqn | r = 91 \cdot 44       | c = centimetres }} {{eqn | r = 0 \cdotp 9144       | c = metres }} {{end-eqn}}


Definition:Imperial/Length/Rod, Pole or Perch

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''rod, pole or perch''' }} {{eqn | r = 5 \tfrac 1 2       | c = yards }} {{eqn | r = 16 \tfrac 1 2       | c = feet }} {{eqn | r = 502 \cdot 92       | c = centimetres }} {{eqn | r = 5 \cdotp 0292       | c = metres }} {{end-eqn}}


Definition:Imperial/Length/Chain

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''chain''' }} {{eqn | r = 4       | c = rods, poles or perches }} {{eqn | r = 22       | c = yards }} {{eqn | r = 66       | c = feet }} {{eqn | r = 20 \cdotp 1168       | c = metres }} {{end-eqn}}


Definition:Imperial/Length/Furlong

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''furlong''' }} {{eqn | r = 10       | c = chains }} {{eqn | r = 40       | c = rods, poles or perches }} {{eqn | r = 220       | c = yards }} {{eqn | r = 660       | c = feet }} {{eqn | r = 201 \cdotp 168       | c = metres }} {{end-eqn}}


Definition:Imperial/Length/International Mile

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''(international) mile''' }} {{eqn | r = 8       | c = furlongs }} {{eqn | r = 80       | c = chains }} {{eqn | r = 320       | c = rods, poles or perches }} {{eqn | r = 1760       | c = yards }} {{eqn | r = 1609 \cdotp 344       | c = metres (exactly) }} {{end-eqn}}


Definition:Imperial/Length/League

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''league''' }} {{eqn | o =        | r = 3       | c = (international) miles }} {{eqn | r = 5280       | c = yards }} {{eqn | o = \approx       | r = 2       | c = leuca }} {{eqn | o = \approx       | r = 3000       | c = paces }} {{eqn | r = 4828 \cdotp 032       | c = metres (exactly) }} {{end-eqn}}


Definition:Palindromic Number

A '''palindromic number (base $b$)''' is a number which is a palindrome when expressed in number base $b$. That is, it reads the same reversed as it does forwards. When the base is not specified, it is commonplace to understand it as being $10$.


Definition:Palindromic Prime

A '''palindromic prime''' is a prime number which is also a palindromic number.


Definition:Palindrome

A '''palindrome''' is a sequence of objects which stays the same when reversed.


Definition:Ulam Number

The sequence of '''Ulam numbers''' is defined as: :It begins $1, 2, 3$ :Each new term is the next to be uniquely the sum of $2$ previous terms in the sequence.


Definition:Ulam Number/Sequence

The sequence of '''Ulam numbers''' begins as follows: :$1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, 48, 53, 57, 62, 69, \ldots{}$


Definition:Lucas Number/Sequence

The '''Lucas sequence''' begins: :$2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, \ldots$


Definition:Continuity/Functional

Let $S$ be a set of mappings. Let $y \in S$ be a mapping. Let $J \sqbrk y: S \to \R$ be a functional. Suppose:    :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \size {y - y_0} < \delta \implies \size {J \sqbrk y - J \sqbrk {y_0} } < \epsilon$ Then $J \sqbrk y$ is said to be a '''continuous functional''' and is continuous at the point $y_0 \in S$.


Definition:Functional/Real

Let $S$ be a set of mappings. Let $J: S \to \R$ be a mapping from $S$ to the real numbers $\R$: :$\forall y \in S: \exists x \in \R: J \sqbrk y = x$ Then $J: S \to \R$ is known as a '''(real) functional''', denoted by $J \sqbrk y$. That is, a '''(real) functional''' is a real-valued function whose arguments are themselves mappings.


Definition:Ordered Set of All Mappings

Let $L = \left({S, \preceq}\right)$ be an ordered set. Let $X$ be a set. The '''ordered set of all mappings''' from $X$ to $L$ is defined by: :$L^X := \left({S^X, \precsim}\right)$ where :$\forall f, g \in S^X: f \precsim g \iff f \preceq g$ :$\preceq$ denotes the ordering on mappings, :$S^X$ denotes the set of all mappings from $X$ into $S$.


Definition:General Fibonacci Sequence

Let $r, s, t, u$ be numbers, usually integers but not necessarily so limited. Let $\sequence {a_n}$ be the sequence defined as: :$a_n = \begin{cases} r & : n = 0 \\ s & : n = 1 \\ t a_{n - 2} + u a_{n - 1} & : n > 1 \end{cases}$ Then $\sequence {a_n}$ is a '''general Fibonacci sequence'''.


Definition:Lucas Number/Definition 1

The '''Lucas numbers''' are a sequence which is formally defined recursively as: :$L_n = \begin{cases} 2 & : n = 0 \\ 1 & : n = 1 \\ L_{n - 1} + L_{n - 2} & : \text{otherwise} \end{cases}$


Definition:Lucas Number/Definition 2

The '''Lucas numbers''' are a sequence defined as: :$L_n = F_{n - 1} + F_{n + 1}$ where $F_k$ is the $k$th Fibonacci number.


Definition:Increment/Functional

Let $S$ be a set of mappings. Let $y, h: \R \to \R$ be real functions. Let $J \sqbrk y: S \to \R$ be a functional defined on a normed vector space. Consider the following difference: :$ \Delta J \sqbrk {y; h} = J \sqbrk {y + h} - J \sqbrk y$ Then $\Delta J \sqbrk {y; h}$ is known as the '''increment of the functional $J$'''.


Definition:Differentiable Functional

Let $S$ be a normed linear space of mappings. Let $y, h \in S: \R \to \R$ be real functions. Let $J \sqbrk y$, $\phi \sqbrk {y; h}$ be functionals. Let $\Delta J \sqbrk {y; h}$ be an increment of the functional $J$ such that: :$\Delta J \sqbrk {y; h} = \phi \sqbrk {y;h} + \epsilon \norm h$ where $\epsilon = \epsilon \sqbrk {y; h}$ is a functional, and $\norm h$ is the norm of $S$. Suppose $\phi \sqbrk {y; h}$ is a linear {{WRT}} $h$ and: :$\ds \lim_{\norm h \mathop \to 0} \epsilon = 0$ Then the functional $J \sqbrk y $ is said to be '''differentiable'''.


Definition:Ordered Set of Increasing Mappings

Let $L = \left({S_1, \preceq_1}\right)$, $K = \left({S_2, \preceq_2}\right)$ be ordered sets. The '''ordered set of increasing mappings''' from $K$ into $L$ is ordered subset of $L^{S_2} = \left({S_1^{S_2}, \preceq'}\right)$ and is defined by :$\operatorname{Increasing}\left({K, L}\right) = \left({X, \precsim}\right)$ where :$X = \left\{ {f:S_2 \to S_1: f}\right.$ is increasing mapping$\left.\right\}$ :$\mathord\precsim = \mathord\preceq' \cap \left({X \times X}\right)$ :$L^{S_2}$ denotes the ordered set of all mappings from $S_2$ into $L$, :$S_1^{S_2}$ denotes the set of all mappings from $S_2$ into $S_1$. $\operatorname{Increasing}\left({K, L}\right)$ as ordered subset of ordered set is ordered set by Ordered Subset of Ordered Set is Ordered Set.


Definition:Prime Decomposition/Linguistic Note

The UK English spelling of '''prime factorization''' is '''prime factorisation'''.


Definition:Prime Decomposition/Multiplicity

For each $p_j \in \left\{ {p_1, p_2, \ldots, p_r}\right\}$, its power $k_j$ is known as the '''multiplicity of $p_j$'''.


Definition:Ordered Subset

Let $R = \struct{S, \preceq}$ be a relational structure. {{Questionable|A relational substructure of a relational structure is not necessarily ordered. So either: (a) $\struct{S, \preceq}$ needs to be ordered; (b) $\struct{S', \preceq'}$ needs to be ordered; or (c) the page needs to be renamed. The theorem Ordered Subset of Ordered Set is Ordered Set and many links to the page suggests (a) is needed.|Leigh.Samphier}} A '''ordered subset''' of $R$ is a relational structure $\struct{S', \preceq'}$ such that :$S'$ is a subset of $S$, :$\mathord\preceq' = \mathord\preceq \cap \paren{S' \times S'}$ That is :$\forall x, y \in S': x \preceq y \iff x \preceq' y$


Definition:Abundant Number/Definition 3

$n$ is '''abundant''' {{iff}} it is smaller than its aliquot sum.


Definition:Deficient Number/Definition 3

$n$ is '''deficient''' {{iff}} it is greater than its aliquot sum.


Definition:Ordered Set of Closure Operators

Let $L = \left({S, \preceq}\right)$ be an ordered set. The '''ordered set of closure operators''' of $L$ is ordered subset of $\operatorname{Increasing} \left({L, L}\right) = \left({X, \preceq'}\right)$ and is defined by :$\operatorname{Closure}\left({L}\right) := \left({Y, \precsim}\right)$ where :$Y = \left\{ {f:S \to S: f}\right.$ is closure operator$\left.\right\}$ :$\mathord\precsim = \mathord\preceq' \cap \left({Y \times Y}\right)$ :$\operatorname{Increasing} \left({L, L}\right)$  denotes the ordered set of increasing mappings from $L$ into $L$. $\operatorname{Closure}\left({L}\right)$ as an ordered subset of an ordered set is an ordered set by Ordered Subset of Ordered Set is Ordered Set.


Definition:Extremum/Functional

Let $S$ be a set of mappings. Let $J \sqbrk y: S \to \R$ be a functional. Let $y, \hat y: \R \to \R$ be real functions. Suppose for $y = \hat y \paren x$ there exists a neighbourhood of the curve $y = \hat y \paren x$ such that the difference $J \sqbrk y - J \sqbrk {\hat y}$ does not change its sign in this neighbourhood. Then $y = \hat y$ is called a ('''relative''') '''extremum''' of the functional $J$. {{Help|Check if "some neighbourhood" could be more precisely defined |user = Julius}}


Definition:Superabundant Number

Let $n \in \Z_{>0}$ be a positive integer. Then $n$ is '''superabundant''' {{iff}}: :$\forall m \in \Z_{>0}, m < n: \dfrac {\map \sigma m} m < \dfrac {\map \sigma n} n$ where $\map \sigma n$ is the $\sigma$ function of $n$.


Definition:Duodecimal Notation

'''Duodecimal notation''' is the technique of expressing numbers in base $12$.


Definition:Imperial/Mass

There are three imperial systems of measurement of mass: * Avoirdupois * Apothecaries' weights * Troy === Grain === {{Definition:Imperial/Mass/Grain}} == Avoirdupois == {{:Definition:Avoirdupois}} == Avoirdupois: Short Variants == {{:Definition:Avoirdupois/Short}} == Apothecaries' Weights == {{:Definition:Apothecaries' Weights and Measures/Mass}} == Troy == {{:Definition:Troy}}


Definition:Avoirdupois/Ounce

:$1$ '''ounce avoirdupois''' $= 16$ drams $= 437 \cdotp 5$ grains $= 28 \cdotp 34952 \, 3125$ grams. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''ounce avoirdupois''' }} {{eqn | r = 16       | c = drams }} {{eqn | r = 437 \cdotp 5       | c = grains }} {{eqn | r = 28 \cdotp 34952 \, 3125       | c = grams }} {{end-eqn}} The gram equivalent is defined to be exact.


Definition:Ounce/Linguistic Note

The word '''ounce''' derives from the Latin word '''uncia''', meaning '''$\dfrac 1 {12}$ part''' (of an '''as'''). Despite the gradual migration to the metric system, the word '''ounce''' still lives on as a rhetorical flourish for '''something small''', for example: :''If you only had an '''ounce''' of sense you would understand that ...''


Definition:Apothecaries' Weights and Measures/Mass/Scruple

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''scruple''' }} {{eqn | r = 20       | c = grains }} {{eqn | r = 1 \cdotp 3       | c = grams }} {{end-eqn}}


Definition:Ounce

=== Ounce Avoirdupois === {{:Definition:Avoirdupois/Ounce}} === Apothecaries' Ounce === {{:Definition:Apothecaries' Weights and Measures/Mass/Ounce}} === Troy Ounce === {{:Definition:Troy/Ounce}} === Uncia (Roman Ounce) === {{:Definition:Roman Weights and Measures/Mass/Uncia}}


Definition:Apothecaries' Weights and Measures

The '''apothecaries' (system of) weights and measures''' is a system of mass and volume units, derived from the Roman system. They were used by physicians and apothecaries for medical recipes, and also sometimes by scientists. Although in some cases bearing the same names as their imperial counterparts, the actual units themselves were subtly different in size.


=== Scruple === {{Definition:Apothecaries' Weights and Measures/Mass/Scruple}} === Drachm === {{Definition:Apothecaries' Weights and Measures/Mass/Drachm}} === Ounce === {{Definition:Apothecaries' Weights and Measures/Mass/Ounce}} === Pound === {{Definition:Apothecaries' Weights and Measures/Mass/Pound}}


Definition:System (Order Theory)

Let $L = \left({S, \preceq}\right)$ be an ordered set. The '''system''' of $L$ is an ordered subset of $L$.


Definition:Infima Inheriting

Let $L = \struct {S, \preceq}$ be an ordered set. Let $R = \struct {T, \preceq'}$ be an ordered subset of $L$. Then $R$ '''inherits infima''' of $L$ {{iff}} :for all subsets $X$ of $T$ if $X$ admits an infimum of $L$, then ::$X$ admits an infimum of $R$ and $\inf_R X = \inf_L X$


Definition:Closure System

Let $L$ be an ordered set. Let $S$ be a system of $L$. Then $S$ is '''closure''' {{iff}} $S$ inherits infima.


Definition:Ordered Set of Closure Systems

Let $L$ be an ordered set. The '''ordered set of closure systems''' of $L$ is a relational structure :$\operatorname{ClSystems}\left({L}\right) = \left({X, \precsim}\right)$ where :$X$ is the set of all closure systems of $L$, :dor all closure systems $S_1 = \left({T_1, \preceq_1}\right), S_2 = \left({T_2, \preceq_2}\right)$ of $L$: $S_1 \precsim S_2 \iff T_1 \subseteq T_2$


Definition:Weak Extremum

Let $S$ be a set of mappings. Let $y, \hat y \in S: \R \to \R$ be real functions. Let $J \sqbrk y: S \to \R$ be a functional. Suppose there exists $\epsilon \in \R_{> 0}$ such that for $\norm {y - \hat y}_1 < \epsilon$ the expression $J \sqbrk y - J \sqbrk {\hat y}$ has the same sign for all $y$. Here $\norm{\, \cdot \,}_1$ denotes the norm of in the space $C^1$. Then $y = \hat y$ is a '''weak extremum''' of the functional $J \sqbrk y$.


Definition:Apothecaries' Weights and Measures/Mass/Ounce

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''apothecaries' ounce''' }} {{eqn | r = 8       | c = drachms }} {{eqn | r = 24       | c = scruples }} {{eqn | r = 480       | c = grains }} {{eqn | r = 1       | c = troy ounce }} {{eqn | r = 31 \cdotp 1       | c = grams }} {{end-eqn}}


Definition:Troy/Ounce

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''troy ounce''' }} {{eqn | r = 20       | c = pennyweight }} {{eqn | r = 480       | c = grains }} {{eqn | r = 1       | c = apothecaries' ounce }} {{eqn | o = \approx       | r = 31 \cdotp 1       | c = grams }} {{end-eqn}}


Definition:Troy

=== Pennyweight === {{Definition:Troy/Pennyweight}} === Ounce === {{Definition:Troy/Ounce}} === Pound === {{Definition:Troy/Pound}} === Talent === {{Definition:Troy/Talent}}


Definition:Roman Weights and Measures/Mass/Uncia

:$1$ '''uncia''' $= \dfrac 1 {12}$ libra $= 24$ scrupuli. Its value in modern units is estimated to be approximately $423$ grains or $27.4$ grams.


Definition:Roman Weights and Measures

The '''Roman system of weights''' is a system of mass units originating from the time of the Roman empire. This system continued to be used throughout Europe, although it evolved in different directions according to use.


The Roman units of mass are as follows: === Calcus === {{Definition:Roman Weights and Measures/Mass/Calcus}} === Siliqua === {{Definition:Roman Weights and Measures/Mass/Siliqua}} === Scrupulus === {{Definition:Roman Weights and Measures/Mass/Scrupulus}} === Uncia === {{Definition:Roman Weights and Measures/Mass/Uncia}} === Libra === {{Definition:Roman Weights and Measures/Mass/Libra}}


Definition:Roman Weights and Measures/Mass/Scrupulus

:$1$ '''scrupulus''' $= \dfrac 1 {24}$ uncia $= 8$ calci. Its value in modern units is estimated to be approximately $17.6$ grains, or approximately $0.14$ grams.


Definition:Roman Weights and Measures/Mass/Calcus

:$1$ '''calcus''' $= \dfrac 1 8$ scrupulus. Its value in modern units is estimated to be approximately $2.2$ grains, or approximately $0.14$ grams.


Definition:Operator Generated by Ordered Subset

Let $L = \left({X, \precsim}\right)$ be an ordered set. Let $S = \left({T, \preceq}\right)$ be an ordered subset of $L$. The '''operator generated by ordered set''' $S$ is defined by :$\forall x \in X:\operatorname{operator}\left({S}\right)\left({x}\right) := \inf_L\left({x^\succeq \cap T}\right)$ where $x^\succeq$ denotes the upper closure of $x$.


Definition:Roman Weights and Measures/Mass/Siliqua

:$1$ '''siliqua''' $= \dfrac 1 6$ scrupulus $= \dfrac 1 {144}$ uncia. Its value in modern units is estimated to be approximately $2.9$ grains, or approximately $0.19$ grams. Its modern equivalent is the carat.


Definition:Roman Weights and Measures/Mass/Libra

:$1$ '''libra''' $= 12$ unciae. Its value in modern units is estimated to be approximately $5 \, 076$ grains or $329$ grams.


Definition:Avoirdupois/Dram

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''dram''' }} {{eqn | r = 27 \cdotp 34375       | c = grains }} {{eqn | r = \dfrac 1 {16}       | c = ounce avoirdupois }} {{eqn | o = \approx       | r = 1 \cdotp 77       | c = grams }} {{end-eqn}}


Definition:Imperial/Mass/Grain

The '''grain''' is the imperial unit of mass which is used as the basis of all three of the imperial weight systems. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''grain''' }} {{eqn | r = \dfrac 1 {437 \cdot 5}       | c = ounce avoirdupois }} {{eqn | o = \approx       | r = 64 \cdotp 8       | c = milligrams }} {{end-eqn}}


Definition:Avoirdupois

=== Dram === {{Definition:Avoirdupois/Dram}} === Ounce === {{Definition:Avoirdupois/Ounce}} === Pound === {{Definition:Avoirdupois/Pound}} === Stone === {{Definition:Avoirdupois/Stone}} === Quarter === {{Definition:Avoirdupois/Quarter}} === Hundredweight === {{Definition:Avoirdupois/Hundredweight}} === Ton === {{Definition:Avoirdupois/Ton}}


Definition:Avoirdupois/Pound

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''pound avoirdupois''' }} {{eqn | r = 16       | c = ounces avoirdupois }} {{eqn | r = 7 \, 000       | c = grains }} {{eqn | r = 453 \cdotp 59237       | c = grams }} {{end-eqn}} The gram equivalent is defined to be exact.


Definition:Avoirdupois/Stone

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''stone''' }} {{eqn | r = 14       | c = pounds avoirdupois }} {{eqn | r = 224       | c = ounces avoirdupois }} {{eqn | o = \approx       | r = 6 \, 350 \cdotp 29318       | c = grams }} {{end-eqn}} The gram equivalent is exact.


Definition:Avoirdupois/Ton

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''ton''' }} {{eqn | r = 20       | c = hundredweight }} {{eqn | r = 80       | c = quarters }} {{eqn | r = 160       | c = stone }} {{eqn | r = 2240       | c = pounds avoirdupois }} {{eqn | o = \approx       | r = 1016       | c = kilograms }} {{end-eqn}}


Definition:Avoirdupois/Hundredweight

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''hundredweight''' }} {{eqn | r = 4       | c = quarters }} {{eqn | r = 8       | c = stone }} {{eqn | r = 112       | c = pounds avoirdupois }} {{eqn | o = \approx       | r = 50 \cdotp 8       | c = kilograms }} {{end-eqn}}


Definition:Avoirdupois/Quarter

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''quarter''' }} {{eqn | r = 2       | c = stone }} {{eqn | r = 28       | c = pounds avoirdupois }} {{eqn | o = \approx       | r = 12 \cdotp 7       | c = kilograms }} {{end-eqn}}


Definition:Avoirdupois/Short

=== Short Hundredweight === {{Definition:Avoirdupois/Short/Hundredweight}} === Short Ton === {{Definition:Avoirdupois/Short/Ton}}


Definition:Avoirdupois/Short/Hundredweight

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''short hundredweight''' }} {{eqn | r = 100       | c = pounds avoirdupois }} {{eqn | o = \approx       | r = 45 \cdotp 359       | c = kilograms }} {{end-eqn}}


Definition:Avoirdupois/Short/Ton

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''short ton''' }} {{eqn | r = 20       | c = short hundredweight }} {{eqn | r = 2000       | c = pounds avoirdupois }} {{eqn | o = \approx       | r = 907 \cdot 2       | c = kilograms }} {{end-eqn}}


Definition:Apothecaries' Weights and Measures/Mass/Drachm

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''drachm''' }} {{eqn | r = 3       | c = scruples }} {{eqn | r = 60       | c = grains }} {{eqn | r = 3 \cdotp 89       | c = grams }} {{end-eqn}}


Definition:Apothecaries' Weights and Measures/Mass/Pound

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''apothecaries' pound''' }} {{eqn | r = 12       | c = apothecaries' ounces }} {{eqn | r = 5 \, 760       | c = grains }} {{eqn | r = 373 \cdotp 24       | c = grams }} {{eqn | r = 1       | c = troy pound }} {{end-eqn}}


Definition:Troy/Pennyweight

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''pennyweight''' }} {{eqn | r = 24       | c = grains }} {{eqn | o = \approx       | r = 1 \cdotp 56       | c = grams }} {{end-eqn}}


Definition:Troy/Pound

{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''troy pound''' }} {{eqn | r = 12       | c = troy ounces }} {{eqn | r = 5 \, 760       | c = grains }} {{eqn | r = 1       | c = apothecaries' pound }} {{eqn | o = \approx       | r = 373 \cdotp 24       | c = grams }} {{end-eqn}}


Definition:Pound/Symbol

The symbol for the pound is $\text{lb}$. This derives from the libra (Roman pound) from which the pound evolved. It always needs to be distinguished from the pound sterling, whose symbol is $\pounds$.


Definition:Ounce/Symbol

The symbol for the ounce is $\text{oz}$. This derives indirectly from the uncia (Roman ounce) via the $15$th century Italian '''onza'''.


Definition:Avoirdupois/Quarter/Symbol

The symbol for the quarter is $\text{qtr}$.


Definition:Troy/Pennyweight/Symbol

The symbol for the pennyweight is $\text{dwt}$. This derives from the Latin '''denarius''', which evolved into the (old) penny.


Definition:Kilogram/Symbol

The symbol for the '''kilogram''' is $\mathrm {kg}$.


Definition:Milligram

:$1$ '''milligram''' is defined as $= \dfrac 1 {1 \, 000}$ of a gram.


Definition:Milligram/Symbol

The symbol for the '''milligram''' is $\mathrm {mg}$.


Definition:Sexagesimal Notation

'''Sexagesimal notation''' is the technique of expressing numbers in base $60$.


Definition:Long Hundred

A '''long hundred''' is another term for the number $120$.


Definition:Short Hundred

A '''short hundred''' is another term for the number $100$.


Definition:Rhombic Dodecahedron

The '''rhombic dodecahedron''' is a Catalan polyhedron with $12$ congruent faces each in the shape of a rhombus.


Definition:Reversal

Let $m = \sqbrk {a_n a_{n - 1} a_{n - 2} \ldots a_2 a_1 a_0}$ be an integer expressed in base $10$. That is: :$m = \displaystyle \sum_{k \mathop = 0}^n a_k 10^k$ Its '''reversal''' $m'$ is the integer created by writing the digits of $m$ in the opposite order: :$m' = \sqbrk {a_0 a_1 a_2 \ldots a_{n - 2} a_{n - 1} a_n}$ That is: :$m' = \displaystyle \sum_{k \mathop = 0}^n a_{n - k} 10^k$


Definition:Emirp

An '''emirp''' is a prime number whose reversal is a different prime number.


Definition:Variational Derivative

Let $\map y x$ be a real function. Let $J = J \sqbrk y$ be a functional dependent on $y$. Let $\map h x$ be a real function, which differs from zero only in the neighbourhood of $x_0$. Consider an increment of functional $\Delta J \sqbrk {y; h}$. Denote the area between $\map y x + \map h x$ and $\map y x$ (or, equivalently, between $\map h x$ and x-axis) as $\Delta \sigma$. Let $\Delta \sigma \to 0$ in such a way, that  $\ds \lim_{\Delta \sigma \mathop \to 0} \map \max {\map h x} = 0$ and the length of interval where $\map h x$ differs from 0 would go to 0. If the ratio $\dfrac {\Delta J \sqbrk {y; h} } {\Delta \sigma}$ converges to a limit as $\Delta \sigma \to 0$, then: $\ds \lim_{\Delta \sigma \mathop \to 0} \frac {\Delta J \sqbrk {y; h} } {\Delta \sigma} = \intlimits {\frac {\delta J} {\delta y} } {x = x_0} {}$ where $\intlimits {\dfrac {\delta J} {\delta y} } {x = x_0} {}$ is called the '''variational derivative''' at the point $x = x_0$ for the function $y = \map y x$. {{Stub|Check if there is a definition not using a special point $x_0$}}


Definition:Euler's Equation for Vanishing Variation

Let $\map y x$ be a real function. Let $\map F {x, y, z}$ be a real function belonging to $C^2$ {{WRT}} all its variables. Let $J \sqbrk y$ be a functional of the form: :$\displaystyle \int_a^b \map F {x, y, y'} \rd x$ Then '''Euler's equation for vanishing variation''' is defined a differential equation, resulting from condition: :$\displaystyle \delta \int_a^b \map F {x, y, y'} \rd x = 0$ In other words: :$\displaystyle F_y - \dfrac \d {\d x} F_{y'} = 0$ {{Stub|Check if conditions can be stricter, add special cases, examples, multidimensional and multiderivative forms}} {{NamedforDef|Leonhard Paul Euler|cat = Euler}}


Definition:Directed Suprema Inheriting

Let $L = \left({X, \preceq}\right)$ be an ordered set. Let $S = \left({Y, \precsim}\right)$ be an ordered subset of $L$. Then $S$ '''inherits directed suprema''' {{iff}} :for all directed subsets $A$ of $Y$: if $A$ admits a supremum in $L$, then $\sup_L A \in Y$


Definition:Ordered Set of Subalgebras

Let $L$ be an ordered set. The '''ordered set of subalgebras''' of $L$ is ordered subset of $\operatorname{ClSystems}\left({L}\right)$ and is defined by :$\operatorname{Subalgeras}\left({L}\right) := \left({X, \precsim}\right)$ where :$X$ is the set of all directed suprema inheriting closure systems on $L$, :$\operatorname{ClSystems}\left({L}\right)$ denotes the ordered set of closure systems on $L$.


Definition:Lattice (Group Theory)/Definition 1

A (point) '''lattice''' is a discrete subgroup of $\R^m$ under addition.


Definition:Lattice (Group Theory)/Definition 2

Let $\R^m$ be the $m$-dimensional real Euclidean space. Let $\set {b_1, b_2, \ldots, b_n}$ be a set of linearly independent vectors of $\R^m$. A '''lattice in $\R^m$''' is the set of all integer linear combinations of such vectors. That is: :$\displaystyle \map \LL {b_1, b_2, \ldots, b_n} = \set {\sum_{i \mathop = 1}^n x_i b_i : x_i \in \Z}$


Definition:Coarser Subset (Order Theory)

Let $L = \left({S, \preceq}\right)$ be a preordered set. Let $X, Y$ be subsets of $S$. Then $X$ is '''coarser (subset)''' than $Y$ {{iff}} :$\forall x \in X: \exists y \in Y: y \preceq x$


Definition:Finer Subset (Order Theory)

Let $L = \left({S, \preceq}\right)$ be a preordered set. Let $X, Y$ be subsets of $S$. Then $X$ is '''finer (subset)''' than $Y$ {{iff}} :$\forall x \in X: \exists y \in Y: x \preceq y$


Definition:Archimedean Polyhedron

An '''Archimedean polyhedron''' is a convex polyhedron with the following properties: :$(1): \quad$ Each of its faces is a regular polygon :$(2): \quad$ It is vertex-transitive :$(3): \quad$ The faces are not all congruent. :$(4): \quad$ It is not a regular prism or a regular antiprism.


Definition:Semiregular Polyhedron

A '''semiregular polyhedron''' is a polyhedron with the following properties: :$(1): \quad$ Each of its faces is a regular polygon :$(2): \quad$ It is vertex-transitive :$(3): \quad$ The faces are not all congruent.


Definition:Inaccessible by Directed Suprema

Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set. Let $X$ be a subset of $S$. Then $X$ is '''inaccessible by directed suprema''' {{iff}} :for all directed subsets $D$ of $S$: $\sup D \in X \implies X \cap D \ne \varnothing$


Definition:Closed under Directed Suprema

Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set. Let $X$ be a subset of $S$. Then $X$ is '''closed under directed suprema''' {{iff}} :for all directed subsets $D$ of $S$: $D \subseteq X \implies \sup D \in X$


Definition:Small Stellated Dodecahedron

The small stellated dodecahedron is a Kepler-Poinsot polyhedron composed of $12$ pentagrammic faces with $5$ pentagrams meeting at each vertex. {{stub|Picture needed for a start}}


Definition:Great Stellated Dodecahedron

The great stellated dodecahedron is a Kepler-Poinsot polyhedron composed of $12$ pentagrammic faces with $3$ pentagrams meeting at each vertex. {{stub|Picture needed for a start}}


Definition:Catalan Polyhedron

A '''Catalan polyhedron''' is a dual polyhedron of an Archimedean polyhedron.


Definition:Contractible Space

=== Definition 1 === {{:Definition:Contractible Space/Definition 1}} === Definition 2 === {{:Definition:Contractible Space/Definition 2}}


Definition:Property (S)

Let $L = \left({S, \preceq}\right)$ be an up-complete ordered set. Let $X$ be a subset of $S$. Then $X$ has '''property (S)''' {{iff}} :for all directed subsets $D$ of $S$: $\sup D \in X \implies \exists y \in D:\forall x \in D: y \preceq x \implies x \in X$


Definition:Stellation

'''Stellation''' is the process of extending the extremities of a geometric figure in a specifically structured way, in order to form new figures from those extensions. The figure so generated is itself known as a '''stellation''' or a '''stellated''' original figure.


Definition:Stellation/Polygon

A '''stellation''' of a polygon $P$ is a plane figure formed by extending the sides of $P$ until they meet.


Definition:Stellation/Polyhedron

A '''stellation''' of a polyhedron $H$ is a solid figure formed by extending the faces of $H$ until they meet.


Definition:Quadratic Gauss Sum

Let $p$ be an odd prime. Let $a$ be an integer. The '''quadratic Gauss sum''' is: :$\displaystyle g \left({a, p}\right) = \sum_{n \mathop = 0}^{p - 1} e^{2 \pi i a n^2 / p}$ Category:Definitions/Number Theory kuh39bp5zd1fa4xw5xigtf5htzh1m2i


Definition:Relational Structure with Topology

A triple $\left({S, \preceq, \tau}\right)$ is a '''relational structure with topology''' where :$\preceq$ is a relation on $S$ and :$\tau$ is a topology on $S$.


Definition:Weierstrass Elementary Factor

Let $n \in \Z_{\ge 0}$ be a positive integer. The '''$n$th (Weierstrass) elementary factor''' is the function $E_n: \C \to \C$ defined as: :$E_n \left({z}\right) = \displaystyle \begin{cases} 1 - z & : n = 0 \\ \left({1 - z}\right) \exp \left({z + \dfrac {z^2} 2 + \cdots + \dfrac{z^n} n}\right) & : \text{otherwise}\end{cases}$


Definition:Absolute Convergence of Product/General Definition/Definition 1

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty \paren{1 + a_n}$ is '''absolutely convergent''' {{iff}} $\displaystyle \prod_{n \mathop = 1}^\infty \paren{1 + \norm{a_n}}$ is convergent.


Let $\sequence {a_n}$ be a sequence in $\C$. === Definition 1 === {{:Definition:Absolute Convergence of Product/Complex Numbers/Definition 1}} === Definition 2 === {{:Definition:Absolute Convergence of Product/Complex Numbers/Definition 2}} === Definition 3 === {{:Definition:Absolute Convergence of Product/Complex Numbers/Definition 3}}


Let $\mathbb K$ be one of the standard number fields $\Q, \R, \C$. === Nonzero Sequence === {{:Definition:Convergent Product/Number Field/Nonzero Sequence}} === Arbitrary Sequence === {{:Definition:Convergent Product/Number Field/Arbitrary Sequence}}


Definition:Scott Topology

Let $T = \left({S, \preceq, \tau}\right)$ be a relational structure with topology where $\left({S, \preceq}\right)$ is an up-complete ordered set. Then $T$ has '''Scott topology''' {{iff}} :$\tau$ is the set of all upper and inaccessible by directed suprema subsets of $S$. {{NamedforDef|Dana Stewart Scott|cat = Scott, Dana}}


Let $\struct {\mathbb K, \norm {\,\cdot\,} }$ be a valued field. === Nonzero Sequence === {{:Definition:Convergent Product/Arbitrary Field/Nonzero Sequence}} === Arbitrary Sequence === {{:Definition:Convergent Product/Arbitrary Field/Arbitrary Sequence}}


Definition:Divergent Product

An infinite product which is not convergent is '''divergent'''. === Divergence to zero === {{:Definition:Divergent Product/Divergence to Zero}}


Definition:Primitive Pythagorean Triangle

A '''primitive Pythagorean triangle''' is a Pythagorean triangle whose sides form a primitive Pythagorean triple.


Definition:O Notation/Big-O Notation/Real/Point

Let $x_0 \in \R$. Let $f$ and $g$ be real-valued or complex-valued functions defined on a punctured neighborhood of $x_0$. The statement: :$f \left({x}\right) = \mathcal O \left({g \left({x}\right)}\right)$ as $x \to x_0$ is equivalent to: :$\exists c \in \R: c \ge 0: \exists \delta \in \R : \delta > 0 : \forall x \in \R : \left({0 < \left\lvert{x - x_0}\right\rvert < \delta \implies \left\lvert{f \left({x}\right)}\right\rvert \le c \cdot \left\lvert{g \left({x}\right)}\right\rvert}\right)$ That is: :$\left\lvert{f \left({x}\right)}\right\rvert \le c \cdot \left\lvert{g \left({x}\right)}\right\rvert$ for all $x$ in a punctured neighborhood of $x_0$.


Definition:O Notation/Big-O Notation/Complex/Infinity

Let $f$ and $g$ be complex functions defined for all complex numbers whose modulus is sufficiently large. The statement: :$f(z) = \mathcal O \left({g(z)}\right)$ as $|z|\to\infty$ is equivalent to: :$\displaystyle \exists c\in \R: c\ge 0 : \exists r_0 \in \R : \forall z \in \C : (|z| \geq r_0 \implies |f(z)| \leq c \cdot |g(z)|)$ That is: :$|f(z)| \leq c \cdot |g(z)|$ for all $z$ in a neighborhood of infinity in $\CC$.


Definition:O Notation/Big-O Notation/Complex/Point

Let $z_0 \in \C$. Let $f$ and $g$ be complex functions defined on an punctured neighborhood of $z_0$. The statement: :$f(z) = \mathcal O \left({g(z)}\right)$ as $z\to z_0$ is equivalent to: :$\displaystyle \exists c \in \R : c \ge 0 : \exists \delta \in \R : \delta > 0 : \forall z \in \C : (0<|z-z_0|<\delta \implies |f(z)| \leq c \cdot |g(z)|)$ That is: :$|f(z)| \leq c \cdot |g(z)|$ for all $z$ in a punctured neighborhood of $z_0$.


Definition:O Notation/Big-O Notation/Real/Infinity

Let $f$ and $g$ be real-valued or complex-valued functions defined on a neighborhood of $+ \infty$ in $\R$. The statement: :$\map f x = \map {\mathcal O} {\map g x}$ as $x \to \infty$ is equivalent to: :$\exists c \in \R: c \ge 0: \exists x_0 \in \R: \forall x \in \R: \paren {x \ge x_0 \implies \size {\map f x} \le c \cdot \size {\map g x} }$ That is: :$\size {\map f x} \le c \cdot \size {\map g x}$ for $x$ sufficiently large. This statement is voiced '''$f$ is big-O of $g$''' or simply '''$f$ is big-O $g$'''.


Let $\left \langle {a_n} \right \rangle$ and $\left \langle {b_n} \right \rangle$ be sequences in $\R$. === Definition 1 === {{:Definition:Asymptotically Equal/Sequences/Definition 1}} === Definition 2 === {{:Definition:Asymptotically Equal/Sequences/Definition 2}} === Definition 3 === {{:Definition:Asymptotically Equal/Sequences/Definition 3}} This is denoted: $a_n \sim b_n$.


Definition:Asymptotically Equal/Functions

Let $f$ and $g$ real functions defined on $\R$. Then: : '''$f$ is asymptotically equal to $g$''' {{iff}}: :$\dfrac {f \left({x}\right)} {g \left({x}\right)} \to 1$ as $x \to +\infty$. That is, the larger the $x$, the closer $f$ gets (relatively) to $g$.


Definition:O Notation/Big-O Notation/General Definition/Point

Let $(X,\tau)$ be a topological space. Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\left\Vert{\,\cdot\,}\right\Vert$ Let $x_0\in X$. Let $f,g:X\setminus\{x_0\}\to V$ be functions. The statement :$f(x) = \mathcal O \left({g(x)}\right)$ as $x\to x_0$ is equivalent to: :$\displaystyle \exists c\in \R: c\ge 0 : \exists U\in\tau: x_0\in U : \forall x\in U\setminus\{x_0\} : \Vert f(x)\Vert \leq c\cdot\Vert g(x)\Vert$ That is: :$\Vert f(x)\Vert\leq c\cdot\Vert g(x)\Vert$ for all $x$ in a punctured neighborhood of $x_0$.


Definition:Asymptotically Equal/General Definition/Point

Let $T = \left({S, \tau}\right)$ be a topological space. Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\left\Vert{\, \cdot \,}\right\Vert$. Let $f, g: S \to V$ be mappings. Let $x_0 \in X$. Then: : $f$ is '''asymptotically equal''' to $g$ as $x \to x_0$ {{iff}}: : $f - g = o \left({g}\right)$ as $x \to x_0$ where $o$ denotes little-O notation.


Let $f$ and $g$ be real-valued or complex-valued functions on a subset of $\R$ containing all sufficiently large real numbers. === Definition 1 === {{:Definition:O Notation/Little-O Notation/Real Functions/Definition 1}} === Definition 2 === {{:Definition:O Notation/Little-O Notation/Real Functions/Definition 2}} This is denoted: :$f = o \left({g}\right)\qquad(x\to\infty)$ </onlyinclude> This statement is voiced '''$f$ is little-o of $g$''' or simply '''$f$ is little-o $g$'''.


Definition:Sequence of Partial Products

The sequence $\sequence {p_N}$ defined as: :$\displaystyle p_N: = \prod_{n \mathop = 1}^N a_n = a_1 \cdot a_2 \cdots a_N$ is the '''sequence of partial products''' of the infinite product $\displaystyle p = \prod_{n \mathop = 1}^\infty a_n$.


Definition:O Notation/Little-O Notation/General Definition/Point

Let $X$ be a topological space. Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\left\Vert{\,\cdot\,}\right\Vert$ Let $f,g:X\to V$ be functions. Let $x_0\in X$. The statement :$f(x) = o \left({g(x)}\right)$ as $x\to x_0$ is equivalent to the statement: :For all $\epsilon>0$, there exists a neighborhood $U$ of $x_0$ such that $\Vert f(x)\Vert\leq \epsilon\cdot\Vert g(x)\Vert$ for all $x\in U$


Definition:Topological Lattice

A '''topological lattice''' is a relational structure with topology $\left({S, \preceq, \tau}\right)$ where $\left({S, \preceq}\right)$ is a lattice and $\left({S, \tau}\right)$ is a topological space.


Definition:Generator for Primitive Pythagorean Triple

The '''generator''' for a primitive Pythagorean triple is an ordered pair: :$G = \left({m, n}\right)$ where $m, n \in \Z$ such that: : $m, n \in \Z_{>0}$ are (strictly) positive integers : $m \perp n$, that is, $m$ and $n$ are coprime : $m$ and $n$ are of opposite parity : $m > n$. The primitive Pythagorean triple which has been '''generated''' by $G$ is: :$\left({2 m n, m^2 - n^2, m^2 + n^2}\right)$


Definition:Generator for Pythagorean Triple

Let $P = \left\{ {a, b, c}\right\}$ be a Pythagorean triple  A '''generator''' for $P$ is an ordered pair: :$G = \left({m, n}\right)$ where $m, n \in \Z$ such that: : $a = 2 m n$ : $b = m^2 - n^2$ : $c = m^2 + n^2$


Definition:Almost Isosceles Pythagorean Triangle

Let $P$ be a Pythagorean triangle whose legs are $a$ and $b$. Then $P$ is '''almost isosceles''' {{iff}} $\size {a - b} = 1$ That is, if the legs of $P$ differ by $1$. === Sequence === {{:Generator for Almost Isosceles Pythagorean Triangle/Sequence}}


=== Neighborhood of Positive Infinity === {{:Definition:Neighborhood of Infinity (Real Analysis)/Positive Infinity}} === Neighborhood of Negative Infinity === {{:Definition:Neighborhood of Infinity (Real Analysis)/Negative Infinity}}


Definition:Neighborhood of Infinity (Real Analysis)/Positive Infinity

A '''neighborhood of $+\infty$''' is a subset of the set of real numbers $\R$ wich contains an interval $(a,+\infty)$ for some $a\in\R$. That is, a subset which contains all sufficiently large real numbers.


Definition:Neighborhood of Infinity (Real Analysis)/Negative Infinity

A '''neighborhood of $-\infty$''' is a subset of the set of real numbers $\R$ wich contains an interval $(-\infty,a)$ for some $a\in\R$. That is, a subset which contains all sufficiently large negative real numbers.


Definition:Nearest Integer Function

The '''nearest integer function''' is defined as: :$\forall x \in \R: \nint x = \begin {cases} \floor {x + \dfrac 1 2} & : x \notin 2 \Z + \dfrac 1 2 \\ x - \dfrac 1 2 & : x \in 2 \Z + \dfrac 1 2 \end{cases}$ where $\floor x$ is the floor function.


Definition:Pythagorean Prime

A '''Pythagorean prime''' is a prime number of the form: :$p = 4 n + 1$ where $n \in \Z_{\ge 0}$ is a positive integer. === Sequence === {{:Definition:Pythagorean Prime/Sequence}}


Definition:Non-Pythagorean Prime

A '''non-Pythagorean prime''' is an odd prime number of the form: :$p = 4 n + 3$ where $n \in \Z_{\ge 0}$ is a positive integer. === Sequence === {{:Definition:Non-Pythagorean Prime/Sequence}}


Definition:O Notation/Big-O Notation/Sequence

Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences of real or complex numbers. '''$a_n$ is big-O of $b_n$''' {{iff}} :$\exists c \in \R: c \ge 0 : \exists n_0 \in \N : \paren {n \ge n_0 \implies \size {a_n} \le c \cdot \size {b_n} }$ That is: :$\size {a_n} \le c \cdot \size {b_n}$ for all sufficiently large $n$.


=== Estimate at infinity === {{:Definition:O Notation/Big-O Notation/General Definition/Infinity}} === Point Estimate === {{:Definition:O Notation/Big-O Notation/General Definition/Point}}


Definition:Neighborhood of Infinity (Topology)

Let $X$ be a non-empty topological space. A '''neighborhood of infinity''' is a subset of $X$ which contains the complement of a closed and compact subset of $X$.


Definition:Hamiltonian

Let $J \sqbrk {\dotsm y_i \dotsm}$ be a functional of the form: :$\displaystyle J \sqbrk {\dotsm y_i \dotsm} = \intlimits {\int_{x_0}^{x_1} \map F {x, \cdots y_i \dotsm, \dotsm y_i \dotsm} \rd x} {i \mathop = 1} {i \mathop = n}$ Then the '''Hamilonian''' $H$ corresponding to $J \sqbrk {\dotsm y_i \dotsm}$ is defined as :$\displaystyle H = -F + \sum_{i \mathop = 1}^n y_i' F_{y_i'}$ {{NamedforDef|William Rowan Hamilton|cat = Hamilton}}


Definition:Canonical Variable

Let $\map {\mathbf y} x : \R \to \R^n$, $n \in \N $ be a vector-valued function. Let $F: \R^{2 n + 1} \to \R$ be a differentiable mapping. Let $J \sqbrk {\mathbf y}$ be a functional of the form: :$\d J \sqbrk {\mathbf y} = \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x$ Consider the (real) variables $x, \mathbf y, \mathbf y'$, and the mapping $F$. Make the following transformation: :$F_{y_i'} = p_i$ where $F_x$ denotes the partial derivative of $F$ {{WRT}} $x$. Let $H$ be the Hamiltonian corresponding to $J \sqbrk {\mathbf y}$. Then the new variables $x, \mathbf y, \mathbf p$, and the mapping $H$ corresponding to $J \sqbrk {\mathbf y}$ are called '''the canonical variables'''.


Definition:Time/Unit/Week

:$1$ '''week''' $=7$ days.


Definition:Time/Unit/Week/Symbol

The symbol for the '''week''' is $\mathrm {week}$ or $\mathrm {wk}$.


Definition:Time/Unit/Fortnight

:$1$ '''fortnight''' $=2$ weeks $=14$ days.


Definition:O Notation/Big-O Notation/Parameter

Let $X$ be a topological space. Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\, \cdot \,}$ Let $x_0 \in X \cup \set \infty$. Let $A$ be a set. Let $X_\alpha$ be a subset of $X$ for every $\alpha \in A$, which, if $x_0 \ne \infty$, contains $x_0$. Let $f_\alpha : X_\alpha \setminus \set {x_0} \to V$ be a mapping for every $\alpha \in A$. Let $g : X \to V$ be a mapping. The statement: :$f_\alpha = \map {\OO_\alpha} g$ as $x \to x_0$ is equivalent to: :$\forall \alpha \in A : f_\alpha = \map \OO g$ as $x \to x_0$ The $\OO$-estimate is said to be '''independent''' of $\alpha \in A$ {{iff}}: :there exists a neighborhood $U$ of $x_0$ in $X$ such that: ::$\exists c \in \R: c \ge 0 : \forall \alpha \in A : \forall x \in \paren {U \setminus \set {x_0} } \cap X_\alpha : \norm {\map {f_\alpha} x} \le c \cdot \norm {\map g x}$ That is, if the implied constant and implied neighborhood can be chosen the same for all $\alpha \in A$.


=== Estimate at infinity === {{:Definition:O Notation/Big-O Notation/Real/Infinity}} === Point Estimate === {{:Definition:O Notation/Big-O Notation/Real/Point}}


=== Estimate at infinity === {{:Definition:O Notation/Big-O Notation/Complex/Infinity}} === Point Estimate === {{:Definition:O Notation/Big-O Notation/Complex/Point}}


Definition:O Notation/Big-O Notation/General Definition/Infinity

Let $\left({X, \tau}\right)$ be a topological space. Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\left\Vert{\,\cdot\,}\right\Vert$. Let $f, g : X \to V$ be functions. The statement: :$f \left({x}\right) = \mathcal O \left({g \left({x}\right)}\right)$ as $x \to \infty$ is equivalent to: :There exists a neighborhood of infinity $U \subset X$ such that: ::$\exists c \in \R: c \ge 0: \forall x \in U: \left\Vert{f \left({x}\right)}\right\Vert \le c \cdot \left\Vert{g \left({x}\right)}\right\Vert$ That is: :$\Vert f \left({x}\right) \Vert \le c \cdot \Vert g \left({x}\right) \Vert$ for all $x$ in a neighborhood of infinity.


Definition:Time/Unit/Month

The '''month''' is measured only approximately in duration, and is not used as a scientific measure. :$1$ '''month''' is between $28$ and $31$ days, depending on which '''month''' it is.


Definition:Time/Unit/Season

The '''season''' is measured only approximately in duration, and is not used as a scientific measure. :$1$ '''season''' $= 3$ months, although exactly which months fall into which '''season''' depends on the locale and the culture.


Definition:Neighborhood of Infinity (Complex Analysis)

A '''neighborhood of $\infty$''' in $\C$ is a subset of the set of complex numbers $\C$ wich contains a set of the form $\{z \in \C : |z| > r\}$ for some $r\in\R$. That is, a subset which contains all complex numbers whose modulus is sufficiently large.


Definition:Covolume of Lattice

Let $L$ be a lattice in $\R^n$. Let $\left({v_1, \ldots, v_n}\right)$ be an ordered basis for $L$. Let $v_i = \left({v_{i 1}, \ldots , v_{i n} }\right)$ for $i \in \left\{ {1, \ldots, n}\right\}$. The '''covolume''' of $L$ is the determinant of the matrix: :$\begin{bmatrix} v_{1 1} & v_{1 2} & \cdots & v_{1 n} \\ v_{2 1} & v_{2 2} & \cdots & v_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ v_{n 1} & v_{n 2} & \cdots & v_{n n} \\ \end{bmatrix}$


Definition:Integer Lattice

Let $n$ be a positive integer. The '''integer lattice''' in $\R^n$ is the lattice $\Z^n$. Category:Definitions/Linear Algebra Category:Definitions/Geometry of Numbers 01q0zkrsaqbk9hph0uhv482i84e1ows


Definition:Fundamental Domain (Lattice)

Let $L \subset \R^n$ be an integral lattice. Let $B = \left({v_1, \ldots, v_n}\right)$ be an ordered basis for $L$. The '''fundamental domain''' of $L$ associated to $B$ is the set: :$\displaystyle \left\{ {\sum_{i \mathop = 1}^n \lambda_i v_i: \forall i: 0 \le \lambda_i < 1}\right\}$ Category:Definitions/Linear Algebra Category:Definitions/Geometry of Numbers nh0m2th0ored6dymzrx9lt7pjpnoojz


Definition:Integral Lattice

Let $n$ be a positive integer. An '''integral lattice''' in $\R^n$ is a subgroup that is isomorphic to the integer lattice $\Z^n$.


Definition:Square Pyramidal Number

'''Square pyramidal numbers''' are those denumerating a collection of objects which can be arranged in the form of a square pyramid.


Definition:Discrete Group

A '''discrete group''' is a topological group whose topology is discrete. Category:Definitions/Topological Groups 2yt2c8qbqpt5dqn8tinr0cwz056jo7p


Definition:Discrete Subgroup

Let $G$ be a topological group. Let $H$ be a subgroup of $G$. Then $H$ is called a '''discrete subgroup''' if it is a discrete group for the induced topology. === Real Numbers === {{:Definition:Discrete Subgroup/Real Numbers}} Category:Definitions/Subgroups Category:Definitions/Topological Groups 84q4o7iejd52d7n595dv1qhcr5fxy1p


Definition:Limit Inferior of Moore-Smith Sequence

Let $\left({S, \preceq}\right)$ be a directed set. Let $L = \left({T, \precsim}\right)$ be a complete lattice. Let $N: S \to T$ be a Moore-Smith sequence in $T$. Then '''limit inferior''' of $N$ is defined as follows: :$\liminf N := \sup_L \left\{ {\inf_L \left({N \left[{\preceq \left({j}\right)}\right]}\right): j \in S}\right\}$ where :$\preceq \left({j}\right)$ denotes the image of $j$ by $\preceq$, :$N \left[{\preceq \left({j}\right)}\right]$ denotes the image of $\preceq \left({j}\right)$ under $N$.


Definition:Discrete Subgroup/Real Numbers

Let $G$ be a subgroup of the additive group of real numbers. Then $G$ is '''discrete''' {{iff}}: :$\forall g \in G : \exists \epsilon > 0: \openint {g - \epsilon} {g + \epsilon} \cap G = \set g$ That is, there exists a neighborhood of $g$ which contains no other elements of $G$.


Definition:Everywhere Dense/Real Numbers

Let $S$ be a subset of the real numbers. Then $S$ is '''(everywhere) dense''' in $\R$ {{iff}}: :$\forall x \in \R : \forall \epsilon \in \R : \epsilon > 0: \exists s \in S: x - \epsilon < s < x + \epsilon$. That is, {{iff}} in every neighborhood of every real number lies an element of $S$.


Definition:Poisson Bracket

Let  :$\map A {x, \mathbf y, \mathbf p}: \R^{2n+1} \to \R$, :$\map B {x, \mathbf y, \mathbf p}: \R^{2n+1} \to \R$, $n \in \N$  be real functions, dependent on canonical variables. The '''Poisson Bracket''' of functions $A$ and $B$ is defined as: :$\displaystyle \sqbrk {A, B} := \sum_{i \mathop = 1}^n \paren {\frac {\partial A} {\partial y_i} \frac {\partial B} {\partial p_i} - \frac {\partial B} {\partial y_i} \frac {\partial A} {\partial p_i} }$ where the notation $\dfrac {\partial A} {\partial y_i}$ denotes partial differentiation. {{NamedforDef|Siméon-Denis Poisson|cat = Poisson}}


Definition:First Integral of System of Differential Equations

Let $S$ be a system of differential equations. Let $g$ be a function which satisfies $S$. Let $f$ be a function. {{explain|Establish the domain and codomain of these functions}} Let $f$ depend on variables (denoted by ordered tuples here) of $S$ independently as well as through $g$ and its derivatives: :$f = \map f {\sequence {x_i} _{1 \mathop \le i \mathop \le n}, \sequence {g^{\paren j} \paren {\sequence {x_i}_{1 \mathop \le i \mathop \le n} }_{0 \mathop \le j \mathop \le k} } }, \quad {n, k} \in \N$ Suppose there exists $g$ such that $f$ is a constant. Then $f$ is the first integral of $S$.