# Help:Editing

You should feel welcome to contribute, even if you think what you write doesn't match the 'house style'.

In that case, this page is more to explain the meaning behind any minor changes that may be applied to your work after it has been submitted.

Over the course of prolonged contribution, it is considered both polite and very desirable that contributors put effort in mastering and adhering to house style, which is located at Help:House Style.

Should there be any questions, Help:Questions is the place to raise issues.

## Page Editing

This reference provides the general outline on creating pages and instructions for specific sections; for the rest, see Help:House Style. Furthermore, this page does not dwell on details pertaining to the MediaWiki architecture; the reference for such things is Help:Wiki Editing.

## Creating a Page

After searching, you may conclude that the page you're looking for does not exist yet. There are several ways to create it.

### By Searching

Search for the page name which you would like to create. As you have found out by now, the page has not already been created. You will see something that looks like this:

Create the page "Proof name" on this wiki!

Clicking on Proof name will open the page editing window, and you can start writing.

### By following a red link

All you have to do is click the red link and start writing.

### By entering a direct URL

If you know for sure that the proof is not on the site, then simply type that name into the URL. For example, if you wanted to prove the Riemann Hypothesis, you would type:

http://www.proofwiki.org/wiki/Riemann Hypothesis


If this page does not exist then, you will get a single line saying:

Clicking create this page will open the editing page where you can edit the page.

## Deleting a page

Don't do this. Only trusted users can delete pages in any event.

If you feel a page needs to be deleted, add {{Delete}} to the top of the page in question.

## Page Naming

Choosing a good page title is important for multiple reasons:

• It makes the page easier to search for.
• The title gives a first impression of what the page is about. A good title adds to the understanding.
• The title is what is used in links. It's always nice to know what a page is about before clicking the link.
• A descriptive title avoids name conflicts.

Some pages are subject to specific naming conventions. See the corresponding help pages for instructions:

Page titles that do not meet the standards can be flagged for renaming.

## Tips to choose a good name

### Be specific

There's no such thing as being too specific! In particular, page names consisting of a single word should generally be avoided: you never know if there are other concepts in mathematics with the same name. That is:

not Definition:Prime Ideal but Definition:Prime Ideal of Ring
not Definition:Supremum or Definition:Supremum of Set but Definition:Supremum of Subset of Real Numbers

The word "of" comes in handy here.

In the same spirit, it is a good practice to always include at least one noun in the title. That is:

not Definition:Differentiable but Definition:Differentiable Mapping
not Definition:Simple but Definition:Simple Group

Note how the shorter page names are always disambiguation pages.

For theorems, the same philosophy applies:

not Uniqueness of Extension but Uniqueness of Analytic Continuation

### Redundant words

Page names should not be started with articles such as "A" or "The", as this makes it significantly harder to find pages alphabetically in their categories. For example: Pythagorean Theorem, not The Pythagorean Theorem.

Similarly, is not necessary to begin the name with "Proof of ...", and this should be avoided. Since $\mathsf{Pr} \infty \mathsf{fWiki}$ is a collection of proofs, it is assumed that each page is going to be a proof of something. This also makes searching for articles much easier.

In general, the words "the", "a" or "an" are strongly discouraged in page names, as they make the name longer than necessary and rarely add clarity to the concept being defined.

The main exception to this rule is for concepts split into subtypes named along the lines "of the $n$th kind" , for example: Definition:Elliptic Integral of the First Kind.

### Descriptiveness

A good page title describes the content accurately. Because theorems are linked to using their exact page title, when reading a proof it is useful when you can guess what a theorem is about without having to visit the page. In particular, if a theorem contains an equivalence proof, the title should makes this clear, by using "iff".

### Don't blindly trust literature

Books, thanks to their limited scope, can afford to use simplified terminology without running into ambiguity problems. At $\mathsf{Pr} \infty \mathsf{fWiki}$ we can not. Thus it is a good idea to check if there is a danger for name conflicts by doing some research. Alternatively, make the name overly specific.

As for theorems, books may call a theorem "Fundamental Property of Homomorphisms" or "Continuity Property". This does not mean that the theorem is everywhere known by that name. While at $\mathsf{Pr} \infty \mathsf{fWiki}$ we do prefer to use as a title the name of a theorem rather than a description of the result, we do so only if there is no ambiguity.

## Naming conventions

### Namespaces

When you would like to create a page for a definition, all you have to do different from naming a proof is to add Definition: in front of the name. So for example if you wanted to create a page for the definition of calculus you would name the page:

Definition:Calculus

Also, after you create the page, be sure to add the definition to the appropriate "Definitions" categories (see Category:Definitions).

The same method that is used for Definitions is also used to name and categorize axiom and symbols pages. Simply substitute Symbol or Axiom for Definition: in the page name and, mutatis mutandis, in the category name. See Help:Categories.

### Capitalizing

Page titles are case-sensitive. For all types of pages, major words in the title of the page should be capitalized. For example: Subring Generated by Unity of Ring with Unity. So as to promote consistency, be informed that in particular, the following words are not considered to be "major" and ought to be used in their lowercase form when naming a page:

• Prepositions, pronouns and conjunctions:
and, around, as, between, by, for, from, if, iff, in, its, minus, no, not, of, on, or, over, plus, such that, that, the, then, to, under, with, with respect to, which, whose
• Short verbs, such as:
are, can, cannot, does, equals, form, has, have, implies, is

These lists are not exhaustive.

Variables in formulas tend not to be capitalized. See #Names with Formulas.

### Names with Formulas

If the name of a page contains mathematical statements, the following formatting practices should be adhered to:

• Put no spaces between numbers/elements that are added, subtracted, divided, or multiplied together.
Example: (1+2)x3
• Do, however, put spaces between elements that are put into equality or inequality with one another. Use != to signify inequality.
Example: 1 = 3/3 != 3/4
• Variables appearing in the formulas need not be capitalized.
Example: Primitive of x squared over a x + b
• A page name is no place for $\LaTeX$ commands.

## Things named after a mathematician

A theorem that is named after a Mathematician gets a call of the {{Namedfor}} template. When done properly, the page is then automatically placed in a corresponding category in Named Theorems, which itself has to be created manually.

Similarly, for named definitions there is the {{NamedforDef}} template.

Disambiguation pages are treated in the same way. Note that theorem disambiguations do not otherwise get categories. See Help:Disambiguation.

A page whose name contains a word named after a Mathematician does not fall under this category. For example, not every theorem about Krull dimension needs a call of {{Namedfor}}.

## Multiple Names

Some mathematical concepts have several names, according to the sources you consult. Which of these names is used in $\mathsf{Pr} \infty \mathsf{fWiki}$ is largely a matter of happenstance.

However, if one of the names of a concept is for a particular mathematician, that name is to be used in preference.

An example of this is Definition:Chebyshev Distance, which is otherwise known as the Definition:Maximum Metric or the Definition:Chessboard Metric.

## Special Characters

### Disallowed Characters

The following characters should not be used in page names:

### No longer supported

The following $\LaTeX$ commands are not supported in MathJax, but may still be present in some pages. When found they need to be replaced.

For $\lor$: \or to be replaced by \lor
For $\land$: \and to be replaced by \land
For $\R$: \reals to be replaced by \R
For $\exists$: \exist to be replaced by \exists

For producing fixed width text in math mode: \texttt needs to be replaced by \mathtt.

If you find any more examples, add them here.

### New commands

New commands can be requested and discussed at Symbols:LaTeX Commands/ProofWiki Specific, transcluded here:

 $\AA$ $\quad:\quad$\AA $\qquad$that is: \mathcal A $\Add$ $\quad:\quad$\Add $\qquad$Addition as a Primitive Recursive Function‎ $\adj {\mathbf A}$ $\quad:\quad$\adj {\mathbf A} $\qquad$Adjugate Matrix $\arccot$ $\quad:\quad$\arccot $\qquad$Arccotangent $\arccsc$ $\quad:\quad$\arccsc $\qquad$Arccosecant $\arcosh$ $\quad:\quad$\arcosh $\qquad$Area Hyperbolic Cosine $\Arcosh$ $\quad:\quad$\Arcosh $\qquad$Complex Area Hyperbolic Cosine $\arcoth$ $\quad:\quad$\arcoth $\qquad$Area Hyperbolic Cotangent $\Arcoth$ $\quad:\quad$\Arcoth $\qquad$Complex Area Hyperbolic Cotangent $\arcsch$ $\quad:\quad$\arcsch $\qquad$Area Hyperbolic Cosecant $\Arcsch$ $\quad:\quad$\Arcsch $\qquad$Complex Area Hyperbolic Cosecant $\arcsec$ $\quad:\quad$\arcsec $\qquad$Arcsecant $\arsech$ $\quad:\quad$\arsech $\qquad$Area Hyperbolic Secant $\Arsech$ $\quad:\quad$\Arsech $\qquad$Complex Area Hyperbolic Secant $\arsinh$ $\quad:\quad$\arsinh $\qquad$Area Hyperbolic Sine $\Arsinh$ $\quad:\quad$\Arsinh $\qquad$Complex Area Hyperbolic Sine $\artanh$ $\quad:\quad$\artanh $\qquad$Area Hyperbolic Tangent $\Artanh$ $\quad:\quad$\Artanh $\qquad$Complex Area Hyperbolic Tangent $\Area$ $\quad:\quad$\Area $\qquad$Area of Plane Figure $\Arg z$ $\quad:\quad$\Arg z $\qquad$Principal Argument of Complex Number $\Aut {S}$ $\quad:\quad$\Aut {S} $\qquad$Automorphism Group $\BB$ $\quad:\quad$\BB $\qquad$that is: \mathcal B $\Bei$ $\quad:\quad$\Bei $\Ber$ $\quad:\quad$\Ber $\Bernoulli {p}$ $\quad:\quad$\Bernoulli {p} $\qquad$Bernoulli Distribution $\BetaDist {\alpha} {\beta}$ $\quad:\quad$\BetaDist {\alpha} {\beta} $\qquad$Beta Distribution $\bigintlimits {\map f s} {s \mathop = 0} {s \mathop = a}$ $\quad:\quad$\bigintlimits {\map f s} {s \mathop = 0} {s \mathop = a} $\qquad$Limits of Integration $\bigvalueat {\delta x} {x \mathop = x_j}$ $\quad:\quad$\bigvalueat {\delta x} {x \mathop = x_j}  $\Binomial {n} {p}$ $\quad:\quad$\Binomial {n} {p} $\qquad$Binomial Distribution $\braket {a} {b}$ $\quad:\quad$\braket {a} {b} $\qquad$Dirac Notation $\bsDelta$ $\quad:\quad$\bsDelta $\qquad$a vector '$\Delta$' $\bsone$ $\quad:\quad$\bsone $\qquad$vector of ones $\bst$ $\quad:\quad$\bst $\qquad$a vector 't' $\bsv$ $\quad:\quad$\bsv $\qquad$a vector 'v' $\bsw$ $\quad:\quad$\bsw $\qquad$a vector 'w' $\bsx$ $\quad:\quad$\bsx $\qquad$a vector 'x' $\bsy$ $\quad:\quad$\bsy $\qquad$a vector 'y' $\bsz$ $\quad:\quad$\bsz $\qquad$a vector 'z' $\bszero$ $\quad:\quad$\bszero $\qquad$vector of zeros $\map \Card {S}$ $\quad:\quad$\map \Card {S} $\qquad$Cardinality $\card {S}$ $\quad:\quad$\card {S} $\qquad$Cardinality $\Cauchy {x_0} {\gamma}$ $\quad:\quad$\Cauchy {x_0} {\gamma} $\qquad$Cauchy Distribution $\CC$ $\quad:\quad$\CC $\qquad$that is: \mathcal C $\Cdm {f}$ $\quad:\quad$\Cdm {f} $\qquad$Codomain of Mapping $\ceiling {11.98}$ $\quad:\quad$\ceiling {11.98} $\qquad$Ceiling Function $30 \cels$ $\quad:\quad$30 \cels $\qquad$Degrees Celsius $\Char {R}$ $\quad:\quad$\Char {R} $\qquad$Characteristic of Ring, etc. $\Ci$ $\quad:\quad$\Ci $\qquad$Cosine Integral Function $\cis \theta$ $\quad:\quad$\cis \theta $\qquad$$\cos \theta + i \sin \theta $\cl {S}$ \quad:\quad\cl {S} \qquadClosure (Topology) $\closedint {a} {b}$ \quad:\quad\closedint {a} {b} \qquadClosed Interval $\cmod {z^2}$ \quad:\quad\cmod {z^2} \qquadComplex Modulus $\condprob {A} {B}$ \quad:\quad\condprob {A} {B} \qquadConditional Probability $\conjclass {x}$ \quad:\quad\conjclass {x} \qquadConjugacy Class $\cont {f}$ \quad:\quad\cont {f} \qquadContent of Polynomial $\ContinuousUniform {a} {b}$ \quad:\quad\ContinuousUniform {a} {b} \qquadContinuous Uniform Distribution $\cosec$ \quad:\quad\cosec \qquadCosecant (alternative form) $\Cosh$ \quad:\quad\Cosh \qquadHyperbolic Cosine $\Coth$ \quad:\quad\Coth \qquadHyperbolic Cotangent $\cov {X, Y}$ \quad:\quad\cov {X, Y} \qquadCovariance $\csch$ \quad:\quad\csch \qquadHyperbolic Cosecant $\Csch$ \quad:\quad\Csch \qquadHyperbolic Cosecant $\curl$ \quad:\quad\curl \qquadCurl Operator $\DD$ \quad:\quad\DD \qquadthat is: \mathcal D $\dfrac {\d x} {\d y}$ \quad:\quad\dfrac {\d x} {\d y} \qquadRoman \d for Derivatives $30 \degrees$ \quad:\quad30 \degrees \qquadDegrees of Arc $\diam$ \quad:\quad\diam \qquadDiameter $\Dic n$ \quad:\quad\Dic n \qquadDicyclic Group $\DiscreteUniform {n}$ \quad:\quad\DiscreteUniform {n} \qquadDiscrete Uniform Distribution $a \divides b$ \quad:\quada \divides b \qquadDivisibility $\Dom {f}$ \quad:\quad\Dom {f} \qquadDomain of Mapping $\dr {a}$ \quad:\quad\dr {a} \qquadDigital Root $\E$ \quad:\quad\E \qquadElementary Charge $\EE$ \quad:\quad\EE \qquadthat is: \mathcal E $\Ei$ \quad:\quad\Ei \qquadExponential Integral Function $\empty$ \quad:\quad\empty \qquadEmpty Set $\eqclass {x} {\RR}$ \quad:\quad\eqclass {x} {\RR} \qquadEquivalence Class $\erf$ \quad:\quad\erf \qquadError Function $\erfc$ \quad:\quad\erfc \qquadComplementary Error Function $\expect {X}$ \quad:\quad\expect {X} \qquadExpectation $\Exponential {\beta}$ \quad:\quad\Exponential {\beta} \qquadExponential Distribution $\Ext {\gamma}$ \quad:\quad\Ext {\gamma} \qquadExterior $\F$ \quad:\quad\F \qquadFalse $30 \fahr$ \quad:\quad30 \fahr \qquadDegrees Fahrenheit $\family {S_i}$ \quad:\quad\family {S_i} \qquadIndexed Family $\FF$ \quad:\quad\FF \qquadthat is: \mathcal F $\Fix {\pi}$ \quad:\quad\Fix {\pi} \qquadSet of Fixed Elements $\floor {11.98}$ \quad:\quad\floor {11.98} \qquadFloor Function $\fractpart {x}$ \quad:\quad\fractpart {x} \qquadFractional Part $\Frob {R}$ \quad:\quad\Frob {R} \qquadFrobenius Endomorphism $\Gal {S}$ \quad:\quad\Gal {S} \qquadGalois Group $\Gaussian {\mu} {\sigma^2}$ \quad:\quad\Gaussian {\mu} {\sigma^2} \qquadGaussian Distribution $\gen {S}$ \quad:\quad\gen {S} \qquadGenerator $\Geometric {p}$ \quad:\quad\Geometric {p} \qquadGeometric Distribution $\GF$ \quad:\quad\GF \qquadGalois Field $\GG$ \quad:\quad\GG \qquadthat is: \mathcal G $\GL {n, \R}$ \quad:\quad\GL {n, \R} \qquadGeneral Linear Group $\grad {p}$ \quad:\quad\grad {p} \qquadGradient $\hav \theta$ \quad:\quad\hav \theta \qquadHaversine $\hcf$ \quad:\quad\hcf \qquadHighest Common Factor $\H$ \quad:\quad\H \qquadSet of Quaternions $\HH$ \quad:\quad\HH \qquadHilbert Space $\hointl {a} {b}$ \quad:\quad\hointl {a} {b} \qquadLeft Half-Open Interval $\hointr {a} {b}$ \quad:\quad\hointr {a} {b} \qquadRight Half-Open Interval $\horectl a b$ \quad:\quad\horectl a b \qquadHalf-Open Rectangle (on the left) $\horectr c d$ \quad:\quad\horectr c d \qquadHalf-Open Rectangle (on the right) $\ideal {a}$ \quad:\quad\ideal {a} \qquadIdeal of Ring $\II$ \quad:\quad\II \qquadthat is: \mathcal I $\map \Im z$ \quad:\quad\map \Im z \qquadImaginary Part $\Img {f}$ \quad:\quad\Img {f} \qquadImage of Mapping $\index {G} {H}$ \quad:\quad\index {G} {H} \qquadIndex of Subgroup $\inj$ \quad:\quad\inj \qquadCanonical Injection $\Inn {S}$ \quad:\quad\Inn {S} \qquadGroup of Inner Automorphisms $\innerprod {x} {y}$ \quad:\quad\innerprod {x} {y} \qquadInner Product $\Int {\gamma}$ \quad:\quad\Int {\gamma} \qquadInterior $\intlimits {\dfrac {\map f s} s} {s \mathop = 1} {s \mathop = a}$ \quad:\quad\intlimits {\dfrac {\map f s} s} {s \mathop = 1} {s \mathop = a} \qquadLimits of Integration $\invlaptrans {F}$ \quad:\quad\invlaptrans {F} \qquadInverse Laplace Transform $\JJ$ \quad:\quad\JJ \qquadthat is: \mathcal J $\KK$ \quad:\quad\KK \qquadthat is: \mathcal K $\laptrans {f}$ \quad:\quad\laptrans {f} \qquadLaplace Transform $\lcm \set {x, y, z}$ \quad:\quad\lcm \set {x, y, z} \qquadLowest Common Multiple $\leadstoandfrom$ \quad:\quad\leadstoandfrom $\leftset {a, b, c}$ \quad:\quad\leftset {a, b, c} \qquadConventional set notation (left only) $\leftparen {a + b + c}$ \quad:\quad\leftparen {a + b + c} \qquadParenthesis (left only) $\len {AB}$ \quad:\quad\len {AB} \qquadLength Function: various $\LL$ \quad:\quad\LL \qquadthat is: \mathcal L $\Ln$ \quad:\quad\Ln \qquadPrincipal Branch of Complex Natural Logarithm $\Log$ \quad:\quad\Log \qquadPrincipal Branch of Complex Natural Logarithm $\map {f} {x}$ \quad:\quad\map {f} {x} \qquadMapping or Function $\MM$ \quad:\quad\MM \qquadthat is: \mathcal M $\Mult$ \quad:\quad\Mult \qquadMultiplication as a Primitive Recursive Function‎ $\NegativeBinomial {n} {p}$ \quad:\quad\NegativeBinomial {n} {p} \qquadNegative Binomial Distribution $\Nil {R}$ \quad:\quad\Nil {R} \qquadNilradical of Ring $\nint {11.98}$ \quad:\quad\nint {11.98} \qquadNearest Integer Function $\NN$ \quad:\quad\NN \qquadthat is: \mathcal N $\norm {z^2}$ \quad:\quad\norm {z^2} \qquadNorm $\O$ \quad:\quad\O \qquadEmpty Set $\OO$ \quad:\quad\OO \qquadthat is: \mathcal O $\On$ \quad:\quad\On \qquadOrdinal Class $\openint {a} {b}$ \quad:\quad\openint {a} {b} \qquadOpen Interval $\Orb S$ \quad:\quad\Orb S \qquadOrbit $\Ord {S}$ \quad:\quad\Ord {S} \qquad$$S$ is an Ordinal $\order {G}$ $\quad:\quad$\order {G} $\qquad$Order of Structure, and so on $\Out {G}$ $\quad:\quad$\Out {G} $\qquad$Group of Outer Automorphisms $\paren {a + b + c}$ $\quad:\quad$\paren {a + b + c} $\qquad$Parenthesis $\Poisson {\lambda}$ $\quad:\quad$\Poisson {\lambda} $\qquad$Poisson Distribution $\polar {r, \theta}$ $\quad:\quad$\polar {r, \theta} $\qquad$Polar Form of Complex Number $\pounds$ $\quad:\quad$\pounds $\qquad$Pound Sterling $\powerset {S}$ $\quad:\quad$\powerset {S} $\qquad$Power Set $\PP$ $\quad:\quad$\PP $\qquad$that is: \mathcal P $\Preimg {f}$ $\quad:\quad$\Preimg {f} $\qquad$Preimage of Mapping $\map {\pr_j} {F}$ $\quad:\quad$\map {\pr_j} {F} $\qquad$Projection $\PV$ $\quad:\quad$\PV $\qquad$Cauchy Principal Value $\QQ$ $\quad:\quad$\QQ $\qquad$that is: \mathcal Q $\radians$ $\quad:\quad$\radians $\qquad$Radian $\Rad$ $\quad:\quad$\Rad $\qquad$Radical of Ideal of Ring $\ds \int \map f x \rd x$ $\quad:\quad$\ds \int \map f x \rd x $\qquad$Roman $\d$ for use in Integrals $\rD$ $\quad:\quad$\rD $\qquad$Differential Operator $y \rdelta x$ $\quad:\quad$y \rdelta x \qquad\delta operator for use in sums $\map \Re z$ \quad:\quad\map \Re z \qquadReal Part $\relcomp {S} {A}$ \quad:\quad\relcomp {S} {A} \qquadRelative Complement $\rem$ \quad:\quad\rem \qquadRemainder $\Res {f} {z_0}$ \quad:\quad\Res {f} {z_0} \qquadResidue $\rightparen {a + b + c}$ \quad:\quad\rightparen {a + b + c} \qquadParenthesis (right only) $\rightset {a, b, c}$ \quad:\quad\rightset {a, b, c} \qquadConventional set notation (right only) $\Rng {f}$ \quad:\quad\Rng {f} \qquadRange of Mapping $\RR$ \quad:\quad\RR \qquadthat is: \mathcal R $\sech$ \quad:\quad\sech \qquadHyperbolic Secant $\Sech$ \quad:\quad\Sech \qquadHyperbolic Secant $\sequence {a_n}$ \quad:\quad\sequence {a_n} \qquadSequence $\set {a, b, c}$ \quad:\quad\set {a, b, c} \qquadConventional set notation $\ShiftedGeometric {p}$ \quad:\quad\ShiftedGeometric {p} \qquadShifted Geometric Distribution $\Si$ \quad:\quad\Si \qquadSine Integral Function $\Sinh$ \quad:\quad\Sinh \qquadHyperbolic Sine $\size {x}$ \quad:\quad\size {x} \qquadAbsolute Value, and so on $\SL {n, \R}$ \quad:\quad\SL {n, \R} \qquadSpecial Linear Group $\Spec {R}$ \quad:\quad\Spec {R} \qquadSpectrum of Ring $\sqbrk {a}$ \quad:\quad\sqbrk {a}  $\SS$ \quad:\quad\SS \qquadthat is: \mathcal S $\Stab x$ \quad:\quad\Stab x \qquadStabilizer $\stratgame {N} {A_i} {\succsim_i}$ \quad:\quad\stratgame {N} {A_i} {\succsim_i} \qquadStrategic Game $\struct {G, \circ}$ \quad:\quad\struct {G, \circ} \qquadAlgebraic Structure $\StudentT {k}$ \quad:\quad\StudentT {k} \qquadStudent's t-Distribution $\SU {n}$ \quad:\quad\SU {n} \qquadUnimodular Unitary Group $\Succ$ \quad:\quad\Succ \qquadSuccessor Function $\Syl {p} {N}$ \quad:\quad\Syl {p} {N} \qquadSylow p-Subgroup $\T$ \quad:\quad\T \qquadTrue $\Tanh$ \quad:\quad\Tanh \qquadHyperbolic Tangent $\tr$ \quad:\quad\tr \qquadTrace $\TT$ \quad:\quad\TT \qquadthat is: \mathcal T $\tuple {a, b, c}$ \quad:\quad\tuple {a, b, c} \qquadOrdered Tuple $\UU$ \quad:\quad\UU \qquadthat is: \mathcal U $\valueat {\dfrac {\delta y} {\delta x} } {x \mathop = \xi}$ \quad:\quad\valueat {\dfrac {\delta y} {\delta x} } {x \mathop = \xi}  $\var {X}$ \quad:\quad\var {X} \qquadVariance $\vers \theta$ \quad:\quad\vers \theta \qquadVersed Sine $\VV$ \quad:\quad\VV \qquadthat is: \mathcal V $\WW$ \quad:\quad\WW \qquadthat is: \mathcal W $\XX$ \quad:\quad\XX \qquadthat is: \mathcal X $\YY$ \quad:\quad\YY \qquadthat is: \mathcal Y $\ZZ$ \quad:\quad\ZZ \qquadthat is: \mathcal Z ## Aligned Equations To include aligned equations, a set of templates has been written: begin-eqn, eqn and end-eqn. For more explanation, see Template:eqn. #### Known issues ## Specific Topics ### Commutative diagrams See Help:Commutative Diagrams ## External references and manuals It may not be exactly the same version of \LaTeX, but I always find this page helpful as a first, quick overview: This is also a good reference page, pertaining to MediaWiki \LaTeX: but be aware that not all commands are supported. This is a link of all the currently supported commands available: ## House Style This page lists the various house style conventions that have been adopted on \mathsf{Pr} \infty \mathsf{fWiki}. Active contributors are expected to gradually master these, but when first starting to contribute, seasoned editors will come in and tidy the pages you write or create. ## Linking Due to the desired standard of rigor on \mathsf{Pr} \infty \mathsf{fWiki}, there are a lot of concepts on any given (proof) page that have their own, dedicated Proof or Definition page on \mathsf{Pr} \infty \mathsf{fWiki}. To ensure ease of reference and maximal clarity and consistency, the following rules for internal reference are to be adhered to. For information on creating links, see this section. ### References to Theorems and Axioms Whenever a theorem is invoked or referred to, be it in a proof or, for example, a clarifying comment, it should be referenced by its full title. Also, for ease of editing, there is no need to change the case of theorem names; the \mathsf{Pr} \infty \mathsf{fWiki} page title will suffice. Thus, for example, a valid reference to the result Union Distributes over Intersection is simply: "By Union Distributes over Intersection, A \cup \left({B \cap C}\right) = \left({A \cup B}\right) \cap \left({A \cup C}\right)." This is achieved by simply putting the title of the page you want to reference between double square brackets, [[ and ]]. The same convention applies to axioms, except that the namespace identifier Axiom: should be removed. The correct way to reference the page Axiom:Axiom of Choice thus is: Axiom of Choice which is produced by: [[Axiom:Axiom of Choice|Axiom of Choice]] ### References to Definitions Whenever a concept (or part of it) is invoked on a page, it should be presented as a link to the definition of that concept. It is preferred that every use of the word for that concept is presented as such a link. These references are made in a non-intrusive way. Thus, we write: Let R be a ring. and not: Let R be a Ring (Abstract Algebra). ## Mathematical Symbols ### Symbol Set The only symbols that are accepted in \mathsf{Pr} \infty \mathsf{fWiki} \LaTeX source code are the standard alphanumeric and punctuation characters that can be found on a standard English-language keyboard. Letters with diacritical marks such as "á" should not be used. If non-English characters are needed within \LaTeX source code (the obvious instance being Greek), then the full \LaTeX definition is to be used. For example: \alpha is to be rendered by the code \alpha. The same applies to specialised mathematical symbols. While it is appreciated that some contributors may have favourite techniques to allow them to place various mathematical symbols directly into the wiki text, such techniques are not portable and cause rendering issues in some browsers. The only exceptions to this rule are: when reporting the name of a mathematician, for example: Wacław Franciszek Sierpiński when explaining the linguistic derivation of a term in, for example, a Language Note section. It is standard, for example, to use Greek characters directly here, rather than their \LaTeX codes. ### Inline Equations Inline equations (that is, those that appear as part of a text sentence) merely need the dollar delimiters. For example: The semilinear wave equation \partial_t U = A U + \map B U is Hamiltonian. is produced by the input: The semilinear wave equation \partial_t U = A U + \map B U is Hamiltonian. ### Displayed Equations Displayed equations should be indented using a single colon, for example, a displayed equation should look like: \ds \map H U = \int_0^{2 \pi} \frac {\paren {\partial_x u}^2} 2 + \frac {v^2} 2 - \map F u \rd x which you can enter as: :\ds \map H U = \int_0^{2 \pi} \frac {\paren {\partial_x u}^2} 2 + \frac {v^2} 2 - \map F u \rd x Note the \mathsf{Pr} \infty \mathsf{fWiki} custom command \ds which is a convenience abbreviation for the full command \displaystyle. Using a format that places the equation on the center of the page:E = m c^2$is discouraged, because with our "short sentence" house style, this breaks up the reading flow. ### Big Operators The \displaystyle command should be used at the front of expressions using the 'big operators' such as \sum and \prod, whether the equation is displayed or inline. This includes (but may not be exclusive to) the commands \frac, \binom, \lim, \bigcup, \bigcap, \int, \sum and \prod. For example:$\sum_{i \mathop = 1}^n\prod_{i \mathop = 1}^n\frac {-b \pm \sqrt {b^2 - 4ac} } {2 a}\lim_{n \to \infty} \frac 1 n$all look better as:$\ds \sum_{i \mathop = 1}^n\ds \prod_{i \mathop = 1}^n\ds \frac {-b \pm \sqrt {b^2 - 4ac} } {2 a}\ds \lim_{n \mathop \to \infty} \frac 1 n$and are produced by, respectively:  \ds \sum_{i \mathop = 1}^n  \ds \prod_{i \mathop = 1}^n  \ds \frac {-b \pm \sqrt {b^2 - 4ac} } {2 a}  \ds \lim_{n \mathop \to \infty} \frac 1 n Furthermore, to improve aesthetic appeal certain characters, such as$=$and$\in$, when used in subscripts of big operators, must be endowed with the \mathop command to enforce appropriate spacing. As a contrast, compare:$\ds \sum_{i = 1}^n \quad \sum_{i \mathop = 1}^n\ds \bigcap_{n \in \N} \quad \bigcap_{n \mathop \in \N}$The \mathop command is to be used in the following manner (the code produces$\ds \sum_{i \mathop = 1}^n$): \ds \sum_{i \mathop = 1}^n #### Abbreviated Symbols Certain symbols have abbreviated forms for their big versions: \dfrac can be used instead of \ds \frac \dbinom can be used instead of \ds \binom where d is for display. Of course, if other big operators are used in the same equation, the \ds command is needed anyway. However, it does no harm to include \dfrac and \dbinom inside a line defined as \ds, and may make refactoring easier. So feel free to develop the habit of using \dfrac and \dbinom throughout. ### The d of Calculus When writing calculus operators, use a non-italic form for the$\d$. To achieve this, write it as \d or \rd (the latter includes a half-space before it, for use in integrals). So you would have:$\dfrac {\d y} {\d x}$which would be produced by: \dfrac {\d y} {\d x} rather than:$\dfrac {d y} {d x}$which would be produced by: \dfrac {d y} {d x} ### Fonts We have several fonts available, many of which have particular conventional uses in mathematics. Examples are: Calligraphy: \mathcal, which produces$\mathcal{ABCDE} ..., \mathcal {1234567890}$(uppercase only, but also digits) Blackboard: \mathbb or (preferably) \Bbb, which produces$\Bbb{ABCDE} ...$(uppercase only, no digits) Script: \mathscr, which produces$\mathscr{ABCDE} ...$(uppercase only, no digits) Sans serif: \mathsf, which produces$\mathsf{ABCDE} ... \mathsf{abcde} ..., \mathsf {1234567890}$Fraktur: \mathfrak, which produces$\mathfrak{ABCDE} ... \mathfrak{abcde} ..., \mathfrak {1234567890}$Fixed Width: \mathtt, which produces$\mathtt{ABCDE} ... \mathtt{abcde} ..., \mathtt {1234567890}$The use of Fraktur and Script are discouraged, as they are not so easy on the eye and can be difficult to decipher on certain browsers. Also note that: \T and \F implement$\T$and$\F$respectively, designed to be used for True and False respectively$\N, \Z, \Q, \R, \C$have their own$\LaTeX$codes: \N, \Z, \Q, \R, \C We also have \GF implemented for$\GF$(note that \F cannot be used here, as it is already used for$\F$) all \mathcal uppercase letters have custom$\mathsf{Pr} \infty \mathsf{fWiki}$shortcut implementations, as \AA, \BB, and so on ### Use of Logical Symbols in Mathematical Exposition This applies mainly to the use of the conjunction symbol$\land$, that is$\text {and}$, and the disjunction symbol$\lor$, that is$\text {or}$. It is convenient sometimes to write a statement in the style:$\forall y \in R: \lambda_y = y * I_{_R} \land \rho_y = I_{_R} * y$However, it may not be immediately obvious to the reader exactly what$\land$means. In the various fields, for example abstract algebra and set theory,$\land$and$\lor$have a number of different meanings, for example meet and join. If the reader has been studying such material, it can be irritating to have to change mental gears and suddenly have to adjust to the fact that$\land$means$\text {and}$. Hence it is strongly recommended that the above statement be written:$\forall y \in R: \lambda_y = y * I_{_R} \text { and } \rho_y = I_{_R} * y$reserving$\land$and$\lor$for their use in the field of logic. ### Punctuation niceties A sentence broken by a displayed equation should be ended with a colon:$\dfrac {\text{display}} {\text{equation}}$for a better presentation. On the other hand, the displayed equation itself should not be ended with a full stop or comma. That is, one should write:$\displaystyle \bigcap_{S \mathop \in \Bbb S} \Bbb U \setminus S = \Bbb U \setminus \bigcup_{S \mathop \in \Bbb S} S$and not:$\displaystyle \bigcap_{S \mathop \in \Bbb S} \Bbb U \setminus S = \Bbb U \setminus \bigcup_{S \mathop \in \Bbb S} S$. In particular, including the full stop inside the$\LaTeX$it terminates is definitely incorrect, for readily apparent reasons. So please do not do this:$\displaystyle \bigcap_{S \mathop \in \Bbb S} \Bbb U \setminus S = \Bbb U \setminus \bigcup_{S \mathop \in \Bbb S} S.$This is a style tip borrowed from Ian Stewart, from his Galois Theory, 3rd ed. of$2004$. Use of commas is discouraged. This sort of structure is considered incorrect: Let$x$be as follows,$x \in S$as commas are reserved in mathematics for separation of elements of a list. ### Q.E.D. To end a proof, the template {{qed}} should be used, which looks like:$\blacksquare$or if you wish to break your page up into subproofs, end those subproofs with {{qed|lemma}}, which looks like:$\Box$In a dash for consistent notation, it is understood that these templates should immediately succeed the last line of the proof, that is: Hence the result. {{qed}}  and not: Hence the result. {{qed}}  Tempting though it is to write "Q.E.D." at the bottom, this is so uncool as to be positively naff. ##$\LaTeX$Code Style There are a few general source code conventions which make your code easier to read and maintain: • Each variable, and each command beginning with a backslash should be preceded by a space, except (for some unexplained result of evolution) when enclosing things in brackets. See some of the above instance for a typical example. • When enclosing brackets around an object, always use the \paren { ... } command, for example \paren {a + b} rather than (a + b). • There should be no need to use the commands \big, \Big, \Bigg etc. for specifying the sizes of parentheses. Using the \paren {...} technique (as above) will almost always automatically size the brackets aesthetically. • Punctuation should appear (if it is really necessary) outside the$\LaTeX$environment, for example: Hence$\map f {\sin x}$. (produced by: $\map f {\sin x}$.) as opposed to: Hence$\map f {\sin x}.$(produced by: $\map f {\sin x}.$) • Single-character parameters to standard$\LaTeX$constructs need not be put in curly braces. That is, \dfrac 1 2 is preferred to \dfrac {1} {2}. They both produce:$\dfrac 1 2$It makes the source code significantly easier to read. Having said that, please do not ignore the rule about spacing. The same effect can also be achieved by \dfrac12 (see, it still looks like$\dfrac12$) but that is significantly harder to interpret visually. ### Aligned Material If an equation includes multiple equalities or inequalities, it is best to place each equality on a new line. For example:$\dfrac \d {\d t} \map H U = \d \map H U \cdot \dot U = \map \Omega {\map {X_H} U, \dot U} = \map \Omega {\map {X_H} U, \map {X_H} U} = 0would look better as an aligned equation. This is done using the following commands: {{begin-eqn}} {{eqn | l = | r = }} ... {{eqn | l = | r = }} {{end-eqn}}  Here, the section following | l = is a\LaTeX$environment, and should contain anything you want to appear to the left of the equals sign. The section following | r = is the same, but will appear to the right of the equals sign. | ll = and | rr = are similar, but produce material in columns further to the left and further to the right respectively. In particular, the ll column is often used for an "implies" or "leads to" sign where the l and r are used for either side of a string of equations. All these$\LaTeXenvironments are already in \displaystyle mode, so there is no need to include that command in your equation. When entering such an {{eqn}} environment, it should globally look like this: {{eqn | l = 1 + 1 | r = 2 }}  That is, it adheres to the following principles: • Every empty column should in general be omitted, except perhaps for | c = sections, which can be left as placeholders for possible future addition of comments • Non-empty columns are entered on separate lines, with the | and = all aligned. These conventions serve to optimize readability. More options and abilities of the {{eqn}} can be found on its page, {{eqn}}. The section following | c = is not a\LaTeX$environment, and can be used to add any comments about the equation at this point. So the example we gave above would be typeset as: {{begin-eqn}} {{eqn | l = \frac \d {\d t} \map H U | r = \d \map H U \cdot \dot U | c = [[Chain Rule for Derivatives]] }} {{eqn | r = \map \Omega {\map {X_H} U, \dot U} | c = Definition of$X_H$}} {{eqn | r = \map \Omega {\map {X_H} U, \map {X_H} U} | c = [[Hamilton's Equations]] }} {{eqn | r = 0 | c = from above:$\Omega$is [[Definition:Skew-Symmetry|skew-symmetric]] }} {{end-eqn}}  The finished result will look like:  $\ds \frac \d {\d t} \map H U$ $=$ $\ds \d \map H U \cdot \dot U$ Chain Rule for Derivatives $\ds$ $=$ $\ds \map \Omega {\map {X_H} U, \dot U}$ Definition of$X_H$$\ds$ $=$ $\ds \map \Omega {\map {X_H} U, \map {X_H} U}$ Hamilton's Equations $\ds$ $=$ $\ds 0$ from above:$\Omega$is skew-symmetric The operator that is displayed in this template can be changed using | o =  to show inequalities, etc. Note the following: • Do not include two consecutive open or close curly braces: {{ or }} anywhere in your {{eqn}} templates. It will break the interpreter. Put spaces in: { { or } } and it will be okay. • Do not include the vertical line | (a.k.a. "pipe") in$\LaTeX$expressions as this also breaks the interpreter. Use \vert (or \lvert and \rvert) instead. • In particular, \| (used to produce$\|$) has the same problem. Use \Vert etc. instead. These caveats apply only within the {{eqn}} environment. Elsewhere on the page such constructs should be fine. To accommodate for the inevitable copy-paste efforts, and for consistency's sake, it is however desirable to always use \vert and \Vert, and to insert a space between adjacent curly braces within$\LaTeX$strings. ## Linguistic Style ### Language This is an English language website, and so all pages are to be presented in English.[1] Where there is a difference between spellings between US and rest-of-world English, the US version is generally used, with a few exceptions (the spelling of metre is under discussion). ### Linguistic Style During the presentation of a mathematical argument, a formal style is preferred. For example: Suppose that ... is preferred to: Let's suppose that ... and: Hence the result. is preferred to: ... and we're done. As an attempt is being made for$\mathsf{Pr} \infty \mathsf{fWiki}$to appeal to as wide an audience as possible worldwide, using colloquial language (except for example when illustrating logical concepts by means of everyday examples) is discouraged. ### "Let" and "Suppose" It is preferred that "Let" is used to introduce the existence of an entity in an argument, as follows: Let$S$be a set. Let$x, y \in S: x \ne y$.$\ldots$However, when introducing an entity whose existence is in question (for example, when constructing a Proof by Contradiction), the word "Suppose" is recommended: Suppose$T \subseteq S$such that$\card T > \card S$.$\ldots$### Abbreviations The difference between "e.g." (exempli gratia - for example) and "i.e." (id est - that is) is sadly falling into obscurity. It is all too common for "i.e." to be used when "for example" is meant, and vice versa. So as to remove all confusion, such abbreviations are discouraged. Also, beware the ubiquitous confusion between its and it's. The full version it is should be used instead of it's in any case, so it's should have no reason to appear. ### Sentence Length During the course of an argument to present a mathematical proof, follow these rules: • Each sentence should be short. • Each sentence should be on a separate line. Compare the presentations:$(1):S$, because of$R$(we know this from Tom's Theorem), because of$Q$(from above) which applies when$P$holds (see Fred's Theorem), but we know$P$holds because it's what we defined in the first place.$(2):$Let$P$hold. From Fred's Theorem, it follows that$Q$. From above,$R$. From Tom's Theorem,$S$. The following is an example of the style of mathematical exposition which we believe has no place in$\mathsf{Pr} \infty \mathsf{fWiki}$, and indeed, the entire universe: The ($\implies$) is shown just the same as above, while the other direction easily follows, since$\mathcal M$satisfying the condition that for every$\mathcal L$-formula$\phi \left({x, \bar v}\right)$and for every$\bar a$in$\mathcal M$, if there is an$n$in$\mathcal N$such that$\mathcal N \models \phi \paren {n, \bar a}$, then there is an$m$in$\mathcal M$such that$\mathcal N \models \phi \paren {m, \bar a}$, is closed under functions (by directly applying the condition to formulae of the form$\phi \paren {x, \bar y} = \paren {x = f \paren {\bar y} }$), and hence the universe of a substructure, which reduces it to the statement above. ### Filler Words Whether or not filler words are needed (it follows that, we have, hence etc.) is a stylistic decision. Fewer words are preferred, but clarity and completeness override every other consideration. The general approach is to try to use as terse a form as possible. Compare: We have that the ordinal subset of an ordinal is an initial segment of it, so it follows that: with: From Ordinal Subset of Ordinal is Initial Segment: The latter form is preferred. ### Empty Statements and Waffle It is tempting to fill a page up with statements that do not actually impart any information, but which make the author look and feel good. Such are to be avoided. Examples: The first part of the proof is easy. We mention for the interested reader ... See also the templates {{handwaving}} and {{explain}}. ## Capital Letters begin Sentences This is raised as a particular point, because it crops up over and over again. The sentence form in question is: Let (such-and-such) hold, where (so-and-so) means (thus and so). When (such-and-such) is a statement in mathematical symbols, placed on its own line (as per house style recommendations), the temptation is to present the above sentence as: Let:$\displaystyle S = \sum_{i \mathop \in \N} \frac 1 {2^i}$Where$\displaystyle \sum$denotes summation. Just because it starts a new line does not mean that "where" is to be written with a capital W. It is the continuation of the previous sentence, which just happens to have, as part of its main clause, a mathematical expression. It should be: Let:$\displaystyle S = \sum_{i \mathop \in \N} \frac 1 {2^i}$where$\displaystyle \sum$denotes summation. Breaking this linguistic rule can lead to confusion, especially when the "where" clause starts to get complicated: Let:$\displaystyle S = \sum_{j \mathop \in \N} \lim_{x \mathop \to \infty} \cos j x + i \sin j x$Where$\displaystyle \sum$denotes summation and$\lim$is the limit as$x$tends to infinity and:$\cos j x + i \sin j x = e^{ijx}$In the above, the reader, thinking that "where" starts the next sentence, and therefore a new thought, is left wondering: "Where this applies, and that means that, and this ... then what?" whereas in fact the only reason for the "where" clause is to amplify the sense of the expression above it. Similarly:  $\ds x \in A \cap \paren {B \cap C}$ $\leadstoandfrom$ $\ds x \in A \land \paren {x \in B \land x \in C}$ By definition of Set Intersection $\ds$ $\leadstoandfrom$ $\ds \paren {x \in A \land x \in B} \land x \in C$ Rule of Association: Conjunction $\ds$ $\leadstoandfrom$ $\ds x \in \paren {A \cap B} \cap C$ By definition of Set Intersection In the above, the "by definition" phrases in the comment column should not start with a capital letter, as they continue the "sentence" started on the left. Thus the above structure is better rendered as:  $\ds x \in A \cap \paren {B \cap C}$ $\leadstoandfrom$ $\ds x \in A \land \paren {x \in B \land x \in C}$ by definition of Set Intersection $\ds$ $\leadstoandfrom$ $\ds \paren {x \in A \land x \in B} \land x \in C$ Rule of Association: Conjunction $\ds$ $\leadstoandfrom$ $\ds x \in \paren {A \cap B} \cap C$ by definition of Set Intersection Better still, lose the redundant filler-word "by", and render the entire structure elegantly as:  $\ds x \in A \cap \paren {B \cap C}$ $\leadstoandfrom$ $\ds x \in A \land \left({x \in B \land x \in C}\right)$ Definition of Set Intersection $\ds$ $\leadstoandfrom$ $\ds \paren {x \in A \land x \in B} \land x \in C$ Rule of Association: Conjunction $\ds$ $\leadstoandfrom$ $\ds x \in \paren {A \cap B} \cap C$ Definition of Set Intersection Now, as there is no filler-word "by", the comment is no longer implicitly part of a sentence, and so the comment is a standalone label which now merits an uppercase presentation. Here, note that a further evolutionary step has been made: to replace the code Definition of [[Definition:Set Intersection|Set Intersection]] with the template construct {{Defof|Set Intersection}} for further streamlining of the source. ## Sources It is good to indicate where the information comes from. This is done in$\mathsf{Pr} \infty \mathsf{fWiki}$in the last of the page in a section called Sources. ## Adding sources If there are multiple sources, they are to be listed first in chronological order, then alphabetically on the name of the (first) author. As stated on Help:Page Editing, the sources should be using a bulleted list, ordered by date of publication of the edition cited, and after that alphabetically, sorted on the surname of the (first) author. For example (an excerpt of the Sources section of Definition:Set Union): == Sources == * {{BookReference|Naive Set Theory|1960|Paul R. Halmos|prev = Union of Singleton|next = Union with Empty Set}}:$\S 4$: Unions and Intersections * {{BookReference|Abstract Algebra|1964|W.E. Deskins|prev = Equality of Sets|next = Definition:Set Intersection}}:$\S 1.1$: Definition$1.2$* {{BookReference|Point Set Topology|1964|Steven A. Gaal|prev = Definition:Set Union/General Definition|next = Union is Commutative}}: Introduction to Set Theory:$1$. Elementary Operations on Sets * {{BookReference|Sets and Groups|1965|J.A. Green|prev = Empty Set Subset of All|next = Intersection Subset Union}}:$\S 1.4$* {{BookReference|Modern Algebra|1965|Seth Warner|prev = Associative and Anticommutative|next = Definition:Set Intersection}}:$\S 3$ ## Types of sources There are several templates that can be used: ### Hardcopy Sources Template:BookReference This is used to reference a specific book which will have been documented in the Books page. The idea of this is that if you have sourced the information for a page directly from a book, then it should be possible to provide the details of that book. Example: which can be found on the page Characteristic times Ring Element is Ring Zero. Template:Citation This is used to reference a specific article in a journal. This is still under development, as the individual Journal entries still need to be worked on. Examples of their use can be found on various Mathematicians pages, for example: • 1908: Mathematical Logic as Based on the Theory of Types (Amer. J. Math. Vol. 30: pp. 222 – 262) which appears on the page for Bertrand Russell. The style of this is still evolving. ### Online Sources There are templates for the following online sources. Each one has been crafted so as to produce a reference in the style requested by the online source in question. Template:MathWorld This provides a direct link to a page on the http://mathworld.wolfram.com/ website. Example: which can be found in the page Area of Sector. Template:Planetmath This provides a direct link to a page on the http://planetmath.org/ website. Example: This article incorporates material from Urysohn's Lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. which can be found in the page Urysohn's Lemma. An alternative format for the same link is: This article incorporates material from Urysohn's Lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. Template:MacTutor Biography This provides a direct link to a page on the http://www-history.mcs.st-andrews.ac.uk/history/index.html website. Example: which can be found in the page for Hanna Neumann. Note that the link presentation is taken from the page the template is invoked from. Template:Khanacademy This provides a direct link to the Khan Academy. Example: which can be found in the page Limit of Sine of X over X/Geometric Proof. WARNING: They have since changed their style of citation linking, so until this is fixed in$\mathsf{Pr} \infty \mathsf{fWiki}\$, we will not be able to link to them using the current template.

This is work in progress.

Template:Metamath
This provides a direct link to Metamath.

Example:

which can be found in the page Principle of Transfinite Recursion.

Template:Mizar
This provides a direct link to Mizar.

Example:

which can be found in the page Characterization of Boundary by Basis.

Template:OEIS
This provides a direct link to the On-Line Encyclopedia of Integer Sequences.

Example:

which can be found in the page Square Root of 2 is Irrational.

Template:TORI
This provides a direct link to the TORI source site.

Example:

WARNING: This link is broken. The TORI site seems to no longer exist.

which can be found in the page Definition:Tetration.

• UPDATE: It is believed that this site may have closed down, as its links now 404.

Template:SpringerOnline
This provides a direct link to a page on the Springer Online Encyclopedia of Mathematics.

Example:

which can be found in the page Definition:Ring (Abstract Algebra).

Template:Stackexchange
This provides a direct link to a page on Mathematics Stack Exchange.

Example:

which can be found in the page Derivative of Sine Function/Proof 5.

### Acceptability of Online Sources

NOTE: The above are currently the ONLY web resources which are to be used as general citation sources.

Others may be added to the above as and when they come to our attention as being particularly useful.

So feel free to challenge this assertion if you find something which appears to be a particularly rich and productive resource.

Scholarly papers which are available online may usually also be cited.

What are not generally acceptable include:

Lecture notes for university courses available online (because they do not stay online forever, and this causes dead links)
Links to pages in homework help forums
Discussion pages in any web forum
Wikipedia -- not because we don't like them, but because as they are self-proclaimed tertiary source, there is no need to do so -- we would rather go to the actual source works. See also Wikipedia:Citing Wikipedia.

## Splitting sources

In some cases it is necessary to split a referenced theorem, proof or definition into multiple pages, because for example:

a theorem contains multiple statements
a proof contains in fact multiple proofs
a definition defines multiple concepts at once.

If so, the source has to be referenced at every page, and its process flow is updated according to the order in which the elements appear in the source.

## References

1. Suggestions have been made as to how we may go about the exercise of internationalization, but progress in that direction is slow due to its perceived low priority.