Editing Sigma (section)

==Uses==

===Language and linguistics===
* In both [[Ancient Greek|Ancient]] and [[Modern Greek]], the sigma represents the [[voiceless alveolar fricative]] {{IPA|el|s|}}. In Modern Greek, this sound is [[voice (phonetics)|voiced]] to the [[voiced alveolar fricative]] {{IPA|el|z|}} when occurring before {{IPA|el|m|}}, {{IPA|el|n|}}, {{IPA|el|v|}}, {{IPA|el|ð|}}, or {{IPA|el|ɣ|}}.
* The uppercase form of sigma ('''Σ''') was re-borrowed into the [[Latin alphabet]]—more precisely, the [[Africa Alphabet|International African Alphabet]]—to serve as the uppercase of modern [[esh (letter)|''esh'']] (lowercase: '''ʃ''').
* In [[phonology]], '''σ''' is used to represent [[syllable]]s.
* In [[linguistics]], '''Σ''' represents the set of symbols that form an alphabet (see also [[Computer Science|computer science]]).
* In [[historical linguistics]], '''Σ''' is used to represent a [[Common Brittonic]] consonant with a sound between {{IPAblink|s}} and {{IPAblink|h}}; perhaps an [[Aspirated consonant|aspirated]] [[Voiceless postalveolar fricative|[ʃʰ<nowiki>]</nowiki>]].<ref>{{Cite web|url=http://dlib.bc.edu/islandora/object/bc-ir:102094|title=Celtic initial consonant mutations - nghath and bhfuil?|first=Kevin M.|last=Conroy|date=21 February 2008|via=dlib.bc.edu}}</ref>

=== Science and mathematics ===

==== Mathematics ====
* In general mathematics, lowercase '''σ''' is commonly used to represent unknown angles, additionally serving as a shorthand for "[[countable set|countably]]", whereas '''Σ''' is regularly used as the [[operator (mathematics)|operator]] for [[summation]], e.g.:<ref name=":0">{{Cite web |last=Weisstein |first=Eric W. |title=Sigma |url=https://mathworld.wolfram.com/Sigma.html |access-date=2025-01-25 |website=mathworld.wolfram.com |language=en}}</ref>
<math display="block">\sum_{k=0}^5k= 0 + 1 + 2 + 3 + 4 + 5 = 15</math>
* In [[mathematical logic]], <math>\Sigma^0_n</math> is used to denote the set of formulae with bounded quantifiers beginning with existential quantifiers, alternating <math>n-1</math> times between existential and universal quantifiers. This notation reflects an indirect analogy between the relationship of summation and products on one hand, and existential and universal quantifiers on the other. See the article on the [[arithmetic hierarchy]].
* In [[statistics]], '''σ''' represents the [[standard deviation]] of population or [[probability distribution]] (where [[Mu (letter)|mu]] or '''μ''' is used for the mean).<ref name=":0" />
* In [[topology]], '''σ-compact''' [[topological space]] is one that can be written as a countable [[union (set theory)|union]] of [[compact space|compact subsets]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Sigma-Compact Topological Space |url=https://mathworld.wolfram.com/Sigma-CompactTopologicalSpace.html |access-date=2025-01-25 |website=mathworld.wolfram.com |language=en}}</ref>
* In [[mathematical analysis]] and in [[probability theory]], there is a type of [[Field of sets|algebra of sets]] known as '''[[sigma-algebra|σ-algebra]]''' (aka '''σ-field'''). Sigma algebra also includes terms such as:
** '''σ(''A'')''', denoting the [[Σ-algebra#σ-algebras generated by families of sets|generated sigma-algebra]] of a set ''A''
** '''[[Σ-finite measure]]''' (see [[Measure (mathematics)|measure theory]])
* In [[number theory]], '''σ''' is included in various [[divisor function]]s, especially the '''sigma function''' or sum-of-divisors function.
* In [[applied mathematics]], '''σ(''T'')''' denotes the spectrum of a [[linear map]] ''T''.
* In [[complex analysis]], '''σ''' is used in the [[Weierstrass functions#Weierstrass sigma-function|Weierstrass sigma-function]].<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Weierstrass Sigma Function |url=https://mathworld.wolfram.com/WeierstrassSigmaFunction.html |access-date=2025-01-25 |website=mathworld.wolfram.com |language=en}}</ref>
* In [[probability theory]] and [[statistics]], '''Σ''' denotes the [[covariance matrix]] of a set of [[random variable]]s, sometimes in the form <math>\;|\!\!\!\Sigma</math> to distinguish it from the summation operator.
* [[Spectral theory|Theoretical spectral analysis]] uses '''σ''' as standard deviation opposed to lowercase [[Mu (letter)|mu]] as the absolute mean value.

====Biology, physiology, and medicine====
* In [[biology]], the [[sigma receptor]] ('''σ–receptors''') is a type of [[cell surface receptor]].
* In [[biochemistry]], the '''[[Sigma factor|σ factor]]''' (or specificity factor) is a protein found in [[RNA polymerase]].
* In [[Bone|bone physiology]], the [[bone remodeling period]]—i.e., the [[Sigma (bone remodeling)|life span]] of a basic multicellular unit—has historically been referred to as the '''sigma period'''.
* In early 20th-century [[physiology]] literature, '''σ''' had been used to represent [[millisecond]]s.<ref name="Hill1935">{{Cite journal | last1 = Hill | first1 = A. V. | title = Units and Symbols | doi = 10.1038/136222a0 | journal = Nature | volume = 136 | issue = 3432 | pages = 222 | year = 1935 | bibcode = 1935Natur.136..222H | s2cid = 4087300 | doi-access = free }}</ref>

====Business, finance, and economics====
* In [[finance]], '''σ''' is the symbol used to represent [[Volatility (finance)|volatility]] of [[stock]]s, usually measured by the [[standard deviation]] of [[logarithmic return]]s.
* In [[accounting]], '''Σ''' indicates the [[Balance (accounting)|balance]] of invoice classes and the overall amount of debts and demands.
* In [[macroeconomics]], '''σ''' is used in equations to represent the [[elasticity of substitution]] between two inputs.
* In the [[machine industry]], [[Six Sigma]] ('''6σ''') is a quality model based on the [[standard deviation]].

====Chemistry====
* [[Sigma bond]]s ('''σ bonds''') are the strongest type of [[covalent]] [[chemical bond]].
* In [[organic chemistry]], '''σ''' symbolizes the sigma constant of [[Hammett equation]].<ref>{{Cite journal |last=Chemistry (IUPAC) |first=The International Union of Pure and Applied |title=IUPAC - σ-constant (C05438) |url=https://goldbook.iupac.org/terms/view/C05438 |access-date=2025-01-25 |website=goldbook.iupac.org |doi=10.1351/goldbook.C05438 |language=en}}</ref>
* In [[alchemy]], '''Σ''' was sometimes used to represent [[sugar]].

====Engineering and computer science====
* In [[Computer Science|computer science]], '''Σ''' represents the set of symbols that form an alphabet (see also [[linguistics]])
* [[Relational algebra]] uses the values <math>\sigma_{a \theta b}( R )</math> and <math>\sigma_{a \theta v}( R )</math> to denote [[Selection (relational algebra)|selections]], which are a type of [[unary operation]].
* In [[machine learning]], '''σ''' is used in the formula that derives the [[Sigmoid function]].
* In [[radar jamming]] or [[electronic warfare]], [[radar cross-section]]s (RCS) are commonly represented as '''σ''' when measuring the size of a target's image on radar.
* In [[signal processing]], '''σ''' denotes the [[damping ratio]] of a system parameter.
* In [[theoretical computer science]], '''Σ''' serves as the [[busy beaver]] function.
* In [[civil engineering]], '''σ''' refers to the normal stress applied on a material or structure.

====Physics====
* In [[nuclear physics|nuclear]] and [[particle physics]], '''σ''' is used to denote [[cross section (physics)|cross sections]] in general (see also [[Radar cross-section|RCS]]), while '''Σ''' represents macroscopic [[Cross section (physics)|cross section]]s [1/length].
* The symbol is to denote the [[Stefan–Boltzmann constant]].<ref>{{Cite web |title=CODATA Value: Stefan-Boltzmann constant |url=https://physics.nist.gov/cgi-bin/cuu/Value?sigma |access-date=2025-01-25 |website=physics.nist.gov}}</ref>
* In relation to fundamental properties of material, '''σ''' is often used to signify [[electrical conductivity]].
* In [[electrostatics]], '''σ''' represents [[surface charge]] density.
* In [[continuum mechanics]], '''σ''' is used to signify normal [[Stress (physics)|stress]].<ref>{{Citation |last=Elert |first=Glenn |title=Special Symbols |date=2023 |work=The Physics Hypertextbook |quote= σ normal stress|url=https://physics.info/symbols/ |access-date=2025-02-01 |publisher=hypertextbook |language=en}}</ref>
* In [[condensed matter physics]], '''Σ''' denotes [[self-energy]].
* The symbol can be used to signify [[surface tension]] (alternatively, ''[[Gamma|γ]]'' or ''[[T]]'' are also used instead).
* In [[quantum mechanics]], '''σ''' is used to indicate [[Pauli matrices]].
* In [[astronomy]], '''σ''' represents [[velocity dispersion]].
* In astronomy, the prefix '''Σ''' is used to designate double stars of the [[Catalogus Novus Stellarum Duplicium]] by [[Friedrich Georg Wilhelm von Struve]].
* In particle physics, '''Σ''' represents a class of [[baryon]]s.

=== Organizations ===
* During the 1930s, an uppercase '''Σ''' was in use as the symbol of the [[Brazilian Integralism|Ação Integralista Brasileira]], a fascist political party in [[Brazil]].
* [[Sigma Corporation]] uses the name of the letter but not the letter itself, but in many Internet forums, photographers refer to the company or its lenses using the letter.
* [[Sigma Aldrich]] incorporate both the name and the character in their logo.