Sigma

Sigma ( ; uppercase Σ, lowercase σ, lowercase in word-final position ς; ) is the eighteenth letter of the Greek alphabet. When used at the end of a letter-case word (one that does not use all caps), the final form (ς) is used. In ' (Odysseus), for example, the two lowercase sigmas (σ) in the center of the name are distinct from the word-final sigma (ς) at the end.

In the system of Greek numerals, sigma has a value of 200. In general mathematics, uppercase Σ is used as an operator for summation. The Latin letter S derives from sigma while the Cyrillic letter Es (С) derives from a lunate form of this letter.

Sigma's original name may have been san, but due to the complicated early history of the Greek epichoric alphabets, san came to be identified as a separate letter in the Greek alphabet, represented as Ϻ. Herodotus reports that "san" was the name given by the Dorians to the same letter called "sigma" by the Ionians.

According to one hypothesis, the name "sigma" may continue that of Phoenician samekh (𐤎), the letter continued through Greek xi, represented as Ξ. Alternatively, the name may have been a Greek innovation that simply meant 'hissing', from the root of (, from Proto-Greek *sig-jō 'I hiss').

Today, it is known as lunate sigma (uppercase Ϲ, lowercase ϲ), because of its crescent-like shape, and is still widely used in decorative typefaces in Greece, especially in religious and church contexts (i.e. the Christogram ΙϹ ΧϹ), as well as in some modern print editions of classical Greek texts.

A dotted lunate sigma (sigma periestigmenon, Ͼ) was used by Aristarchus of Samothrace (220–143 BC) as an editorial sign indicating that the line marked as such is at an incorrect position. Similarly, a reversed sigma (antisigma, Ͻ), may mark a line that is out of place. A dotted antisigma (antisigma periestigmenon, Ͽ) may indicate a line after which rearrangements should be made, or to variant readings of uncertain priority.

In Greek inscriptions from the late first century BC onwards, Ͻ was an abbreviation indicating that a man's father's name is the same as his own name, thus Dionysodoros son of Dionysodoros would be written Διονυσόδωρος Ͻ (Dionysodoros Dionysodorou).

• In both Ancient and Modern Greek, the sigma represents the voiceless alveolar fricative . In Modern Greek, this sound is voiced to the voiced alveolar fricative when occurring before , , , , or .
• The uppercase form of sigma (Σ) was re-borrowed into the Latin alphabet—more precisely, the International African Alphabet—to serve as the uppercase of modern esh (lowercase: ʃ).
• In phonology, σ is used to represent syllables.
• In linguistics, Σ represents the set of symbols that form an alphabet (see also computer science).
• In historical linguistics, Σ is used to represent a Common Brittonic consonant with a sound between and ; perhaps an aspirated [ʃʰ<nowiki>]</nowiki>.

• In general mathematics, lowercase σ is commonly used to represent unknown angles, additionally serving as a shorthand for "countably", whereas Σ is regularly used as the operator for summation, e.g.:

• In mathematical logic, is used to denote the set of formulae with bounded quantifiers beginning with existential quantifiers, alternating 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 or μ is used for the mean).
• In topology, σ-compact topological space is one that can be written as a countable union of compact subsets.
• In mathematical analysis and in probability theory, there is a type of algebra of sets known as σ-algebra (aka σ-field). Sigma algebra also includes terms such as:
• σ(A), denoting the generated sigma-algebra of a set A
• Σ-finite measure (see measure theory)
• In number theory, σ is included in various divisor functions, 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 sigma-function.
• In probability theory and statistics, Σ denotes the covariance matrix of a set of random variables, sometimes in the form to distinguish it from the summation operator.
• Theoretical spectral analysis uses σ as standard deviation opposed to lowercase mu as the absolute mean value.

• In biology, the sigma receptor (σ–receptors) is a type of cell surface receptor.
• In biochemistry, the σ factor (or specificity factor) is a protein found in RNA polymerase.
• In bone physiology, the bone remodeling period—i.e., the 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 milliseconds.

• In finance, σ is the symbol used to represent volatility of stocks, usually measured by the standard deviation of logarithmic returns.
• In accounting, Σ indicates the 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.

• Sigma bonds (σ bonds) are the strongest type of covalent chemical bond.
• In organic chemistry, σ symbolizes the sigma constant of Hammett equation.
• In alchemy, Σ was sometimes used to represent sugar.

• In computer science, Σ represents the set of symbols that form an alphabet (see also linguistics)
• Relational algebra uses the values and to denote 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-sections (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.

• In nuclear and particle physics, σ is used to denote cross sections in general (see also RCS), while Σ represents macroscopic cross sections [1/length].
• The symbol is to denote the Stefan–Boltzmann constant.
• 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.
• In condensed matter physics, Σ denotes self-energy.
• The symbol can be used to signify surface tension (alternatively, 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 baryons.

• During the 1930s, an uppercase Σ was in use as the symbol of the 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.

• Antisigma
• Greek letters used in mathematics, science, and engineering
• Sampi
• Sho (letter)
• Stigma (letter)
• Sibilant consonant
• Summation (Σ)
• Combining form "sigm-" (e.g. sigmodon, sigmurethra, etc.)
• Derivative "sigmoid" (e.g. sigmoid sinus, sigmoid colon, sigmoidoscopy, etc.)