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Omega (majuscule: Ω, minuscule: ω; Greek Ωμέγα) is the 24th and last letter of the Greek alphabet. In the Greek numeric system, it has a value of 800. The word literally means "great O" (ō mega, mega meaning 'great'), as opposed to Omicron, which means "little O" (o mikron, micron meaning "little"). This name is Byzantine; in Classical Greek, the letter was called ō (ὦ), whereas the Omicron was called ou (οὖ). The form of the lowercase letter derives from a double omicron, which came to be written open at the top.
Phonetically, the Ancient Greek Ω is a long open-mid o [ɔː], similar to the vowel of English raw as pronounced in New York. In Modern Greek Ω represents the same sound as omicron. The letter omega is transcribed ō or simply o.
Omega (the last letter of the Greek alphabet) is often used to denote the last, the end, or the ultimate limit of a set, in contrast to Alpha, the first letter of the Greek alphabet. In the New Testament book of Revelation, Christ is declared to be the "alpha and omega, the beginning and the end, the first and the last".
As the symbol of the Sumerian goddess Ninhursag, the omega has been depicted in art from around 3000 BC, though more generally from the early second millennium. It appears on some boundary stones on the upper tier, which indicates her importance.
- In physics:
- For ohm — SI unit of electrical resistance; formerly also used upside down (℧) to represent mho, the old name for the inverse of an ohm (now siemens with symbol S) used for electrical conductance. Unicode has a separate code point for the ohm sign (U+2126, Ω), but it is only included for backwards compatibility and the Greek uppercase omega character (U+03A9, Ω) is preferred. 
- For angular velocity in circular motion
- In statistical mechanics Ω refers to the multiplicity (number of microstates) in a system.
- The solid angle or the rate of precession in a gyroscope.
- In particle physics to represent the Omega baryons.
- In astronomy (cosmology) Ω refers to the density of the universe, also called the density parameter.
- In mathematics/computer science:
- In mathematics
- In complex analysis, the Omega constant, a solution of Lambert's W function
- A variable for a 2-dimensional region in calculus, usually corresponding to the domain of a double integral.
- In set theory, the first uncountable ordinal number (more commonly written as ω1)
- In topos theory, the (codomain of the) subobject classifier of an elementary topos.
- In combinatory logic, the looping combinator, (λ x. x x) (λ x. x x)
- In group theory, the omega and agemo subgroups of a p-group, Ω(G) and ℧(G)
- In statistics, it is used as the symbol for the sample space, or total set of possible outcomes.
- As part of logo or trademark:
- In fictional worlds:
- In the fictional world of Star Trek, the Omega symbol is used in conjunction with the Omega Directive.
- As the logo of the popular PlayStation franchise God of War.
- In the fictional universe Warhammer 40,000, the symbol for the Sons of Orar Chapter and, if turned upside down, the symbol for the Ultramarine Chapter of the Space Marines (Warhammer 40,000).
- In the popular web series Red vs Blue the Blood Gulch Chronicles Omega (or o'mally) is freelancer Tex's AI implant and the main antagonist of the Blood Gulch Chronicles
The symbol ω (lower case letter)
The minuscule letter ω is used as a symbol:
- Biochemistry and Chemistry:
- Denotes the carbon atom furthest from the carboxyl group of a fatty acid.
- In biochemistry, for one of the RNA Polymerase subunits.
- In biochemistry, for the dihedral angle associated with the peptide group, involving the backbone atoms Cα-C'-N-Cα
- In genomics, as a measure of evolution at the protein level (also denoted as dN/dS or Ka/Ks ratio).
- Computer science:
- The first transfinite ordinal number, often identified with the set of natural numbers including 0
- The complex cube roots of 1
- The Wright Omega function
- A generic differential form.
- In number theory, an arithmetic function
- In combinatory logic, the self-application combinator, (λ x. x x)
- In mathematical finance, the elasticity of financial options.
- angular velocity or angular frequency
- Computational fluid dynamics : the specific turbulence dissipation rate
- In circuit analysis and signal processing to represent natural frequency, related to frequency f by ω = 2πf.
- In astronomy, as a ranking of a star's brightness
- In particle physics to represent the omega meson
- Used in place of ん in Japanese typing shorthand.
- In linguistics, the phonological word
- In textual criticism, the archetype of a manuscript tradition
- In phonology, for a prosodic (or phonological) word.
- In sociology, used to refer to the lowest ranking member of a group.
- As an Asian-style emoticon for a happy cat: ^ω^
- ^ The Greek Alphabet
- ^ Herbert Weir Smyth. A Greek Grammar for Colleges. §1
- ^ Revelation 22:13, KJV, and see also 1:8, Greek ἐγὼ τὸ ἄλφα καὶ τὸ ὦ, ὁ πρῶτος καὶ ὁ ἔσχατος, ἡ ἀρχὴ καὶ τὸ τέλος. Or in Revelation 1:8 as seen in the Latin Vulgate Bible, the Greek is shown, surrounded by Latin: "ego sum α et ω principium et finis dicit Dominus Deus qui est et qui erat et qui venturus est Omnipotens"
- ^ Excerpts from the The Unicode Standard, Version 4.0, accessed 11 October 2006