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List of submarines of the Royal Navy

This is a list of Royal Navy submarines, arranged chronologically. Submarines that are currently active and commissioned are shown below in bold.

Pre-First World War

 * Holland-class submarine
 * HMS Holland 1, launched: 2 October 1901, decommissioned: 5 November 1913
 * HMS Holland 2
 * HMS Holland 3
 * HMS Holland 4
 * HMS Holland 5
 * A-class submarine (1903)
 * HMS A1, launched: 9 July 1902
 * HMS A2
 * HMS A3
 * HMS A4
 * HMS A5
 * HMS A6
 * HMS A7
 * HMS A8
 * HMS A9
 * HMS A10
 * HMS A11
 * HMS A12
 * HMS A13
 * B class
 * 11 boats, 1904–1906
 * C class
 * 38 boats, 1906–1910

First World War

 * D class
 * E class
 * F class
 * S class
 * V class
 * W class
 * G class
 * H class
 * Nautilus class
 * Swordfish class
 * J class
 * K class
 * HMS K1
 * HMS K2
 * HMS K3
 * HMS K4
 * HMS K5
 * HMS K6
 * HMS K7
 * HMS K8
 * HMS K9
 * HMS K10
 * HMS K11
 * HMS K12
 * HMS K13
 * HMS K14
 * HMS K15
 * HMS K16
 * HMS K17
 * HMS K26
 * L class
 * M class
 * HMS M1
 * HMS M2
 * HMS M3
 * HMS M4
 * R class
 * HMS K15
 * HMS K16
 * HMS K17
 * HMS K26
 * L class
 * M class
 * HMS M1
 * HMS M2
 * HMS M3
 * HMS M4
 * R class
 * R class

Interwar years

 * HMS X1
 * Odin class
 * Parthian class
 * HMS Pandora (N42)
 * HMS Parthian (N75)
 * HMS Perseus (N36)
 * HMS Phoenix (N96)
 * HMS Poseidon (P99)
 * HMS Proteus (N29)
 * Rainbow class
 * S (Swordfish) class
 * River or Thames class
 * HMS Thames (N71)
 * HMS Severn (N57)
 * HMS Clyde (N12)
 * Grampus class
 * T (Triton) class
 * U (Undine) class
 * HMS Clyde (N12)
 * Grampus class
 * T (Triton) class
 * U (Undine) class
 * U (Undine) class
 * U (Undine) class

Second World War

 * S (safari) class (War Emergency Programme)
 * T (Tempest) class (War Emergency Programme)
 * U (Umpire) class (War Emergency Programme)
 * P611 class
 * HMS P611
 * HMS P612
 * HMS P614
 * HMS P615
 * United States R-class submarine
 * HMS P511
 * HMS P512
 * HMS P514
 * U (Uproar) class
 * V (Vampire) class
 * HMS P514
 * U (Uproar) class
 * V (Vampire) class
 * V (Vampire) class

To mid-1950s

 * Amphion class
 * Amphion (P439/S39/S43) (laid down as Anchorite but name changed before launch)
 * Astute (P447/S47/S45)
 * Auriga (P419/S19/S69)
 * Aurochs (P426/S26/S62)
 * Alcide (P415/S15/S65)
 * Alderney (P416/S16/S66)
 * Alliance (P417/S17/S67)
 * Ambush (P418/S18/S68)
 * Anchorite (P422/S22/S64)
 * Andrew (P423/S23/S63)
 * Affray (P421)
 * Aeneas (P427/S27/S72/SSG72)
 * Alaric (P441/S41)
 * Artemis (P449/S39/S49)
 * Artful (P456/S56/S96)
 * Acheron (P411/S11/S61)
 * Ace (P414)
 * Achates (P433)
 * Explorer class
 * HMS Explorer (submarine)
 * HMS Excalibur
 * Stickleback class (midget submarines)
 * X51 Stickleback
 * X52 Shrimp
 * X53 Sprat
 * X54 Minnow
 * Foreign Built
 * Graph (P715) – German Type VII U-boat
 * Meteorite – German Type XVII U-boat
 * N2 – German Type VII-C/41 U-boat
 * X2 – Italian, Archimede-class submarine

Late 1950s to late 1980s
From HMS Porpoise Royal Navy submarines were given their own "S" pennant numbers.
 * Porpoise class (Diesel-electric hunter-killer)
 * Porpoise, commissioned/decommissioned: 1958–1982
 * Narwhal, c/d: 1959–1977
 * Finwhale, c/d: 1960–1987 (harbour service from 1979)
 * Cachalot, c/d: 1959–1979
 * Sealion, c/d: 1961–1987
 * Walrus, c/d: 1961–1986
 * Grampus, c/d: 1958–1978
 * Rorqual, c/d: 1958–1976
 * Oberon class (Diesel-electric hunter-killer)
 * Oberon, commissioned/decommissioned: 1961–1986
 * Onslaught, c/d: 1962–1990
 * Orpheus, c/d: 1960–1990 (harbour service from 1987)
 * Odin, c/d: 1962–1990
 * Otter, c/d: 1962–1991
 * Olympus, c/d: 1961–1989
 * Oracle, c/d: 1963–1993
 * Ocelot, c/d: 1964–1991
 * Otus, c/d: 1963–1991
 * Opossum, c/d: 1964–1993
 * Opportune, c/d: 1964–1993
 * Osiris, c/d: 1964–1989
 * Onyx, c/d: 1967–1991
 * Dreadnought (Nuclear-powered hunter-killer), commissioned/decommissioned: 1963–1980
 * Valiant class (PWR1 nuclear-powered hunter-killer)
 * Valiant, commissioned/decommissioned: 1966–1994
 * Warspite, c/d: 1967–1991
 * Resolution class (PWR1 nuclear-powered ballistic missile)
 * Resolution, commissioned/decommissioned: 1967–1994
 * Repulse, c/d: 1967–1996
 * Renown, c/d: 1967–1996
 * Revenge, c/d: 1969–1992
 * Churchill class (PWR1 nuclear-powered hunter-killer)
 * Churchill, commissioned/decommissioned: 1970–1991
 * Conqueror, c/d: 1971–1990
 * Courageous, c/d: 1971–1992
 * Swiftsure class (PWR1 nuclear-powered hunter-killer)
 * Swiftsure, commissioned/decommissioned: 1973–1992
 * Sovereign, c/d: 1974–2006
 * Superb, c/d: 1976–2008
 * Sceptre, c/d: 1978–2010
 * Spartan, c/d: 1979–2006
 * Splendid, c/d: 1981–2004
 * Trafalgar class (PWR1 nuclear-powered hunter-killer)
 * Trafalgar, commissioned/decommissioned: 1983–2009
 * Turbulent, c/d: 1984–2012
 * Tireless, c/d: 1985–2014
 * Torbay, c/d: 1987–2017
 * Trenchant, c/d: 1989-2022
 * Talent, c/d: 1990-2022
 * Triumph, commissioned: 1991

1990s to present

 * Upholder class (Diesel-electric hunter-killer) Transferred to Canada as the Victoria-class submarine
 * Upholder, commissioned/decommissioned: 1990–1994
 * Unseen, c/d: 1991–1994
 * Ursula, c/d: 1992–1994
 * Unicorn, c/d: 1993–1994
 * Vanguard class (PWR2 nuclear-powered ballistic missile)
 * Vanguard, commissioned: 1993
 * Victorious, commissioned: 1995
 * Vigilant, commissioned: 1996
 * Vengeance, commissioned: 1999
 * Astute class (PWR2 nuclear-powered hunter-killer)
 * Astute, commissioned: 2010
 * Ambush, commissioned: 2013
 * Artful, commissioned: 2016
 * Audacious, commissioned: 2020
 * Anson, commissioned: 2022
 * Agamemnon, construction began: 2013/Expected commission: 2024
 * Agincourt, confirmed, steel cut/Expected commission: 2026
 * Dreadnought class (PWR3 nuclear-powered ballistic missile)
 * Dreadnought, under construction - first elements of construction underway in 2016
 * Valiant, under construction
 * Warspite, under construction
 * King George VI, ordered

title: List of submarines of the Royal Navy


text:
Dichotomy

A dichotomy is a partition of a whole (or a set) into two parts (subsets). In other words, this couple of parts must be
 * jointly exhaustive: everything must belong to one part or the other, and
 * mutually exclusive: nothing can belong simultaneously to both parts.

If there is a concept A, and it is split into parts B and not-B, then the parts form a dichotomy: they are mutually exclusive, since no part of B is contained in not-B and vice versa, and they are jointly exhaustive, since they cover all of A, and together again give A.

Such a partition is also frequently called a bipartition. The two parts thus formed are complements. In logic, the partitions are opposites if there exists a proposition such that it holds over one and not the other. Treating continuous variables or multicategorical variables as binary variables is called dichotomization. The discretization error inherent in dichotomization is temporarily ignored for modeling purposes.

Etymology
The term dichotomy is from the Greek language διχοτομία dichotomía "dividing in two" from δίχα dícha "in two, asunder" and τομή tomḗ "a cutting, incision".

Usage and examples

 * In set theory, a dichotomous relation R is such that either aRb, bRa, but not both.
 * A false dichotomy is an informal fallacy consisting of a supposed dichotomy which fails one or both of the conditions: it is not jointly exhaustive and/or not mutually exclusive. In its most common form, two entities are presented as if they are exhaustive, when in fact other alternatives are possible. In some cases, they may be presented as if they are mutually exclusive although there is a broad middle ground (see also undistributed middle).
 * One type of dichotomy is dichotomous classification – classifying objects by recursively splitting them into two groups. As Lewis Carroll explains, "After dividing a Class, by the Process of Dichotomy, into two smaller Classes, we may sub-divide each of these into two still smaller Classes; and this Process may be repeated over and over again, the number of Classes being doubled at each repetition. For example, we may divide "books" into "old" and "new" (i.e. "not-old"): we may then sub-divide each of these into "English" and "foreign" (i.e. "not-English"), thus getting four Classes."
 * In statistics, dichotomous data may only exist at first two levels of measurement, namely at the nominal level of measurement (such as "British" vs "American" when measuring nationality) and at the ordinal level of measurement (such as "tall" vs "short", when measuring height). A variable measured dichotomously is called a dummy variable.
 * In computer science, more specifically in programming-language engineering, dichotomies are fundamental dualities in a language's design. For instance, C++ has a dichotomy in its memory model (heap versus stack), whereas Java has a dichotomy in its type system (references versus primitive data types).
 * In astronomy dichotomy is when the Moon or an inferior planet is exactly half-lit as viewed from Earth. For the Moon, this occurs slightly before one quarter Moon orbit and slightly after the third quarter of the Moon's orbit at 89.85° and 270.15°, respectively. (This is not to be confused with quadrature which is when the Sun-Earth-Moon/superior planet angle is 90°.)
 * In botany, branching may be dichotomous or axillary. In dichotomous branching, the branches form as a result of an equal division of a terminal bud (i.e., a bud formed at the apex of a stem) into two equal branches. This also applies to root systems as well.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Dichotomy


text:
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text:
Dragonet

Dragonets are small, percomorph, marine fish of the diverse family Callionymidae (from the Greek kallis, "beautiful" and onyma, "name") found mainly in the tropical waters of the western Indo-Pacific. They are benthic organisms, spending most of their time near the sandy bottoms, at a depth of roughly two hundred meters. There exist 139 species of the fish, in nineteen genera.

Due to similarities in morphology and behavior, dragonets are sometimes confused with members of the goby family. However, male dragonets can be differentiated from the goby by their very long dorsal fins, and females by their protruding lower jaws. The Draconettidae may be considered a sister family, whose members are very much alike, though rarely seen.

Genera
The following genera are classified within the Callionymidae:

 * Anaora J. E. Gray, 1835
 * Bathycallionymus Nakabo, 1982
 * Callionymus Linnaeus, 1758 (including Calliurichthys)
 * Diplogrammus Gill, 1865 (including Chalinops)
 * Draculo Snyder, 1911
 * Eleutherochir Bleeker, 1879
 * Eocallionymus Nakabo, 1982
 * Foetorepus Whitley, 1931
 * Neosynchiropus Nalbant, 1979
 * Paracallionymus Barnard, 1927
 * Protogrammus Fricke, 1985
 * Pseudocalliurichthys Nakabo, 1982
 * Repomucenus Whitley, 1931
 * Spinicapitichthys Fricke, 1980
 * Synchiropus Gill, 1859
 * Tonlesapia Motomura & Mukai, 2006

Description
These "little dragons" are generally very colorful and possess cryptic patterns. Their bodies are elongated and scaleless. A large preopercular spine is characteristic of this fish, and has been reported to be venomous in some species. All fins are large, showy and elongated; the first high dorsal fin usually has four spines; in males, the first of these spines may be further adorned with filamentous extensions. Dragonets have flattened, triangular heads with large mouths and eyes; their tail fins are fan-shaped and tapered.

The largest species, the longtail dragonet (Callionymus gardineri) reaches a length of 30 cm. At the other end of the spectrum, the Saint Helena dragonet (Callionymus sanctaehelenae) reaches a length of just 2 cm. Many species exhibit marked sexual dimorphism: males and females are coloured and patterned differently from each other, and (in addition to the spine filament) males have a much higher dorsal fin. This difference is extreme in the high-finned dragonet (Synchiropus rameus).

Reproduction
Dragonet spawning occurs during late afternoons, right before the sun sets. The fish's spawning behavior is divided into four distinctive stages: courtship display, pairing, ascending, and the release of eggs and milt. Both male and female dragonets have been observed displaying and courting each other, although the practice is much more frequent in the males. Females only do so when they are ready to spawn and are in need of a mate. Both sexes display by spreading their pectoral and caudal fins, and moving around or by the side of the other sex. Males will sometimes also spread their dorsal fins, repeatedly open and close their mouths, and position themselves on top of the females and rub their abdomens with their bodies. If a female accepts a male for spawning, they form a pair. Occasionally, another male might intrude upon the pair as they are mating and attempt to sneak fertilizations with the female. Such an act would result in aggression by the original male.

Prior to spawning, a male and female dragonet pair will ascend approximately 0.7-1.2 meters up a water column from the sand at the bottom of the ocean. The male assumes a parallel position to the female, touching the female's side with the part of its body near its ventral fin. The pair rises slowly up the water column, moving in a semicircular manner by swimming with their pectoral fins. The ascent occurs in two phases. During the first phase, the dragonet pair moves upward about fifteen centimeters and rests for around five seconds. Then it proceeds with its second rise. During this second phase of the ascent, the male and female flex their bodies and move their genital papillae toward each other. The male releases its ejaculate and the female releases its eggs. The release of eggs occurs singly and continuously for approximately five seconds. The eggs are pelagic, floating freely in the water column. The female releases a high number of eggs during each spawning, and the dragonets do not guard their offspring. The eggs are buoyant, so they intermingle with plankton and get swept away by the ocean current. After the spawning, the dragonet pair parts from each other and swims back down to the ocean floor. Male dragonets are polygynous, and will begin to search for other females to repeat the mating process with. They generally spawn with several different females within one reproductive day. Dragonets are very sexually dimorphic, with the males being much larger and having longer fins than the females. This sexual dimorphism may have evolved in males in response to female mate choice, male-male competition, or both.

Competition/aggression
Male dragonets form dominance hierarchies and act extremely aggressively towards each other. They are often observed chasing and biting, which occurs primarily when two males are close to a female during courtship and pairing. Fights can be very intense; when one male recognizes another male near its breeding site, it will rush toward it and bite at its rival's mouth. The two may bite at each other and twist their bodies around one another for longer than a minute. As a result of this behavior, male dragonets suffer higher mortality rates than females do after attaining maturation. The highest mortality rates in adult males occur during breeding. Males have evolved larger bodies, as well as longer spines and rays, in order to achieve dominance in reproduction. They have also developed bright colors so as to more effectively compete for female attention. These secondary sex characteristics further reduce the survival potential of male dragonets, as they increase the risk of predation, require greater energy costs, and escalate the risk of suffering injuries.

Feeding
Feeding by the dragonet occurs throughout the day, including the intervals between courtships and spawning. The fish feeds entirely on benthic sources, primarily copepods, amphipods, and other small invertebrates living on blades of sea grass. Species of dragonets from different locations show variations in specific food preference, attributable to the different availabilities and abundances of food organisms in those places. All of them feed by extending their highly protractible jaws toward their food and drawing it into the mouths, frequently followed by the expulsion of sand. No evidence suggests that dragonets are territorial. Individuals do not defend specific areas of substrate, as well as any resources that might be present on them, from intrusion by conspecifics or other fish species.

Among Calliurichthys japonicus and Repomucenus huguenini, the two most abundant dragonet species, amphipods are the most plentiful prey during the spring and winter months. The fish also supplement their diets with polychaetes, bivalves, and gastropods in these periods. During the summer, the dragonets feed primarily on ophiuroids and amphipods. In this season, ophiuroids are the most dominant in number. Finally, in the fall, the two species predominantly consume polychaetes, amphipods, and gastropods, with polychaetes contributing the highest amount.

Locomotion
Four types of swimming are observed in the dragonet. The first is burst swimming, the most common of the four, and utilized during foraging. The dragonet uses its pelvic fins to propel its body off of a substrate, and then its pectoral fins to guide itself forward. The second is continuous swimming, often utilized by males when approaching a potential mate or retreating during an aggressive encounter with another male. The dragonet uses its pectoral fins to propel its body forward, and its pelvic fins to lift and guide itself. The third type of swimming is rapid swimming, which is observed when the dragonet is attacking or fleeing. The fish primarily uses its caudal fins to achieve a quick speed. Finally, the fourth type is vertical swimming, utilized by the dragonet during spawning when it ascends. The pectoral fins are used to propel the fish's body up the water column.

Defense
In defense against its predators, the dragonet rapidly buries itself under the sand at the bottom of the ocean so that only its eyes remain visible. Many species of the fish also are capable of producing and secreting foul-tasting and -smelling substances that may ward off any potential predators.

title: Dragonet


text:
1996 in LGBT rights

This is a list of notable events in the history of LGBT rights that took place in the year 1996.

Events

 * Homosexuality is decriminalised in Bosnia and Herzegovina, although Republika Srpska did not remove homosexuality from its criminal code until 1998.

May

 * 8 – South Africa becomes the first nation in the world to explicitly prohibit discrimination based on sexual orientation in its constitution.
 * 20 – In Romer v. Evans, the Supreme Court of the United States rules 6–3 that Colorado's Amendment 2, which would have voided existing gay rights laws in Colorado and prevented any new gay rights laws from taking effect, is unconstitutional.
 * 21 – Hungary legalizes same-sex unions, first among the ex-communist countries, and fifth country worldwide. The Parliament voted 207–73 in favor for the bill.

June

 * 17 – The United States Supreme Court, in light of its decision in Romer v. Evans, vacates and remands the United States Court of Appeals for the Sixth Circuit in Equality Foundation of Greater Cincinnati, et al. v. The City of Cincinnati, which had found the city's anti-gay Issue 3 constitutional.

July

 * 1 – The United States Court of Appeals for the Second Circuit vacates and remands the district court in Able v. United States of America, which had ruled the military's gay-exclusionary "don't ask, don't tell" policy unconstitutional.

September

 * 10 – The United States Senate passes the Defense of Marriage Act (85–14) and rejects prohibiting employment discrimination based on sexual orientation in the private sector (49–50).
 * 15 – The European Parliament calls for an end to anti-gay discrimination.
 * 21 – United States President Bill Clinton signs the Defense of Marriage Act (DOMA) into law. DOMA allows individual states to refuse to recognize same-sex marriages performed in other jurisdictions and creates federal definitions for the terms "marriage" and "spouse".

December

 * 3 – In Baehr v. Miike Hawaii judge Kevin S.C. Chang rules in favor of the plaintiffs and enjoins the state of Hawaii from refusing to issue marriage licenses to same-sex couples. The following day Chang stays his ruling, acknowledging the "legally untenable" position couples would be in should the Hawaii Supreme Court reverse him on appeal.

Deaths

 * November 18 – Evelyn Hooker, whose research helped lead to declassifying homosexuality as a mental illness (b. 1907)

title: 1996 in LGBT rights


text:
Wikipedia talk:Requested articles/Mathematics

HTML4?
Character entity is mentioned at Table of mathematical symbols as having something to do with HTML4 - someone knows where else to put the request? Mikez 02:43, 28 Feb 2004 (UTC)

topology
strong topology is dealt with in the weak topology entry

--Mathmuncher 17:40, 22 Mar 2004 (UTC)

 * Not sure that exhausts the possible usages, though. Charles Matthews 18:53, 10 Apr 2004 (UTC)

Copyvio
Copyvio problems: the Jean Morlet article should be relisted. Charles Matthews 11:58, 17 Apr 2004 (UTC)

redirections
Minimax theorem has been redirected; but perhaps not very appropriately. Charles Matthews 16:58, 13 May 2004 (UTC)

lp sequence space
Regarding the requested article "lp sequence space": there is a section on them in Lp space. So I wanted to make lp sequence space a redirect to Lp space. Unfortunately, it is not possible to have a title start with a lower-case letter. Now I don't know what to do; any suggestions? -- Jitse Niesen 13:41, 21 May 2004 (UTC)

For a redirect, Little-lp space would be OK. On a page, it might look a bit silly, but one can hide it in a piped link! Charles Matthews 14:18, 21 May 2004 (UTC)

As I understand it, the purpose of redirects is that if a user types "xyz" in the go-to box (in the upper right corner), and xyz is actually defined in the page abc, then the redirect xyz -> abc brings the user automatically to the page abc. However, it's rather unlikely that somebody looking for information on \ell^p spaces types "Little-lp space", so it seems rather useless to create the redirect "Little-lp space" -> "Lp space". Note that I only want to be able to delete "lp sequence space" from the list of requested articles. It might be argued that one can already perform this deletion now, as there is an article on the \ell^p spaces, albeit hard to find; what do you think? -- Jitse Niesen 17:09, 21 May 2004 (UTC)

Actually, one can see redirects in backlinks, which can sometimes be helpful. Also, the policy is to create ample redirects, since they are rather cheap in resource terms, may cut down the increasing amount of duplication. But I don't mind, really. We have already used many more bytes discussing this!

Charles Matthews 17:57, 21 May 2004 (UTC)

You're right. I created the redirect. -- Jitse Niesen 20:38, 21 May 2004 (UTC)

space subset
space subset was linked only from Leopold Vietoris. I think it refers to spaces of subsets (see Vietoris topology on topological space). The relevant passage was added by someone interested in supercentenarians, so may well have been incorrectly copied. Can the link on this page simply be deleted? Rvollmert 16:50, 2004 Jul 26 (UTC)

Reconstructive surgery
I have (a) replaced super-almost-huge cardinal since the page was deleted, and (b) put back jet bundle, which was a frivolous (IMHO) redirect. Charles Matthews 09:21, 22 Aug 2004 (UTC)

..."Wiener measure"?
Is there REALLY a theorem in the study of Probabilities called "Wiener measure"? I literally laughed myself to tears when I saw this. Better now than in class, I guess (halfway through a university semester of Intro to Probabilities).

 * Errr - try growing up? Charles Matthews 11:28, 24 Oct 2004 (UTC)

 * I do wonder about a lot of these things - can anyone prove these things exist? I think that someone has just been editing putting random mathematical words together for their own amusement. There are also a load which have a name but are rather vague. Definition (mathematics) for example. What did whoever put that there expect? mattbuck 22:37, 28 February 2007 (UTC)

One article requested

 * V. International Mathematical Conference (Cambridge, 1912)

Gubbubu

Quantum Mechanics
Why is QM listed under Mathematics and not physics? Even the most abstract mathematical physics should be listed under physics because real mathematicians can see there is still a lot of hand-waving going on. --ub3rm4th 21:05, 23 Feb 2005 (UTC)

signature of a knot
Just created signature of a knot, based on MathWorld

Lagrange error bound
Regarding the request
 * Lagrange error bound - Unknown definition, featured on question 6(Part c) of the 2004 AP Calculus BC test

added by an anonymous editor: Does anybody know what the question is? My guess is that Lagrange error bound refers to the Lagrange form of the remainder term in Taylor's series (see Taylor's theorem). -- Jitse Niesen 11:20, 28 Apr 2005 (UTC)

Lagrange Error Bound is used to determine the accuracy of an approximation by a Taylor Polynomial. I wouldn't know how to explain via Wiki, however.

The Lagrage Error Bound is the error between the Taylor series approximation and the actual function value. The Lagrange Error Bound is also called the remainder for a Taylor series. The official definition is the difference between the actual value of the function and the nth partial sum that approximates the function. You can find the term by using this:

Rn(x) = (fn+1(c) * (x-a)n+1) / (n+1)! Where c is an unknown value on the interval [x, a].

This is part of the AP BC Test and should be included.

Boundary logic
I've taken these off the page: they were anyway in the wrong place.

Boundary logic axiomatisations - J. Spencer-Brown axiomatisation - W. Bricken axiomatisation - L. H. Kauffman axiomatisation -

These relate to the laws of form, so-called; which is at best marginal mathematics; and articles on different ways to present the system seem to me over the top. Charles Matthews 16:25, 25 May 2005 (UTC)

Mathematicians (alphabetization)
What's standard with non-capitalized parts of surnames -- de Bruijn, von Neumann, etc? My understanding was that de Bruijn goes under "B", von Neumann under "N". I guess the Category tag would have to be a rather unnatural-looking, but that's OK. --Trovatore 16:00, 23 July 2005 (UTC)

 * I think it depends on context: a Dutch phone book would alphabetise me as Sanden, Hugo van der, but the UK phone book puts me at van der Sanden, Hugo. The fact this is the english language WikiPedia is an argument for the latter approach. Hv 10:56, 16 September 2005 (UTC)

 * I have the same kind of issues at List of mathematicians, with the added difficulty that I need to teach a bot to alphabetize automatically. I also eventually settled to van der Sanden, Hugo (listed however at S), but I am still not sure about many Arab names and East Asian. So, if in the future you run into the List of mathematicians, and notice some improper split into "Last name, First name" format, or improper alphabetization, please lend a hand and fix it. :) (My bot does not attempt to change names already formattted in "Last, First" manner, it is concerned only with newer names showing up, so it will not modify your changes.) Oleg Alexandrov 16:26, 16 September 2005 (UTC)

Discrete mathematics?
Is there any reason why the category Discrete mathematics is missing? The closest existing category I could find was Set theory, which is not quite the same. Normally, I would just be bold and add it in, but this is a special project page. Vonkje 11:27, 12 September 2005 (UTC)

 * Go ahead. I at least have no qualms about adding another section. -- Jitse Niesen (talk) 12:09, 12 September 2005 (UTC)

 * Is there any actual discrete-math topic on which you want to request an article, or are you just planning to put an empty category there? Not that I mind either way.  But I wouldn't assume there'll be a rush of requests just because you add the section.  --Trovatore 16:41, 12 September 2005 (UTC)

 * Discrete mathematics can include combinatorics, graph theory, and order theory (at least the finite part), each of which has its own category here. On the other hand, I don't think of discrete mathematics and set theory as having much in common, because the hard parts of set theory deal with large infinities. --JWWalker (talk) 06:06, 29 November 2008 (UTC)

MathWorld?
Would it make sense to take the index of MathWorld and add all non-existant entries to this list of requested articles? —BenFrantzDale 20:07, 15 September 2005 (UTC)

 * Generating a list of articles on MarthWorld and checking it against Wikipedia for missing articles seems like a great idea to me. I don't think that the results would be best placed here though. There are other such lists for other reference works, for example see WikiProject_Missing_encyclopedic_articles and its subprojects. In mathematics see also WikiProject_Mathematics/PlanetMath_Exchange. A good place to discuss this further might be at Wikipedia talk:WikiProject Mathematics. Paul August ☎ 20:55, 15 September 2005 (UTC)

 * Yes, generating the index would be nice. I would like to see it when it is done. But I agree that putting all those articles as requested articles is not a good idea. As Paul says, see WikiProject_Mathematics/PlanetMath_Exchange where we attempt to review/copy articles from a different website (which is free, unlike mathworld). Oleg Alexandrov 23:32, 15 September 2005 (UTC)

 * Such a list, were it to exist as independent entity, is one of a class of things that have recently been agreed upon as running a significant risk of being a copyright violation (Wolfram's copyright covers their index even independently of the content of each topic) and similar lists have been deleted by administrative fiat (i.e. Jimbo wanted it done). However incorporating a list of topics that they have and we want to have along with similar lists from other sources or requests people have independently made is presently accepted (though some dispute the validity of even this).  So, if you wanted to add their topic list to this page, it would be okay under the current thinking.  One of the issues here is that we do not want to be seen to be republishing their index or any mechanically generated large subset of it.  Dragons flight 00:35, 16 September 2005 (UTC)

 * I've put up #, A, B, and C of the index at WikiProject Mathematics/PlanetMath Exchange/MathWorld. Unfortunately, the list is so long that Wikipedia won't let me post all of it – it times out. I also included the Perl script to generate the list. —BenFrantzDale 01:36, 16 September 2005 (UTC)
 * I think it is good to split it into several smaller lists. Wikipedia is slow recently, and 250 K of text is way too much for a single page. By the way, a lot of those links show up red because they are made up of capitals, like Orthogonal Group for example. Doing some lowercasing (Perl's lc function) would take care of that.

 * I will now put an announcement at Wikipedia talk:WikiProject Mathematics. Oleg Alexandrov 03:19, 16 September 2005 (UTC)

I'm positive that I've already seen this list somewhere on wikipedia. A long list of red links that have extant mathworld articles. As I recall, capitalization made a lot of false positives. I don't remember where I saw the list though, someone's user page. Maybe Charles Matthew's? I'll see if I can find it. -Lethe | Talk 03:57, 16 September 2005 (UTC)

 * Visit one of those red links, and do a "what links here". That should give the answer, if the answer exists. Oleg Alexandrov 04:03, 16 September 2005 (UTC)

 * Take a look at Missing science topics. It looks like the list was there, but was since deleted, according to some policy decision. -Lethe | Talk 04:27, 16 September 2005 (UTC)

Did no one read my comment? This list falls squarely under Jimbo's don't steal lists of articles from our competitors prohibition, which is why the seperate lists of articles in Encarta, EB, and other encyclopedias were deleted. As in that case, it is generally believed that we are reasonably protected as long as missing article lists are not single source, hence the Encyclopedia lists were recreated as a merged topic list. You can ask that to happen here as well, but simply republishing Mathworld's index is a copyvio of their index (and yes the index by itself is subject to copyright protection). Dragons flight 04:22, 16 September 2005 (UTC)

 * Crap, I apparently missed it. Interesting. I had no idea there was prescident for this. By that standard, the list is obviously not OK. It seems interesting since there is nothing stoping someone from requesting, or starting stubs of, every page from MathWorld not on Wikipedia. The problem seems to be that the index itself is copyrighted, not that copyright somehow prevents me, personally, from using MathWorld's index to find holes in Wikipedia. Hmmm... (PS, thanks for notifying me directly; I just missed your edit above.) —BenFrantzDale 04:44, 16 September 2005 (UTC)

 * I have now killed the list. Dragons flight 04:52, 16 September 2005 (UTC)

removed misplaced requests
I have removed the following requests from the order theory section:
 * Composite Trapeziod Methods - Composite Simpson's Methods - Adaptive Simpson's Methods

They were added by 141.151.173.98. They belong, if anywhere, in the "Numerical analysis" section --Trovatore 22:36, 24 October 2005 (UTC)
 * I see that Jitse's taken care of this. --Trovatore 22:38, 24 October 2005 (UTC)

Conway's game of life in reverse
Is this even a useful article to want/have? From what I know of CA, isn't Conway's game non-reversible? --PeruvianLlama(spit) 21:30, 21 November 2005 (UTC)
 * Exact phrase has only 18 hits in Google! DFH 20:59, 15 February 2007 (UTC)

 * A single live cell and no live cells both transition to no live cells. It definitely isn't reversible so I'm fairly sure that that article shouldn't exist.109.150.134.159 (talk) 15:15, 12 August 2012 (UTC)

Multiple Versions
I think that for most mathematical topics, we should have multiple versions for people with different backgrounds. People can then choose to see either more generalized or more concrete explanations by clicking on links. Do you guys think this would be worth the community's efforts? --anon

It would be a nightmare to maintain, and I think there is a policy about not having multiple articles on the same topic. The current consensus is to start articles simple, and have them become more complicated as one reads along. Oleg Alexandrov (talk) 03:25, 11 March 2006 (UTC)

 * There is the Simple English Wikipedia, these articles are addressed to lower levels of English and mathematical usage see for example Algebra. --Salix alba (talk) 08:30, 11 March 2006 (UTC)

E9 (Lie algebra)
Theres a request for E9 (Lie algebra). Does this actually exist, I though E6, E7, E8 were exceptional lie algebras and there is no other example in the sequence? --Salix alba (talk) 14:46, 29 March 2006 (UTC)
 * There is an E9, as an affine Lie algbra. One classification that ends at E8 is that of the finite-dimensional simple complex Lie algebras - if you are prepared to extend the class you're talking about, E9 can make sense! QuantumGroupie 19:01, 2 November 2006 (UTC)

Information theory
shouldn't this have a section? Or does it go under physics? (or computer science for that matter)

RA formatting standards
Please see: Wikipedia_talk:Requested_articles. Dragons flight 18:49, 15 July 2006 (UTC)

Date/place wikilinks
I haven't checked who did it, but, although the addition of dates and places for mathematicians may be helpful in determining which person of that name we are talking about, it breaks the bots which maintain WikiProject_Mathematics/Current_activity.

Lemmas
I have just copied most of the redlinks from the list of lemmas article. There are some that I could not categorize. DFH 20:57, 15 February 2007 (UTC)

Perfect forward security/secrecy
The page reports that "Perfect forward security" is a requested article in the Cryptography section. I think it's likely that anyone wanting that topic is really after Perfect forward secrecy.

John Y 10:31, 7 April 2007 (UTC)

Watchout for further attempts at spreading the Digg vs. AACS dispute
The previous revert was sensible. The requested article title was probably the code on Digg which the AACS sent a "cease and desist" about. It's currently a "hot potato" in the news. Watchout against further attempts to spread the dispute into Wikipedia. DFH 18:35, 2 May 2007 (UTC)

10-polytope
I am surprised that 10-polytope does not exist because Simplex, Hypercube, Cross-polytope, 10-simplex, 10-cube and Category:10-polytopes links to it. I do not create it because I do not have related knowledge and reference. Therefore, I would like to ask somebody to create it. QQ (talk) 22:04, 1 March 2008 (UTC)

Rigorization of analysis
I think this requested title is a synonym for Arithmetization of analysis, which already exists as an article. 207.241.239.70 (talk) 04:24, 18 February 2009 (UTC)

Bluelinks
What's the accepted policy on removing entries from the list for which articles have been created? I seem to recall someone making the point that keeping them in the list is helpful, to keep track of progress. (I ask because some have recently been removed from the list.) Artie P.S. (talk) 15:53, 25 April 2009 (UTC)

Numbers, Specifically the "Teens"
Hello, all. I was doing a search for the "Teens" because I wanted to know why their naming convention differed from other higher order numbers (ie, thirteen instead of ten-three, compared to forty-three) but I was unable to find any articles on the numbers 13 through 19 inclusive.

I cannot imagine that nobody has ever created such an article. Before I try to create a stub for it, is there a way to find out if such an article was deleted?

Additionally, what would be the category to request an article on a specific number (or numbers)? --Bertrc (talk) 19:10, 27 May 2009 (UTC)

Article introducing complex numbers
I would like to request a new article introducing the concept of complex numbers. The current article does not introduce them in a way that's accessible to someone who does not already have a great deal of mathematical knowledge. After looking up educational resources elsewhere on the web I found the concept fairly straightforward and logical but I'm not qualified to write it myself.

Just the introduction to 'Complex number' contains 27 links to other articles, of which at least half are similarly dense and inscrutable. As it stands there's no way for someone to develop an understanding of these concepts from reading the wikipedia because there's no starting point, you just wind up clicking between articles full of thick and unelaborated jargon.

FTA: Complex numbers form a closed field somehow with real numbers. OK, so what's a closed field? Don't know, go to the article. OK, it's some type of field, what's a field? Go to the article, and before I've left the introduction I'm wondering what 'quintic relations' are or an 'integral domain' and if I'd only read the wiki I still wouldn't know what a complex number is or what it has to do with anything. Now I'm not averse to learning all this, I'd love to understand it, but clicking from article to article isn't helping. It's frustrating in a way that other areas of the wikipedia aren't, I don't experience this in the physics or computer science sections for example, if understanding one area depends on understanding another one can usually just click through and read the prerequisite article without falling down the rabbit hole. 196.209.232.87 (talk) 15:18, 21 December 2009 (UTC)

L-space - article already exists?
Under Set-theoretic topology, one of the requests is L-space (mathematics). Is this the same thing as Lp space? CarrieVS (talk) 21:58, 7 May 2012 (UTC)
 * No. In Set-theoretic topology, an L-space is a regular space that is hereditarily Lindelöf but not hereditarily separable. An S-space, also requested there, is the other way around. Ntsimp (talk) 17:43, 31 May 2012 (UTC)

Spiegelungssatz
Requested entries Leopoldt's Spiegelungssatz and Leopoldt reflection theorem already existed at Reflection theorem: I have created the redirects. Deltahedron (talk) 20:22, 29 August 2012 (UTC)

Average-fold error (AFE)
There was a request for a page of this name, based on its use in this JEPT article. I looked at the article in question, and in my opinion this is not a statistical term that needs a separate article. As the author and others in pharmacology have used it, it is merely the geometric mean of a set of observed ratios. Nothing special or noteworthy. If anyone disagrees, then please write a stub, or reinstate the request for an article. — Aetheling (talk) 18:55, 29 December 2012 (UTC)
 * The exact phrase has 51 hits in Google Books and 163 in Google Scholar, not to mention 28,600 in Google Web, mainly relating to its use in pharmacology and going back at least to 1997. I think that would justify at least a mention.  Deltahedron (talk) 19:19, 29 December 2012 (UTC)

Which Coleman?
For the requested article of Coleman's principle there are different Coleman's with different principles. If a first name was given that would be greatly appreciated. Littlelago123 (talk) 01:44, 3 January 2013 (UTC)

Stephinity
This has been in the uncategorized section since this edit four years ago. Looking at the draft article here, and noting the word has no hits on ZMATH and none relevant on Google Books, I am inclined to say Wikipedia is not for things made up one day. Spectral sequence (talk) 18:28, 27 May 2013 (UTC)

 * I agree, the draft article looks like original research. Other than folks using the term as a cute online alias, there is a hit at the Urban dictionary, but this is of course not a reliable source. I'll be bold and remove the entry. (Ah, looks like you have already done so.) --Mark viking (talk) 19:27, 27 May 2013 (UTC)

East Gawley overlap algorithm
I can find no evidence for the existence of this, added in 2007. Can anyone verify it? Spectral sequence (talk) 18:49, 6 June 2013 (UTC)

New sections
I would like to propose new sections on Field Theory, taking in Galois Theory as a subsection, and Rings, Modules and Algebras. The current Abstract Algebra section is getting rather long. Spectral sequence (talk) 08:45, 23 June 2013 (UTC)

Brand quotient
The request for Brand quotient has been here since 2008 under Coding Theory. I can find nothing to suggest that there is such a concept outside marketing: there seems to be nothing in GBooks or ZMATH. Can anyone provide a reference to show that this concept exists? Spectral sequence (talk) 21:29, 24 June 2013 (UTC)

 * Perhaps Brandt semigroup is intended? See   Deltahedron (talk) 21:24, 26 February 2014 (UTC)

New "requirements"
I have reverted this imposition of new "requirements" on requested articles pending a consensus on their provenance and applicability. Spectral sequence (talk) 19:28, 2 August 2013 (UTC)
 * I have replied on my talk page. Thanks, Mat  ty  .  007  19:36, 2 August 2013 (UTC)
 * Which is not the right place for a discussion of what happens on this page. There is a discussion in progress at Wikipedia talk:Requested articles on the general principle.  Spectral sequence (talk) 19:51, 2 August 2013 (UTC)
 * Well, you posted on my talk page, and here, and on the RA talk. Please decide which one you want to speak at, rather than posting on all three. Thanks, Mat  ty  .  007  09:43, 3 August 2013 (UTC)

Mutation algebra
The request for mutation algebra needs to be unpacked. There are three distinct concepts of mutation algebra — one due to Santilli, an algebra derived from an associative algebra by defining a twisted multiplication $$x * y = x a y + \lambda y a x$$; a baric algebra with endomorphism, which is indeed a train algebra ; and a homotope of a Jordan algebra. Deltahedron (talk) 19:39, 3 February 2014 (UTC)

 * See Mutation (algebra). Deltahedron (talk) 21:15, 26 February 2014 (UTC)

1058 (number)
There seems no special reason to add this number rather than any other, so I'm removing the suggestion pending some pointers that would establish a reason to have an article on it. By the way, it would seem to be a good practice to adhere to in general, to accompany a suggestion with some further information of the kind. Deltahedron (talk) 21:13, 26 February 2014 (UTC)

Abstract comunitronics
I removed Abstract comunitronics, added 2006-09-27T19:10:40Z, as I see no reason to believe it exists: no relevant hits on Google Books or ZMATH. Deltahedron (talk) 20:24, 27 February 2014 (UTC)

Organisations
I thought that a section on organisations was appropriate, given that there's one on mathematicians. Deltahedron (talk) 20:08, 17 May 2014 (UTC)

Rings and Modules
I boldly split out a separate section on this topic as it seems to me that the section of Abstract Algebra is getting overgrown, and that this is an identifiable subtopic. This has been reverted with the comment don't fine-tune the list. Very well then, why not, and what if anything should we do about sections that are getting overlong? Deltahedron (talk) 19:57, 20 June 2014 (UTC)

 * I think I get the feeling the subsection is getting too long, but it doesn't have to stay that way in the future: "ideally", we should have very few (or even none) requested articles; at least that's the goal we're aiming at. In other words, lists of this type are maintained in the manner that is different from those in the wikipedia main space; the lists of the latter type are meant to be forever--long live wikipedia! Anyway, I don't insist on my revert. If you really feel stronger about the break-up and in particular if you suspect the subsection will see a further growth in the future, you can go ahead with the break-up. -- Taku (talk) 22:29, 20 June 2014 (UTC)

 * Ideally the list would be very short indeed, but right now it is as it is. Abstract algebra is the largest section with 99 entries.  Rings and modules is not represented, but there's even a subsection on Galois theory with 3 items.  It seems to me that breaking out a major topic with 18 entries, which would be in the top third of sections  by length, is an improvement.  Deltahedron (talk) 07:09, 21 June 2014 (UTC)

 * I don't see the breaking-up as an "improvement" in the sense of simplification; you're reducing the number of items in a section by increasing the number of overall sections. It's more of a matter of preference. My view/preference is that something like a subsection with 3 items should be merged into some other section. To elaborate on the point I was making, the important question is who is an audience of this list? Certainly, not general Wikipedia readers not even general Wikipedia editors; it's not their modus operandi to consult the list before they create new articles At least that's not my MO. The point of lists like this is more of a link farm. It's very useful to have red links in some form. Indeed, there are even several user-maintained list of red links. How is this an elaboration on "don't fine-tune"? -- Taku (talk) 11:44, 21 June 2014 (UTC)

 * (Not that's we're on it). Maybe my position is even antagonistic against lists such as this one; that is, is this page really that useful? There are better ways to do link farm. For example, I think "glossary of such and such" is much a better way to have red links since we can give some context or definitions with a reference in the wikipedia mainspace. Especially in math articles most important thing is to give definitions and generally try to contextualize the concept (e.g., it generalizes this and that). List of prize winner type articles are also better substitutes than the list of requested mathematicians; notability for one thing. -- Taku (talk) 12:00, 21 June 2014 (UTC)

 * It may not be useful to everybody, but some potential contributors use it to find suggestions for articles to work on, and it is useful to them to be able to navigate it according to their fields of interest and expertise. If you don't use the list, why the concern of how it is organised?  Deltahedron (talk) 13:10, 21 June 2014 (UTC)

 * I understand that's the idea and I'm not certainly proposing that we scrap this page. I'm just questioning the effectiveness. And I'm very much interested in the productivity of us as math editors. Generally speaking, anything that is in the wikipedia main page is more useful to the readers than any maintainance-type lists. -- Taku (talk) 15:20, 22 June 2014 (UTC)

 * I'm quite interested in the notion of using the various Glossary pages. I have been following the practice at this page of trying to support each suggestion with a link to a reliable source that at least mentions the existence of the missing content.  If we allowed ourselves to do that on Glossary pages that might be useful.  Deltahedron (talk) 15:48, 22 June 2014 (UTC)

Non-commutative plane
I reverted this edit which changed the redlink Non-commutative plane to a piped link to Quantum_spacetime. That did not seem to fit with the purpose of this page, which is to list articles that don't exist. If that's a suitable target for a redirect, then in general either say so on the same line, or create a redirect in article space from Non-commutative plane. But don't do the latter here, as it doesn't seem to be a particularly good target, since the proposed target paragraph does not mention non-commutative planes at all. Natural boundary (talk) 06:43, 15 September 2014 (UTC)

 * Thanks for explaining your reversion. The wikilink was meant to be a hint as to what this topic might be about. But I agree, this would be better expressed as a separate comment. I've only ever seen non-commutative planes mentioned in the context of a quantum model of spacetime, but perhaps that is just my physics bias. --Mark viking (talk) 12:39, 15 September 2014 (UTC)

Helmut Hofer
The Helmut Hofer cited in the IAS announcement is the same who has already an article, co-founder of symplectic topology.--Claude J (talk) 09:07, 13 January 2016 (UTC)

Cartan–Tits fixed point theorem
I believe this theorem is stated, under the name Bruhat-Tits fixed point theorem, on the page CAT(0) inequality. If nobody objects I will make a redirect for this. jraimbau (talk) 11:18, 25 July 2016 (UTC)

"Cubinder" draft needs attention. Unlisted in project.
Hello fellow Wikipedians. I am in the process of creating an article about the 4D shape, the “cubinder”. It was previously red linked on other articles, and I was surprised to see it was not already an item listed for creation by Wiki Projects Mathematics, as the duocylinder and spheriender are already articles. I require help to improve the draft, as I require more formulae, sources, and additional information to create this article. You can access this page at User:Darnburn98/Cubinder, please come on over and help improve this article to get into the main space! Darnburn98 (talk) 22:33, 20 January 2017 (UTC)

What do I do after creating a requested Article?
Do I delete the article from the listing in this Wiki page? Chaos1618 (talk) 20:00, 14 October 2018 (UTC)
 * This is a good question! It would be great for someone to reply, and add this information to the project page. Cheers! Doctormatt (talk) 18:11, 27 March 2019 (UTC)

Fewnomial
If I remember correctly the talks from Khovanskii, I think the name for the article requested should be better either Fewnomial theory, or Theory of fewnomials. I might be wrong, but I think there is not a concept/object called a 'fewnomial'. Instead 'fewnomial theory' is that body of results that bound the number of solutions of polynomial equations by a function depending on the number of monomials appearing in the polynomial, and their many applications. The 'few' referring to the philosophy that when there are few monomials, then the geometry of the solutions is simpler.

I could attempt starting the article, but wanted first a feedback about the name. Cactus0192837465 (talk) 01:44, 2 January 2019 (UTC)


text:
Sector light

A sector light is a man-made pilotage and position fixing aid that consists of strictly delineated horizontal angle light beams to guide water-borne traffic through a safe channel at night in reasonable visibility. Sector lights are most often used for safe passage through shallow or dangerous waters. This may be when leaving or entering harbour. Nautical charts (paper and electronic) give all the required information.

Sectors of colored glass (or plastic) are placed in the lanterns of these lights. The light will then show these colors when observed certain bearings. Bearings referring to a sector are given in degrees true, as observed from sea. Though the colors of the light will change, the characteristics will not. The change of color is not abrupt. The transition is made through an arc of uncertainty of 2° or greater.

The colors that are used, are conform to the IALA Maritime Buoyage system that is designed by the International Association of Lighthouse Authorities: A ship that is sailing in safe water and then sees the red (or green) color of the light has to make an alteration in course.
 * white - this sector is in the middle of the safe channel
 * red - indicates the port edge of the channel for vessels approaching the light source
 * green - indicates the starboard edge of the channel for vessels approaching the light source.

The world has different navigation standards managed by IALA (International Association of Lighthouse Authorities). For example, the United States uses a signalling standard which is the opposite of Europe. In USA, the red light indicates the starboard side of the channel for harbour bound vessels, while the green light indicates the port side of the same channel. An expression to remind of this is "red on right returning".

Examples of sector lights
An example of a sector light is the light of the Fisgard Lighthouse in British Columbia. The lighthouse was built to guide ships through the entrance of Esquimalt harbour. The white sector is an isophase light of 2s from 322° to 195°. If the ship sees this white light, it can pass safely. The rest shows a red light from 195 to 322°. If a vessel sees this light, it should alter its course.

Another example is the PEL sector light at Diego Garcia. The PEL sector light was constructed to guide US Navy vessels into the Diego Garcia port through a narrow entrance in the lagoon. The PEL sector light produces 3.5 million candela and is visible for 10 nautical miles (18.5 km) by day. The beam with 3 colors is narrow at 1.6°. This was required because the PEL sector light was built over 7NM (13 km) from the atoll entrance and it had to illuminate a safe entrance which is only 228 meters wide.

A more complex example is Kolnesholmane, on the southern approach to Risavika in SW Norway. This has multiple white green and red sectors, indicating safe passage from several directions.

title: Sector light


text:
Talk:European Convention on Human Rights

Disputed/invaded terorteses
On the map it shows the Crimea and eastern Ukraine as signatories is this still correct?

Anything on protection of liberty of movement and transport vs speed limitations
All nice to put speed limitations up to 30km/h if this is in the interest of public safetly. But limiting the speed on roads leading to villages and communes (not in the village and in the commune) to 30 km/h even at night? We are we supposed to deliver medicines during the night - but we cannot do that anymore for the actual prices if we have to reduce our speed in the night to 30km/h? Thy, SvenAERTS (talk) 15:51, 27 November 2022 (UTC)

Ukraine's Suspended Adherance
Ukraine suspended it's adherence to the European Convention on Human Rights in order to effectively enforce martial law in its territory on the 29th. The article, and map, should reflect this. Kingofmapps (talk) 14:47, 1 May 2024 (UTC)


text:
Leading lights

Leading lights, also known as range lights in the United States, are a pair of light beacons used in navigation to indicate a safe passage for vessels entering a shallow or dangerous channel; they may also be used for position fixing. At night, the lights are a form of leading line that can be used for safe navigation. The beacons consist of two lights that are separated in distance and elevation, so that when they are aligned, with one above the other, they provide a bearing. Range lights are often illuminated day and night.

In some cases the two beacons are unlighted, in which case they are known as a range in the United States or a transit in the UK. The beacons may be artificial or natural.

Operation
Two lights are positioned near one another. One, called the front light, is lower than the one behind, which is called the rear light. At night when viewed from a ship, the two lights only become aligned vertically when a vessel is positioned on the correct bearing.

During the day, the lights may not easily be seen and therefore leading lights are often fitted with secondary visual aids, e.g. large red flags with wide black lines running down them. When both red flags and black lines line up, the navigator knows that the vessel is on the correct bearing. The structures are usually painted to make them more prominent.

Some major rivers, such as the Elbe River in Germany, have a series of leading lines. When it is necessary to make a turn, the navigator lines up the next pair of leading lights. This provides guidance from Hamburg to the sea, using successive pairs of leading lights.

Leading lights were used in Great Britain as early as 1763 to mark the Port of Liverpool. The first set of range lights in the United States were privately established by subscription at Newburyport Harbor in Massachusetts in 1788.

Leading lights are sometimes designed to be movable, allowing their position to be shifted in the event of a change in the safe channel; these include one at Hilton Head, South Carolina, the original Chatham Light, and the Nantucket Beacon, predecessor to the Nantucket Harbor Range shown above.

title: Leading lights


text:
Lewy body

Lewy bodies are the inclusion bodies — abnormal aggregations of protein — that develop inside neurons affected by Parkinson's disease (PD), the Lewy body dementias (Parkinson's disease dementia and dementia with Lewy bodies (DLB)), and some other disorders. They are also seen in cases of multiple system atrophy, particularly the parkinsonian variant (MSA-P).

Lewy bodies appear as spherical masses in the cytoplasm that displace other cell components. For instance, some Lewy bodies tend to displace the nucleus to one side of the cell. There are two main kinds of Lewy bodies — classical and cortical. Lewy bodies may be found in the midbrain (within the substantia nigra) or within the cortex. A classical Lewy body is an eosinophilic cytoplasmic inclusion consisting of a dense core surrounded by a halo of 10 nm wide radiating fibrils, the primary structural component of which is alpha-synuclein (α-synuclein).

History
In 1910, Fritz Heinrich Lewy was studying in Berlin for his doctorate. He was the first doctor to notice some unusual proteins in the brain, comparing them to earlier findings by Gonzalo Rodríguez Lafora. In 1913, Lafora described another case, and acknowledged Lewy as the discoverer, naming them cuerpos intracelulares de Lewy (intracellular Lewy bodies). Konstantin Nikolaevich Trétiakoff found them in 1919 in the substantia nigra of PD brains, called them corps de Lewy and is credited with the eponym. In 1923, Lewy published his findings in a book, ''The Study on Muscle Tone and Movement. Including Systematic Investigations on the Clinic, Physiology, Pathology, and Pathogenesis of Paralysis agitans''. Eliasz Engelhardt, who is in the neurology department at Federal University of Rio de Janeiro, argued in 2017 that Lafora should be credited with the eponym, because he named them six years before Trétiakoff. Nonetheless, Trétiakoff is still the primary figure acknowledged for coining the term, "Lewy bodies".

According to the Journal of the History of the Neurosciences, Dr. Lewy became interested in studying more about the brain (neurology), because of the discovery that Alois Alzheimer made in 1906. The article mentions that the third reported case of Alzheimer's disease had histological structures that happened to be similar to Lewy body histology slides, but the contribution was not given to Lewy's finding.

Cell biology
A Lewy body is composed of the protein α-synuclein associated with other proteins, such as ubiquitin, neurofilament protein, and alpha B crystallin. Tau proteins may also be present, and Lewy bodies may occasionally be surrounded by neurofibrillary tangles. Lewy bodies and neurofibrillary tangles can occasionally exist in the same neuron, particularly in the amygdala.

Alpha-synuclein modulates DNA repair processes, including repair of DNA double-strand breaks (DSBs) by the process of non-homologous end joining The repair function of alpha-synuclein appears to be greatly reduced in Lewy body bearing neurons, and this reduction may trigger cell death. Mutations are the reason behind their damaged repair function. One mutation in particular, in the gene encoding for presynaptic alpha-synuclein, was found to have been passed down from family members with PD. Similarly in regards to DLB, Lewy bodies retrieved from DLB brains were found to contain alpha-synuclein proteins that were shortened by mutations.

Lewy bodies are believed to represent an aggresome response in the cell. When misfolded proteins aggregate, or clump together, many diseases are more likely to develop, including those that are associated with Lewy bodies. Aggregation is believed to occur when there is a high amount of misfolded proteins in the ubiquitin-proteasome pathway, which are then brought to a resulting aggresome so they can be organized into one place. Since Lewy bodies are made of ubiquitinated proteins that would be handled in the ubiquitin-proteasome pathway, they may be made from this or a similar process if the pathway capacity is indeed exceeded by misfolded proteins that aggregate together. Accordingly, the aggresome, where the damaged proteins fully aggregate, is akin to the Lewy body.

Despite their differences, there is evidence that a particular protein family, called 14-3-3, plays a role in the formation of both cortical and classical Lewy bodies. This makes it an important protein family in regards to Lewy body-associated diseases, and there are at least 7 forms of it that have been clearly identified in mammals.

Cortical Lewy bodies are also composed of alpha-synuclein fibrils, but are less defined and lack halos. This kind of Lewy body is one of those aforementioned that regularly displaces the nucleus. In histopathology, cortical Lewy bodies are a distinguishing feature for dementia with Lewy bodies, but may occasionally be seen in ballooned neurons characteristic of behavioural variant frontotemporal dementia and corticobasal degeneration, as well as in patients with other tauopathies.

Lewy neurites
Lewy neurites are abnormal neurites in diseased neurons, containing granular material and abnormal α-synuclein filaments similar to those found in Lewy bodies. Like Lewy bodies, Lewy neurites are a feature of α-synucleinopathies such as dementia with Lewy bodies, Parkinson's disease, and multiple system atrophy. They are also found in the CA2-3 region of the hippocampus in Alzheimer's disease.

title: Lewy body


text: Template:Upholder/Victoria class submarine


text:
Steve Troxler

Steve W. Troxler (born 1952) is an American farmer and Republican politician. He serves as the Commissioner of Agriculture and Consumer Services for North Carolina. Because of a contested election due to lost ballots in the November 2004 statewide election, he was not sworn in until February 8, 2005. He was re-elected in 2008, 2012, 2016 and 2020.

Biography
Troxler was born and grew up in Browns Summit, North Carolina, where he attended local schools. He studied conservation at North Carolina State University, earning a bachelor's degree in 1974.

Troxler originally leased farmland in Browns Summit before he purchased some after he got his degree and primarily grew tobacco. After he was elected to office, he planted less tobacco in favor of wheat and produce.

Political career
Troxler was originally a member of the Democratic Party, but left the organization in the late 1990s, feeling that Democrats were hostile to tobacco production. Jerry Blackwelder, a Republican Party consultant, recruited him as a candidate for the office of North Carolina Commissioner of Agriculture. Troxler ran in 2000, losing to Meg Scott Phipps. Due to a financial corruption scandal, Phipps was replaced by Democrat Britt Cobb in 2003.

Troxler ran again for the same position in the 2004 Council of State election, finishing 2,287 votes ahead of incumbent Cobb. However, because more than 4,000 votes were lost in Carteret County, the race was unresolved for more than three months. After extended legal challenges between Cobb and Troxler, Cobb conceded the race on February 4, 2005.

Troxler was sworn in as North Carolina Agriculture Commissioner on February 8, 2005. He was the first Republican to be elected as Agriculture Commissioner in the state's history, reflecting the ascendancy of the party in the state. After his inauguration at the North Carolina State Fairgrounds, Troxler rode a tractor to his downtown offices. Troxler was re-elected in 2008 as Commissioner of Agriculture and Consumer Services for the state of North Carolina. As of 2021 he is currently serving his fifth term.

Because of a state law passed in 2013 that expanded venues for carrying legally permitted weapons, in 2014 a gun owners group challenged the no-guns policy of the North Carolina State Fair. Troxler enforced the ban that year while court cases were litigated. A Wake County Superior Court upheld the ban at the fair later that year. By early 2015, some legislators introduced bills to allow guns, while others introduced a bill to maintain the ban on carrying guns at the fair. The ban continues as of 2023.

title: Steve Troxler


text:
2000 in LGBT rights

This is a list of notable events in the history of LGBT rights that took place in the year 2000.

Events

 * Voters in US state of Maine reject a proposal to reinstate the ban on sexual orientation discrimination in the private sector, with 50.5 per cent against and 49.5 per cent for.

January

 * 12 – The ban on lesbians and gay men serving in the United Kingdom armed forces is lifted.

February

 * 14 – Jerrold Nadler, congressman from New York, introduces the Permanent Partners Immigration Act (H.R 3650) in the United States Congress.

March

 * 7 – California voters approve Proposition 22, a preemptive measure stating that California will not recognize same-sex marriages, even if the marriages took place in states that permitted them.
 * 15 – Equality Mississippi is formed in response to a hate-crime murder and attempts in the U.S. state to ban adoption by same-sex couples.

April

 * 28—30 – Millennium March on Washington in the United States.

June

 * 21 – The parliament of Scotland repeals Section 28, the law that banned local authorities from "promoting homosexuality". It is the first part of the United Kingdom to do so.

July

 * 1
 * In the U.S. state of Vermont, civil union law goes into effect, granting most state-level marriage rights to registered same-sex couples.
 * Mississippi becomes the third U.S. state to ban adoption by same-sex couples.

November

 * Montana governor Marc Racicot issues an executive order banning sexual orientation discrimination in the public sector.
 * The Connecticut Commission on Human Rights and Opportunities, in In re John/Jane Doe, rules that gender identity discrimination is included in the existing ban on sex discrimination in the private sector.

December

 * The age of consent across the United Kingdom is equalised at 16. Previously, the age of consent was 18 for homosexual acts and 16 for heterosexual acts. The equalisation took place after a long struggle, in which the Labour party eventually invoked the Parliament Acts 1911 and 1949 after the House of Lords repeatedly blocked the bill.
 * Tom Vilsack, governor of the U.S. state of Iowa, rescinds an executive order he issued in 1999 banning discrimination based on sexual orientation or gender identity in the public sector. Vilsack would later reinstate the order for sexual orientation only.
 * 15 – Delaware governor Thomas Carper issues an executive order banning sexual orientation discrimination in the public sector.

Deaths

 * December 18 – Hal Call, American businessman and homophile leader (Mattachine Society).

title: 2000 in LGBT rights


text:
Talk:List of physiology topics: respiration

This page is to broad. For a page like this to be helpful it must dichotomize the physiology of respiration. This seems to have the opposite effect. Bensaccount 22:24, 27 Feb 2004 (UTC)

This is just unmanageable...Bensaccount 22:30, 27 Feb 2004 (UTC)

Ok heres a solution: This page becomes Respiration (physiology) and branches respiration into gas exhange and control of gas exchange (and maybe a couple more topics that break apart respiration). There is no need to broaden the subject by adding types of diving and general anatomy.

About the anatomy section:
 * Sure anatomy (aka organization) is necessary to understand physiology (function), but the purpose of making a specific page for the physiology of respiration should be to exclude anatomy (as much as possible). Ideally you should be able to talk about the physiology of respiration and let the text hyperlinks take the reader to the appropriate places to fill in the background knowledge. (this goes for stuff like diving too).

These ideas are good but I still think theres something lacking...Bensaccount 22:44, 27 Feb 2004 (UTC)

My final thoughts:
 * To provide the best organization, it should be that the physiology page links to this page renamed as respiration (physiology). This page should link to the standard (non-physiology) respiration page, which is currently called gas-exchange, and to other pages which subdivide the physiology of respiration, such as various physiological mechanisms of respiration. E.g. mechanisms for the control or respiration. Any subjects that do not subdivide the physiology of respiration should be removed. Bensaccount 22:52, 27 Feb 2004 (UTC)


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Control

Control may refer to:

Economics and business

 * Control (management), an element of management
 * Control, an element of management accounting
 * Comptroller (or controller), a senior financial officer in an organization
 * Controlling interest, a percentage of voting stock shares sufficient to prevent opposition
 * Foreign exchange controls, regulations on trade
 * Internal control, a process to help achieve specific goals typically related to managing risk

Mathematics and science

 * Control (optimal control theory), a variable for steering a controllable system of state variables toward a desired goal
 * Controlling for a variable in statistics
 * Scientific control, an experiment in which "confounding variables" are minimised to reduce error
 * Control variables, variables which are kept constant during an experiment
 * Biological pest control, a natural method of controlling pests
 * Control network in geodesy and surveying, a set of reference points of known geospatial coordinates
 * Control room, a room where a physical facility can be monitored
 * Process control in continuous production processes
 * Security controls, safeguards against security risks

Medicine

 * Control, according to the ICD-10-PCS, in the Medical and Surgical Section (0), is the root operation (# 3) that means stopping, or attempting to stop, post-procedural bleeding
 * Chlordiazepoxide, also sold under the trade name Control
 * Lorazepam, sold under the trade name Control

Systems engineering, computing and technology

 * Automatic control, the application of control theory for regulation of processes without direct intervention
 * Control character, or non-printing character, in a character set; does not represent a written symbol, but is used to control the interpretation or display of text
 * Unicode control characters, characters with no visual or spatial representation
 * Control engineering, a discipline of modeling and controlling of systems
 * Control system, the ability to control some mechanical or chemical equipment
 * Control theory, the mathematical theory about controlling dynamical systems over time
 * Control flow, means of specifying the sequence of operations in computer programs
 * Control variables in programming, which regulate the flow of control
 * Control key, on a computer keyboard
 * GUI widget (control or widget), a component of a graphical user interface
 * Input device (control), a physical user interface to a computer system

Society, psychology and sociology

 * Control (psychology)
 * Locus of control, an extent to which individuals believe that they can control events that affect them
 * Self-control, the ability to control one's emotions and desires
 * Power (social and political), the ability to control others
 * Social control, mechanisms that regulate social behavior
 * Abusive power and control
 * Control freak, a person who attempts to dictate
 * Mind control, the use of manipulative methods to persuade others
 * Civilian control of the military

Other basic uses

 * Control point (orienteering), a marked waypoint in orienteering and related sports
 * Control (linguistics), a relation between elements of two clauses

Geography

 * Control, Alberta

Books

 * Control (novel), a 1982 novel by William Goldman
 * Control (fictional character), in the 1974 British spy novel Tinker, Tailor, Soldier, Spy

Films

 * Control (1987 film) or Il Giorno prima, a 1987 made-for-television film starring Burt Lancaster
 * Control (2004 film), starring Ray Liotta, Willem Dafoe and Michelle Rodriguez
 * Control (2007 film), a film about Joy Division singer Ian Curtis, directed by Anton Corbijn
 * Control (2013 film), a Chinese–Hong Kong film written and directed by Kenneth Bi
 * Control (2023 film), a British film directed by Gene Fallaize and featuring the voice of Kevin Spacey
 * Control (upcoming film), directed by Robert Schwentke and starring James McAvoy
 * Kontroll, a 2003 Hungarian film, released as Control internationally
 * Control, a UK comedy short by Frank Miller

TV

 * Control (House), a 2005 episode of the television series House
 * Control, a Spanish-language series aired on Univision
 * Control, a recurring character in the sketch programme A Bit of Fry & Laurie
 * Control, a character on the science fiction crime drama Person of Interest
 * [C] - The Money of Soul And Possibility Control, or [C] - Control, a 2011 anime
 * Ctrl (web series), an American comedy web series
 * CONTROL (Get Smart), a fictional counter-espionage agency

Games

 * Control and control-bid, features of the game contract bridge
 * Control (video game), a 2019 video game by Remedy Entertainment

Music

 * Control (Starlight Express), a character from the rock musical

Albums

 * Control (GoodBooks album), 2007
 * Control (Janet Jackson album), 1986
 * Control (Pedro the Lion album), 2002
 * Control, a 2011 album by Abandon
 * Control, a 2014 album by The Brew
 * Control, a 1981 album by Conrad Schnitzler
 * Control, a 2013 EP by Disclosure
 * Control, a 1994 album by Hellnation
 * Control, a 2012 EP by The Indecent
 * Control, a 1971 album by John St Field
 * Control, a 2012 album by Uppermost
 * Control, a 2003 album by Where Fear and Weapons Meet
 * Ctrl (SZA album), 2017

Songs

 * "Control" (Big Sean song), 2013
 * "Control" (Garbage song), 2012
 * "Control" (Janet Jackson song), 1986
 * "Control" (Kid Sister song), 2007
 * "Control" (Matrix & Futurebound song), 2013
 * "Control" (Metro Station song), 2007
 * "Control" (Mutemath song), 2004
 * "Control" (Poe song), 1998
 * "Control" (Puddle of Mudd song), 2001
 * "Control" (Traci Lords song), 1994
 * "Control" (Zoe Wees song), 2020
 * "Control", by Basement from Colourmeinkindness, 2012
 * "Control", by the Black Dahlia Murder from Everblack, 2013
 * "Control", by Delta Goodrem from Child of the Universe, 2012
 * "Control", by Disclosure from The Face, 2012
 * "Control", by División Minúscula, 2008
 * "Control", by Doja Cat from Purrr!, 2014
 * "Control", by Earshot from Two, 2004
 * "Control", by Feder, 2018
 * "Control", by Ghost9 from Now: Who We Are Facing, 2021
 * "Control", by Halsey from Badlands, 2015
 * "Control", by London Grammar from Truth Is a Beautiful Thing, 2017
 * "Control", by Playboi Carti from Whole Lotta Red, 2020
 * "Control", by Poe from Haunted, 2000
 * "Control", by Stabbing Westward from Ungod, 1994
 * "Control", by Wisin from El Regreso del Sobreviviente, 2014
 * "Control (Somehow You Want Me)", by Tenth Avenue North from Followers, 2016

title: Control


text:
Thue–Morse sequence

In mathematics, the Thue–Morse or Prouhet–Thue–Morse sequence is the binary sequence (an infinite sequence of 0s and 1s) that can be obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. It is sometimes called the fair share sequence because of its applications to fair division or parity sequence. The first few steps of this procedure yield the strings 0, 01, 0110, 01101001, 0110100110010110, and so on, which are the prefixes of the Thue–Morse sequence. The full sequence begins:

 * 01101001100101101001011001101001....

The sequence is named after Axel Thue and Marston Morse.

Definition
There are several equivalent ways of defining the Thue–Morse sequence.

Direct definition

To compute the nth element tn, write the number n in binary. If the number of ones in this binary expansion is odd then tn = 1, if even then tn = 0. That is, tn is the even parity bit for n. John H. Conway et al. deemed numbers n satisfying tn = 1 to be odious (intended to be similar to odd) numbers, and numbers for which tn = 0 to be evil (similar to even) numbers.

Fast sequence generation
This method leads to a fast method for computing the Thue–Morse sequence: start with $t_{0} = 0$, and then, for each n, find the highest-order bit in the binary representation of n that is different from the same bit in the representation of $n − 1$. If this bit is at an even index, tn differs from $t_{n − 1}$, and otherwise it is the same as $t_{n − 1}$.

In Python: The resulting algorithm takes constant time to generate each sequence element, using only a logarithmic number of bits (constant number of words) of memory.

Recurrence relation
The Thue–Morse sequence is the sequence tn satisfying the recurrence relation

 * $$\begin{align}

t_0 &= 0,\\ t_{2n} &= t_n,\\ t_{2n+1} &= 1 - t_n, \end{align}$$

for all non-negative integers n.

L-system

The Thue–Morse sequence is a morphic word: it is the output of the following Lindenmayer system:

Characterization using bitwise negation
The Thue–Morse sequence in the form given above, as a sequence of bits, can be defined recursively using the operation of bitwise negation. So, the first element is 0. Then once the first 2n elements have been specified, forming a string s, then the next 2n elements must form the bitwise negation of s. Now we have defined the first 2n+1 elements, and we recurse.

Spelling out the first few steps in detail:
 * We start with 0.
 * The bitwise negation of 0 is 1.
 * Combining these, the first 2 elements are 01.
 * The bitwise negation of 01 is 10.
 * Combining these, the first 4 elements are 0110.
 * The bitwise negation of 0110 is 1001.
 * Combining these, the first 8 elements are 01101001.
 * And so on.

So
 * T0 = 0.
 * T1 = 01.
 * T2 = 0110.
 * T3 = 01101001.
 * T4 = 0110100110010110.
 * T5 = 01101001100101101001011001101001.
 * T6 = 0110100110010110100101100110100110010110011010010110100110010110.
 * And so on.

Infinite product
The sequence can also be defined by:
 * $$ \prod_{i=0}^{\infty} \left(1 - x^{2^i}\right) = \sum_{j=0}^{\infty} (-1)^{t_j} x^j,$$

where tj is the jth element if we start at j = 0.

Properties
The Thue–Morse sequence contains many squares: instances of the string $$XX$$, where $$X$$ denotes the string $$A$$, $$\overline{A}$$, $$A \overline{A}A$$, or $$\overline{A}A \overline{A}$$, where $$A=T_{k}$$ for some $$k\ge 0$$ and $$\overline{A}$$ is the bitwise negation of $$A$$. For instance, if $$k=0$$, then $$A=T_{0}=0 $$. The square $$A \overline{A}AA \overline{A}A = 010010$$ appears in $$T$$ starting at the 16th bit. Since all squares in $$T$$ are obtained by repeating one of these 4 strings, they all have length $$2^n$$ or $$3\cdot2^n$$ for some $$n\ge 0$$. $$T$$ contains no cubes: instances of $$XXX$$. There are also no overlapping squares: instances of $$0X0X0$$ or $$1X1X1$$. The critical exponent of $$T$$ is 2.

The Thue–Morse sequence is a uniformly recurrent word: given any finite string X in the sequence, there is some length nX (often much longer than the length of X) such that X appears in every block of length nX. Notably, the Thue-Morse sequence is uniformly recurrent without being either periodic or eventually periodic (i.e., periodic after some initial nonperiodic segment).

The sequence T2n is a palindrome for any n. Furthermore, let qn be a word obtained by counting the ones between consecutive zeros in T2n. For instance, q1 = 2 and q2 = 2102012. Since Tn does not contain overlapping squares, the words qn are palindromic squarefree words.

The Thue–Morse morphism μ is defined on alphabet {0,1} by the substitution map μ(0) = 01, μ(1) = 10: every 0 in a sequence is replaced with 01 and every 1 with 10. If T is the Thue–Morse sequence, then μ(T) is also T. Thus, T is a fixed point of μ. The morphism μ is a prolongable morphism on the free monoid {0,1}∗ with T as fixed point: T is essentially the only fixed point of μ; the only other fixed point is the bitwise negation of T, which is simply the Thue–Morse sequence on (1,0) instead of on (0,1). This property may be generalized to the concept of an automatic sequence.

The generating series of T over the binary field is the formal power series

 * $$t(Z) = \sum_{n=0}^\infty T(n) Z^n \ . $$

This power series is algebraic over the field of rational functions, satisfying the equation

 * $$Z + (1+Z)^2 t + (1+Z)^3 t^2 = 0 $$

In combinatorial game theory
The set of evil numbers (numbers $$n$$ with $$t_n=0$$) forms a subspace of the nonnegative integers under nim-addition (bitwise exclusive or). For the game of Kayles, evil nim-values occur for few (finitely many) positions in the game, with all remaining positions having odious nim-values.

The Prouhet–Tarry–Escott problem
The Prouhet–Tarry–Escott problem can be defined as: given a positive integer N and a non-negative integer k, partition the set S = { 0, 1, ..., N-1 } into two disjoint subsets S0 and S1 that have equal sums of powers up to k, that is:
 * $$ \sum_{x \in S_0} x^i = \sum_{x \in S_1} x^i$$ for all integers i from 1 to k.

This has a solution if N is a multiple of 2k+1, given by:
 * S0 consists of the integers n in S for which tn = 0,
 * S1 consists of the integers n in S for which tn = 1.

For example, for N = 8 and k = 2,
 * 0 + 3 + 5 + 6 = 1 + 2 + 4 + 7,
 * 02 + 32 + 52 + 62 = 12 + 22 + 42 + 72.

The condition requiring that N be a multiple of 2k+1 is not strictly necessary: there are some further cases for which a solution exists. However, it guarantees a stronger property: if the condition is satisfied, then the set of kth powers of any set of N numbers in arithmetic progression can be partitioned into two sets with equal sums. This follows directly from the expansion given by the binomial theorem applied to the binomial representing the nth element of an arithmetic progression.

For generalizations of the Thue–Morse sequence and the Prouhet–Tarry–Escott problem to partitions into more than two parts, see Bolker, Offner, Richman and Zara, "The Prouhet–Tarry–Escott problem and generalized Thue–Morse sequences".

Fractals and turtle graphics
Using turtle graphics, a curve can be generated if an automaton is programmed with a sequence. When Thue–Morse sequence members are used in order to select program states:

 * If t(n) = 0, move ahead by one unit,
 * If t(n) = 1, rotate by an angle of π/3 radians (60°)

The resulting curve converges to the Koch curve, a fractal curve of infinite length containing a finite area. This illustrates the fractal nature of the Thue–Morse Sequence.

It is also possible to draw the curve precisely using the following instructions:

 * If t(n) = 0, rotate by an angle of π radians (180°),
 * If t(n) = 1, move ahead by one unit, then rotate by an angle of π/3 radians.

Equitable sequencing
In their book on the problem of fair division, Steven Brams and Alan Taylor invoked the Thue–Morse sequence but did not identify it as such. When allocating a contested pile of items between two parties who agree on the items' relative values, Brams and Taylor suggested a method they called balanced alternation, or ''taking turns taking turns taking turns. . . '', as a way to circumvent the favoritism inherent when one party chooses before the other. An example showed how a divorcing couple might reach a fair settlement in the distribution of jointly-owned items. The parties would take turns to be the first chooser at different points in the selection process: Ann chooses one item, then Ben does, then Ben chooses one item, then Ann does.

Lionel Levine and Katherine E. Stange, in their discussion of how to fairly apportion a shared meal such as an Ethiopian dinner, proposed the Thue–Morse sequence as a way to reduce the advantage of moving first. They suggested that “it would be interesting to quantify the intuition that the Thue–Morse order tends to produce a fair outcome.”

Robert Richman addressed this problem, but he too did not identify the Thue–Morse sequence as such at the time of publication. He presented the sequences Tn as step functions on the interval [0,1] and described their relationship to the Walsh and Rademacher functions. He showed that the nth derivative can be expressed in terms of Tn. As a consequence, the step function arising from Tn is orthogonal to polynomials of order n − 1. A consequence of this result is that a resource whose value is expressed as a monotonically decreasing continuous function is most fairly allocated using a sequence that converges to Thue–Morse as the function becomes flatter. An example showed how to pour cups of coffee of equal strength from a carafe with a nonlinear concentration gradient, prompting a whimsical article in the popular press.

Joshua Cooper and Aaron Dutle showed why the Thue–Morse order provides a fair outcome for discrete events. They considered the fairest way to stage a Galois duel, in which each of the shooters has equally poor shooting skills. Cooper and Dutle postulated that each dueler would demand a chance to fire as soon as the other's a priori probability of winning exceeded their own. They proved that, as the duelers’ hitting probability approaches zero, the firing sequence converges to the Thue–Morse sequence. In so doing, they demonstrated that the Thue–Morse order produces a fair outcome not only for sequences Tn of length 2n, but for sequences of any length.

Thus the mathematics supports using the Thue–Morse sequence instead of alternating turns when the goal is fairness but earlier turns differ monotonically from later turns in some meaningful quality, whether that quality varies continuously or discretely.

Sports competitions form an important class of equitable sequencing problems, because strict alternation often gives an unfair advantage to one team. Ignacio Palacios-Huerta proposed changing the sequential order to Thue–Morse to improve the ex post fairness of various tournament competitions, such as the kicking sequence of a penalty shoot-out in soccer. He did a set of field experiments with pro players and found that the team kicking first won 60% of games using ABAB (or T1), 54% using ABBA (or T2), and 51% using full Thue–Morse (or Tn). As a result, ABBA is undergoing extensive trials in FIFA (European and World Championships) and English Federation professional soccer (EFL Cup). An ABBA serving pattern has also been found to improve the fairness of tennis tie-breaks. In competitive rowing, T2 is the only arrangement of port- and starboard-rowing crew members that eliminates transverse forces (and hence sideways wiggle) on a four-membered coxless racing boat, while T3 is one of only four rigs to avoid wiggle on an eight-membered boat.

Fairness is especially important in player drafts. Many professional sports leagues attempt to achieve competitive parity by giving earlier selections in each round to weaker teams. By contrast, fantasy football leagues have no pre-existing imbalance to correct, so they often use a “snake” draft (forward, backward, etc.; or T1). Ian Allan argued that a “third-round reversal” (forward, backward, backward, forward, etc.; or T2) would be even more fair. Richman suggested that the fairest way for “captain A” and “captain B” to choose sides for a pick-up game of basketball mirrors T3:  captain A has the first, fourth, sixth, and seventh choices, while captain B has the second, third, fifth, and eighth choices.

Hash collisions
The initial $2^{k}$ bits of the Thue–Morse sequence are mapped to 0 by a wide class of polynomial hash functions modulo a power of two, which can lead to hash collisions.

Riemann zeta function
Certain linear combinations of Dirichlet series whose coefficients are terms of the Thue–Morse sequence give rise to identities involving the Riemann Zeta function (Tóth, 2022 ). For instance:
 * $$ \begin{align}

\sum_{n\geq1} \frac{5 t_{n-1} + 3 t_n}{n^2} &= 4 \zeta(2) = \frac{2 \pi^2}{3}, \\ \sum_{n\geq1} \frac{9 t_{n-1} + 7 t_n}{n^3} &= 8 \zeta(3),\end{align}$$ where $$(t_n)_{n\geq0}$$ is the $$n^{\rm th}$$ term of the Thue-Morse sequence. In fact, for all $$s$$ with real part greater than $$1$$, we have
 * $$ (2^s+1) \sum_{n\geq1} \frac{t_{n-1}}{n^s} + (2^s-1) \sum_{n\geq1} \frac{t_{n}}{n^s} = 2^s \zeta(s).$$

History
The Thue–Morse sequence was first studied by Eugène Prouhet in 1851, who applied it to number theory. However, Prouhet did not mention the sequence explicitly; this was left to Axel Thue in 1906, who used it to found the study of combinatorics on words. The sequence was only brought to worldwide attention with the work of Marston Morse in 1921, when he applied it to differential geometry. The sequence has been discovered independently many times, not always by professional research mathematicians; for example, Max Euwe, a chess grandmaster and mathematics teacher, discovered it in 1929 in an application to chess: by using its cube-free property (see above), he showed how to circumvent the threefold repetition rule aimed at preventing infinitely protracted games by declaring repetition of moves a draw. At the time, consecutive identical board states were required to trigger the rule; the rule was later amended to the same board position reoccurring three times at any point, as the sequence shows that the consecutive criterion can be evaded forever.

title: Thue–Morse sequence