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Isospin

In nuclear physics and particle physics, isospin (I) is a quantum number related to the up- and down quark content of the particle. Isospin is also known as isobaric spin or isotopic spin. Isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions of baryons and mesons.

The name of the concept contains the term spin because its quantum mechanical description is mathematically similar to that of angular momentum (in particular, in the way it couples; for example, a proton–neutron pair can be coupled either in a state of total isospin 1 or in one of 0 ). But unlike angular momentum, it is a dimensionless quantity and is not actually any type of spin.

Before the concept of quarks was introduced, particles that are affected equally by the strong force but had different charges (e.g. protons and neutrons) were considered different states of the same particle, but having isospin values related to the number of charge states. A close examination of isospin symmetry ultimately led directly to the discovery and understanding of quarks and to the development of Yang–Mills theory. Isospin symmetry remains an important concept in particle physics.

Isospin invariance
To a good approximation the proton and neutron have the same mass: they can be interpreted as two states of the same particle. These states have different values for an internal isospin coordinate. The mathematical properties of this coordinate are completely analogous to intrinsic spin angular momentum. The component of the operator, $$\hat{T}_3$$, for this coordinate has eigenvalues +$1⁄2$ and −$1⁄2$; it is related to the charge operator, $$\hat{Q}$$: $$\hat{Q} = e(\hat{T}_3 + \frac{1}{2}) $$ which has eigenvalues $$e$$ for the proton and zero for the neutron. For a system of n nucleons, the charge operator depends upon the mass number A: $$\hat{Q} = e(\hat{T}_3 + \frac{1}{2}A) $$ Isobars, nuclei with the same mass number like 40K and 40Ar, only differ in the value of the $$\hat{T}_3$$ eigenvalue. For this reason isospin is also called "isobaric spin".

The internal structure of these nucleons is governed by the strong interaction, but the Hamiltonian of the strong interaction is isospin invariant. As a consequence the nuclear forces are charge independent. Properties like the stability of deuterium can be predicted based on isospin analysis. However, this invariance is not exact and the quark model gives more precise results.

Relation to hypercharge
The charge operator can be expressed in terms of the projection of isospin $$T_3$$ and hypercharge, $$Y$$: $$Q=\frac{1}{2}Y+T_3, \ \ \ \ T_3=T, T- 1,...,-T.$$ This is known as the Gell-Mann-Nishijima formula. The hypercharge is the center of splitting for the isospin multiplet: $$\frac{1}{2}Y = \frac{1}{2}(Q_{\textrm{min}}+Q_{\textrm{max}}).$$ This relation has an analog in the weak interaction where T is the weak isospin.

Quark content and isospin
In the modern formulation, isospin ($I$) is defined as a vector quantity in which up and down quarks have a value of $I$ = $1⁄2$, with the 3rd-component ($I$3) being +$1⁄2$ for up quarks, and −$1⁄2$ for down quarks, while all other quarks have $I$ = 0. Therefore, for hadrons in general, where $n$u and $n$d are the numbers of up and down quarks respectively,
 * $$I_3 = \frac{1}{2}(n_u - n_d).$$

In any combination of quarks, the 3rd component of the isospin vector ($I$3) could either be aligned between a pair of quarks, or face the opposite direction, giving different possible values for total isospin for any combination of quark flavours. Hadrons with the same quark content but different total isospin can be distinguished experimentally, verifying that flavour is actually a vector quantity, not a scalar (up vs down simply being a projection in the quantum mechanical $z$ axis of flavour space).

For example, a strange quark can be combined with an up and a down quark to form a baryon, but there are two different ways the isospin values can combine – either adding (due to being flavour-aligned) or cancelling out (due to being in opposite flavour directions). The isospin-1 state (the Sigma0) and the isospin-0 state (the Lambda0) have different experimentally detected masses and half-lives.

Isospin and symmetry
Isospin is regarded as a symmetry of the strong interaction under the action of the Lie group SU(2), the two states being the up flavour and down flavour. In quantum mechanics, when a Hamiltonian has a symmetry, that symmetry manifests itself through a set of states that have the same energy (the states are described as being degenerate). In simple terms, the energy operator for the strong interaction gives the same result when an up quark and an otherwise identical down quark are swapped around.

Like the case for regular spin, the isospin operator I is vector-valued: it has three components Ix, Iy, Iz, which are coordinates in the same 3-dimensional vector space where the 3 representation acts. Note that this vector space has nothing to do with the physical space, except similar mathematical formalism. Isospin is described by two quantum numbers: $I$ – the total isospin, and $I$3 –  an eigenvalue of the Iz projection for which flavor states are eigenstates. In other words, each $I$3 state specifies certain flavor state of a multiplet. The third coordinate ($z$), to which the "3" subscript refers, is chosen due to notational conventions that relate bases in 2 and 3 representation spaces. Namely, for the spin-$1⁄2$ case, components of I are equal to Pauli matrices divided by 2, and so Iz = $1⁄2$ $τ$3, where

 * $$\tau_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$

While the forms of these matrices are isomorphic to those of spin, these Pauli matrices only act within the Hilbert space of isospin, not that of spin, and therefore is common to denote them with τ rather than σ to avoid confusion.

Although isospin symmetry is actually very slightly broken, SU(3) symmetry is more badly broken, due to the much higher mass of the strange quark compared to the up and down. The discovery of charm, bottomness and topness could lead to further expansions up to SU(6) flavour symmetry, which would hold if all six quarks were identical. However, the very much larger masses of the charm, bottom, and top quarks means that SU(6) flavour symmetry is very badly broken in nature (at least at low energies), and assuming this symmetry leads to qualitatively and quantitatively incorrect predictions. In modern applications, such as lattice QCD, isospin symmetry is often treated as exact for the three light quarks (uds), while the three heavy quarks (cbt) must be treated separately.

Hadron nomenclature
Hadron nomenclature is based on isospin.
 * Particles of total isospin $3⁄2$ are named Delta baryons and can be made by a combination of any three up or down quarks (but only up or down quarks).
 * Particles of total isospin 1 can be made from two up quarks, two down quarks, or one of each:
 * certain mesons – further differentiated by total spin into pions (total spin 0) and rho mesons (total spin 1)
 * with an additional quark of higher flavour – Sigma baryons
 * Particles of total isospin $1⁄2$ can be made from:
 * a single up or down quark together with an additional quark of higher flavour – strange (kaons), charm (D meson), or bottom (B meson)
 * a single up or down quark together with two additional quarks of higher flavour – Xi baryon
 * an up quark, a down quark, and either an up or a down quark – nucleons. Note that three identical quarks would be forbidden by the Pauli exclusion principle due to requirement of anti-symmetric wave function
 * Particles of total isospin 0 can be made from
 * a neutral quark-antiquark pair: $$\mathrm{u\bar{u}}$$ or $$\mathrm{d\bar{d}}$$ – eta mesons
 * one up quark and one down quark, with an additional quark of higher flavour – Lambda baryons
 * anything not involving any up or down quarks

Origin of isospin
In 1932, Werner Heisenberg introduced a new (unnamed) concept to explain binding of the proton and the then newly discovered neutron (symbol n). His model resembled the bonding model for molecule Hydrogen ion, H2+: a single electron was shared by two protons. Heisenberg's theory had several problems, most notable it incorrectly predicted the exceptionally strong binding energy of He+2, alpha particles. However, its equal treatment of the proton and neutron gained significance when several experimental studies showed these particles must bind almost equally. In response, Eugene Wigner used Heisenberg's concept in his 1937 paper where he introduced the term "isotopic spin" to indicate how the concept is similar to spin in behavior.

The particle zoo
These considerations would also prove useful in the analysis of meson-nucleon interactions after the discovery of the pions in 1947. The three pions could be assigned to an isospin triplet with $I = 1$ and $I_{3} = +1, 0 or −1$. By assuming that isospin was conserved by nuclear interactions, the new mesons were more easily accommodated by nuclear theory.

As further particles were discovered, they were assigned into isospin multiplets according to the number of different charge states seen: 2 doublets $I = 1⁄2$ of K mesons, , a triplet $I = 1$ of Sigma baryons , a singlet $I = 0$ Lambda baryon , a quartet $I = 3⁄2$ Delta baryons , and so on.

The power of isospin symmetry and related methods comes from the observation that families of particles with similar masses tend to correspond to the invariant subspaces associated with the irreducible representations of the Lie algebra SU(2). In this context, an invariant subspace is spanned by basis vectors which correspond to particles in a family. Under the action of the Lie algebra SU(2), which generates rotations in isospin space, elements corresponding to definite particle states or superpositions of states can be rotated into each other, but can never leave the space (since the subspace is in fact invariant). This is reflective of the symmetry present. The fact that unitary matrices will commute with the Hamiltonian means that the physical quantities calculated do not change even under unitary transformation. In the case of isospin, this machinery is used to reflect the fact that the mathematics of the strong force behaves the same if a proton and neutron are swapped around (in the modern formulation, the up and down quark).

An example: Delta baryons
For example, the particles known as the Delta baryons – baryons of spin $3⁄2$ –  were grouped together because they all have nearly the same mass (approximately $1,232 MeV/c2$) and interact in nearly the same way.

They could be treated as the same particle, with the difference in charge being due to the particle being in different states. Isospin was introduced in order to be the variable that defined this difference of state. In an analogue to spin, an isospin projection (denoted $I_{3}$) is associated to each charged state; since there were four Deltas, four projections were needed. Like spin, isospin projections were made to vary in increments of 1. Hence, in order to have four increments of 1, an isospin value of $3⁄2$ is required (giving the projections $I_{3} = +3⁄2, +1⁄2, −1⁄2, −3⁄2$). Thus, all the Deltas were said to have isospin $I = 3⁄2$, and each individual charge had different $I_{3}$ (e.g. the was associated with $I_{3} = +3⁄2$).

In the isospin picture, the four Deltas and the two nucleons were thought to simply be the different states of two particles. The Delta baryons are now understood to be made of a mix of three up and down quarks – uuu, uud , udd , and ddd ; the difference in charge being difference in the charges of up and down quarks (+$2⁄3$ e and −$1⁄3$ e respectively); yet, they can also be thought of as the excited states of the nucleons.

Gauged isospin symmetry
Attempts have been made to promote isospin from a global to a local symmetry. In 1954, Chen Ning Yang and Robert Mills suggested that the notion of protons and neutrons, which are continuously rotated into each other by isospin, should be allowed to vary from point to point. To describe this, the proton and neutron direction in isospin space must be defined at every point, giving local basis for isospin. A gauge connection would then describe how to transform isospin along a path between two points.

This Yang–Mills theory describes interacting vector bosons, like the photon of electromagnetism. Unlike the photon, the SU(2) gauge theory would contain self-interacting gauge bosons. The condition of gauge invariance suggests that they have zero mass, just as in electromagnetism.

Ignoring the massless problem, as Yang and Mills did, the theory makes a firm prediction: the vector particle should couple to all particles of a given isospin universally. The coupling to the nucleon would be the same as the coupling to the kaons. The coupling to the pions would be the same as the self-coupling of the vector bosons to themselves.

When Yang and Mills proposed the theory, there was no candidate vector boson. J. J. Sakurai in 1960 predicted that there should be a massive vector boson which is coupled to isospin, and predicted that it would show universal couplings. The rho mesons were discovered a short time later, and were quickly identified as Sakurai's vector bosons. The couplings of the rho to the nucleons and to each other were verified to be universal, as best as experiment could measure. The fact that the diagonal isospin current contains part of the electromagnetic current led to the prediction of rho-photon mixing and the concept of vector meson dominance, ideas which led to successful theoretical pictures of GeV-scale photon-nucleus scattering.

The introduction of quarks

The discovery and subsequent analysis of additional particles, both mesons and baryons, made it clear that the concept of isospin symmetry could be broadened to an even larger symmetry group, now called flavor symmetry. Once the kaons and their property of strangeness became better understood, it started to become clear that these, too, seemed to be a part of an enlarged symmetry that contained isospin as a subgroup. The larger symmetry was named the Eightfold Way by Murray Gell-Mann, and was promptly recognized to correspond to the adjoint representation of SU(3). To better understand the origin of this symmetry, Gell-Mann proposed the existence of up, down and strange quarks which would belong to the fundamental representation of the SU(3) flavor symmetry.

In the quark model, the isospin projection (I3) followed from the up and down quark content of particles; uud for the proton and udd for the neutron. Technically, the nucleon doublet states are seen to be linear combinations of products of 3-particle isospin doublet states and spin doublet states. That is, the (spin-up) proton wave function, in terms of quark-flavour eigenstates, is described by

$$\vert \mathrm{p}\uparrow \rangle = \frac 1{3\sqrt 2}\left(\begin{array}{ccc} \vert \mathrm{duu}\rangle & \vert \mathrm{udu}\rangle & \vert \mathrm{uud}\rangle \end{array}\right) \left(\begin{array}{ccc} 2 & -1 & -1\\ -1 & 2 & -1\\ -1 & -1 & 2 \end{array}\right) \left(\begin{array}{c} \left\vert\downarrow\uparrow\uparrow\right\rangle\\ \left\vert\uparrow\downarrow\uparrow\right\rangle\\ \left\vert\uparrow\uparrow\downarrow\right\rangle \end{array}\right)$$

and the (spin-up) neutron by

$$\vert \mathrm{n}\uparrow \rangle = \frac 1{3\sqrt 2}\left(\begin{array}{ccc} \vert \mathrm{udd}\rangle & \vert \mathrm{dud}\rangle & \vert \mathrm{ddu}\rangle \end{array}\right) \left(\begin{array}{ccc} 2 & -1 & -1\\ -1 & 2 & -1\\ -1 & -1 & 2 \end{array}\right) \left(\begin{array}{c} \left\vert\downarrow\uparrow\uparrow\right\rangle\\ \left\vert\uparrow\downarrow\uparrow\right\rangle\\ \left\vert\uparrow\uparrow\downarrow\right\rangle \end{array}\right).$$

Here, $$\mathrm{\vert u \rangle}$$ is the up quark flavour eigenstate, and $$\mathrm{\vert d \rangle}$$ is the down quark flavour eigenstate, while $$\left\vert\uparrow\right\rangle$$ and $$\left\vert\downarrow\right\rangle$$ are the eigenstates of $$S_z$$. Although these superpositions are the technically correct way of denoting a proton and neutron in terms of quark flavour and spin eigenstates, for brevity, they are often simply referred to as "uud" and "udd". The derivation above assumes exact isospin symmetry and is modified by SU(2)-breaking terms.

Similarly, the isospin symmetry of the pions are given by:

$$\begin{align} \vert \pi^+\rangle &= \vert \mathrm{u\overline {d}}\rangle \\ \vert \pi^0\rangle &= \tfrac{1}{\sqrt{2}}\left(\vert \mathrm{u\overline {u}}\rangle - \vert \mathrm{d \overline{d}} \rangle \right) \\ \vert \pi^-\rangle &= -\vert \mathrm{d\overline {u}}\rangle. \end{align}$$

Although the discovery of the quarks led to reinterpretation of mesons as a vector bound state of a quark and an antiquark, it is sometimes still useful to think of them as being the gauge bosons of a hidden local symmetry.

Weak isospin
In 1961 Sheldon Glashow proposed that as relation similar to the Gell-Mann–Nishijima formula for charge to isospin would also apply to the weak interaction: $$Q = T_3 + \frac{1}{2}Y_w.$$ Here the charge $$Q$$ is related to the projection of weak isospin $$T_3$$ and the weak hypercharge $$Y_w$$. Isospin and weak isospin are related to the same symmetry but for different forces. Weak isospin is the gauge symmetry of the weak interaction which connects quark and lepton doublets of left-handed particles in all generations; for example, up and down quarks, top and bottom quarks, electrons and electron neutrinos. By contrast (strong) isospin connects only up and down quarks, acts on both chiralities (left and right) and is a global (not a gauge) symmetry.

title: Isospin


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Dean Headley

Dean Warren Headley (born 27 January 1970) is a former English professional cricketer who played as a right-arm fast bowler for the England cricket team. Domestically he played for Middlesex and Kent County Cricket Clubs. Headley is now an ECB match referee and the current director of cricket at Blundell's School.

Playing career
Headley was born in Stourbridge and comes from a famous cricketing family, being the son of Ron Headley and grandson of George Headley. He was the first Test cricketer to be both the son and grandson of Test cricketers. He was educated at Old Swinford Hospital and then Royal Grammar School Worcester where he excelled at rugby union and cricket.

He played for Worcestershire's second team in 1989, and played as a professional for Leycett Cricket Club based in Leycett just outside Newcastle-under-Lyme, Staffordshire, but his senior debut was in 1991 after he had moved to Middlesex. Clive Lloyd helped him a lot in those early years. In 1993 he moved counties again, this time to Kent. Whilst at Kent he was selected for England, playing Test and ODI cricket between 1996 and 1999. One of his greatest moments was when he won the Man of the Match award for his performance in the 4th Ashes Test in 1998 at Melbourne: Australia was making light work of chasing a target of 175, before Headley took four wickets in three overs to reduce Australia from 130/3 to 140/7; he finished with 6/60 in the innings and England won by 12 runs, their only Test victory on that tour. Due to injury he retired from cricket in 2001.

He played club cricket for Upchurch Cricket Club in Upchurch, Kent and worked as a cricket coach at Stamford School.

World record
On 14 September 1996, Dean Headley equalled a world record by taking three hat-tricks in a single English county season. He achieved this feat against Hampshire in Canterbury. He equalled the world record earlier set by Charlie Parker in 1924 and J.S. Rao in 1963–64.

Personal life
The father of three children, Headley enjoys rugby, football, hockey, and golf in his spare time. He's also a fan of English football side West Bromwich Albion.

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

title: Dean Headley


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Doug Williams (wrestler)

Douglas Clayton Durdle (born 1 September 1972), known by his ring name Doug Williams, is an English professional wrestler. He is best known for his work in Total Nonstop Action Wrestling, where he was a one–time Television Champion, two–time TNA X Division Champion and a two-time tag team champion, having held the TNA World Tag Team Championship and the IWGP Tag Team Championship once each as part of The British Invasion. He also worked as a trainer for TNA's developmental territory Ohio Valley Wrestling (OVW), where he was a one-time OVW Heavyweight Champion. Throughout his career, Williams has also worked in various promotions in the United Kingdom and abroad, including the Frontier Wrestling Alliance (FWA), One Pro Wrestling (1PW) and Pro Wrestling Noah.

Early career
Williams studied judo from the age of 11. Using this as his background, he began training to become a professional wrestler at the age of 16. At 19 Williams started working with NWA UK Hammerlock and became an active member of the roster during the mid-1990s, before moving on to FWA in 2001.

While competing in FWA's King of England tournament, Williams wrestled and won a match against Eddie Guerrero. Williams would also go on to become the promotion's top champion on two occasions. Williams wrestled against Martin Stone in the main event of the promotion's comeback show "New Frontiers" in August 2009, and has since become embroiled in the ongoing feud between Stone's faction "The Agenda" and the FWA Resistance.

By 2004, Williams would begin touring with Pro Wrestling Noah in Japan where he would become a Global Honored Crown Tag Team Champion with Scorpio.

Williams continued to win many foreign and local championships, even defeating "American Dragon" Bryan Danielson in the final of a tournament in Rethel, France on 6 May 2005 to become the first International Catch Wrestling Alliance World Heavyweight Champion.

While World Wrestling Entertainment was touring England in November 2006, Williams made a dark match appearance where he lost to Carlito.

Williams would also appear occasionally on television throughout 2007 and 2008 with British wrestling promotions whilst working with TNA Wrestling, due to the ruling their contracted wrestlers can appear in independent promotions that aren't televised in the USA.

Ring of Honor (2002–2007, 2015)
Williams went on a tour around the United States on several occasions making notable appearances in Ring of Honor over the course of several years. He debuted for the promotion on 22 June 2002, defeating American Dragon. This victory earned Doug a shot at the ROH World Championship on 27 July 2002, which was fought under ironman rules in a four-way contest, which also featured Christopher Daniels, Low Ki and Spanky.

His return to Ring of Honor saw Williams regain the FWA British Heavyweight Championship from Christopher Daniels, on 22 March 2003, before unsuccessfully challenging Samoa Joe for the ROH World Championship on 26 April 2003. On 19 July 2003, Reckless Youth defeated Doug Williams in a pure wrestling challenge.

After another stint away, Williams returned on 14 February 2004 to defeat Chris Sabin in the pure wrestling tournament, but was later knocked out by CM Punk. On 17 July, in Elizabeth, New Jersey, Williams defeated Jay Lethal, Nigel McGuinness and R. J. Brewer, to qualify for the Pure Wrestling final. The final was contested later that evening, and saw Williams defeat Alex Shelley to capture the Pure Wrestling Championship. Williams had two successful defences of the Pure Wrestling Championship, defeating Shelley in a rematch on 23 July, and Austin Aries the following day. He would go on to lose the title to John Walters on 28 August.

On 11 September, Williams had another unsuccessful shot at the World Championship, which was also against Samoa Joe. On 13 May 2005, Williams once again wrestled in ROH when he teamed with Colt Cabana to defeat the team of Nigel McGuiness and Chad Collyer, before losing to Homicide the next day.

During ROH's tour of the United Kingdom in August 2006, Williams teamed with Jody Fleisch as "Team UK", defeating "Team Noah" (SUWA and Go Shiozaki). The second show of the tour took place in Broxbourne and saw Williams defeat Jimmy Rave.

Williams' last appearances in ROH were in 2007. On 13 April, Williams teamed with regular partner Nigel McGuinness to face then-World Champion Takeshi Morishima and Chris Hero in a tag team match. McGuinness pinned Hero to win the match for his team. The next night at Fighting Spirit on 14 April, Doug made it two for two by defeating Colt Cabana with his Chaos Theory finisher.

In November 2015, Williams was booked as part of a series of "Supershow" cross-promotion events between Preston City Wrestling and ROH in Preston, England. Williams was victorious against Silas Young on the first night but unsuccessful the second night in a match against Jay Lethal for the ROH World Championship.

Sporadic appearances (2003, 2008)
Williams wrestled James Storm on 26 March 2003 during TNA Xplosion. It was not until Williams was confirmed to have signed a contract with TNA Wrestling from 12 June 2008 that he became part of their roster to take part in TNA Wrestling's World X Cup.

Williams was part of TNA's UK tour, wrestling in all four shows. On 12 June 2008, in Liverpool, Williams teamed with Gail Kim, though they fell short in defeat to the team of Awesome Kong and "Cowboy" James Storm. On 13, 14 and 15 June, TNA held a second show in Liverpool, along with others in Coventry and Brentwood. Doug also wrestled on these shows, with multiple 3 Way Dances against Jay Lethal and A.J. Styles. Styles was victorious on all three occasions.

Williams represented Team International during the 2008 World X Cup tournament, where he defeated Team Japan's Masato Yoshino in his Impact! debut on 10 July 2008. He then wrestled in a 12-Man World X Cup elimination contest, at Victory Road. Williams returned to TNA at Turning Point where he competed in a X Division seeding match. He eliminated Jimmy Rave from the ten-man match, but in the end the eventual winner Eric Young eliminated him placing him third in the X Division rankings. Williams competed on TNA's 2009 Maximum Impact Tour in X Division matches.

British Invasion and World Elite (2009–2010)

Williams re-appeared on TNA television on 30 April 2009, episode of Impact!, as a member of The British Invasion faction, along with Brutus Magnus and Rob Terry. The British Invasion got to the final of the Team 3D Invitational Tag Team Tournament but lost to Beer Money, Inc. (James Storm and Robert Roode) at Sacrifice. In the dark match of Slammiversary, the British Invasion defeated Rhino and Eric Young. Later that evening, they interfered in the TNA World Tag Title match. At Victory Road, Team 3D retained their IWGP World Tag Team Titles against the British Invasion. On 25 July episode of Impact!, the British Invasion, Sheik Abdul Bashir and Kiyoshi defeated the TNA Originals after Eric Young turned on his team. After the match, Young, the British Invasion, Bashir and Kiyoshi formed World Elite. On 30 July episode of Impact!, Williams and Magnus won the IWGP Tag Team Championship by defeating Team 3D in a Tables Match. However, New Japan Pro-Wrestling would state that they still recognised Team 3D as the official champions since TNA did not have proper sanction to make the title change. On 10 August 2009, NJPW formally recognised the title switch. At Bound for Glory the British Invasion lost the IWGP Tag Team Titles back to Team 3D, but managed to win the TNA World Tag Team Titles in a four way Full Metal Mayhem Tag Team match against Booker T and Scott Steiner, Team 3D and Beer Money. The following month at Turning Point Williams and Magnus successfully defended the titles in a three-way match against Beer Money and The Motor City Machineguns (Alex Shelley and Chris Sabin). At Final Resolution they were once again able to retain their titles in a match against Shelley and Sabin. At Genesis the British Invasion lost the TNA World Tag Team Titles to Matt Morgan and Hernandez.

X Division and Television Champion (2010–2011)
On 28 January episode of Impact! Williams cashed in Rob Terry's "Feast or Fired" briefcase and used it to defeat Amazing Red for the TNA X Division Championship. Two days later Williams competed in his first Ultimate X match at a house show in London, successfully defending his new title against Daniels, Suicide, Amazing Red and Chris Sabin. On 18 February episode of Impact! Rob Terry grew tired of his stable mates' abuse and attacked Magnus, thus completing his face turn. At Destination X Williams successfully defended the TNA X Division Championship against Shannon Moore after hitting him with a brick and afterwards cut a promo, where he claimed to hate the current X Division, full of acrobats calling themselves wrestlers, before announcing that he was going to re-establish the division as a place for pure catch-as-catch-can wrestlers like himself. The British Invasion quietly disbanded after Magnus was taken off television following the event. On 5 April episode of Impact! Williams began using the ring name Douglas Williams. At Lockdown Williams was scheduled to defend his title in a three-way Steel Cage match against both Kazarian and Moore, but was forced to miss the event due to being unable to fly out of England after the country's airspace was closed following a volcano eruption in Iceland. At the event TNA announced that as a result of missing the show, Williams was stripped of the TNA X Division Championship. After missing two more weeks of Impact!, Williams returned on 3 May episode of Impact! and refused to hand the Championship belt over to the new champion, Kazarian, and claimed that he had to beat him for it at Sacrifice, to get it. Two weeks later at Sacrifice Williams defeated Kazarian to regain the TNA X Division Championship. The following month at Slammiversary VIII Williams successfully defended his TNA X Division Championship against Brian Kendrick, using a high flying manoeuvre, a tornado DDT, despite his previous anti–high flying statements.

On 3 July 2010, Dixie Carter announced that Durdle had signed a new long–term contract with TNA which also led to him relocating from the United Kingdom to the United States. On 11 July at Victory Road Williams successfully defended his TNA X Division Championship against Brian Kendrick, this time in an Ultimate X submission match. On 5 August episode of Impact! Williams interfered in a Street Fight between Jay Lethal and Ric Flair, helping Flair win the match. The following week Williams joined Flair's Fortune (originally spelled Fourtune) stable, as he, A.J. Styles, Kazarian, Robert Roode, James Storm and Matt Morgan attacked EV 2.0, a stable consisting of former Extreme Championship Wrestling performers. At No Surrender Williams defeated EV 2.0 member Sabu to retain the X Division Championship. On 6 September, at the tapings of 16 September episode of Impact!, Williams lost the X Division Championship to Jay Lethal. Williams received his rematch for the title at Bound for Glory. After failing to finish Lethal with his pure wrestling, Williams resorted to going for a high flying manoeuvre, a Frankensteiner, which ended up backfiring as Lethal countered it into a sunset flip for the victory. On the following episode of Impact! Fortune formed an alliance with Hulk Hogan's and Eric Bischoff's new stable, Immortal.

On 28 October episode of Impact! tensions rose between Williams and the rest of Fortune, in particular his former X Division rival Kazarian, who thought he had a bad attitude. Williams then replaced Kazarian in a tag team match, where he and A.J. Styles defeated EV 2.0 members Rob Van Dam and Raven, after Styles blind tagged Williams and stole his pinfall. Williams then began showing signs of a possible face turn by standing up for Matt Morgan, who was kicked out of Fortune for showing concern for Mr. Anderson, while also voicing his own concern for not being able to spotlight his talent, while a member of Fortune. The following week Flair ordered Williams and Kazarian to settle their differences in the ring. Kazarian won the match between the members of Fortune, after a miscommunication between Williams and James Storm. After the match Flair made Williams and Kazarian shake hands. At Turning Point the seemingly re–united Fortune defeated EV 2.0 in a ten-man tag team match and, as a result, EV 2.0's Sabu was released from TNA. On 11 November episode of Impact! Williams, Roode and Storm faced former Fortune member Matt Morgan in a three–on–one handicap match. During the match Roode and Storm abandoned Williams and left him to be pinned by Morgan. The following week Fortune interfered in a match between Morgan and Ric Flair, however, Williams refused to take part in beating Morgan down. When confronted by Styles, Williams turned face for the first time in his TNA career by attacking the rest of Fortune, helping Morgan clear the ring and costing Flair the match. On 5 December at Final Resolution Williams defeated Styles to win the TNA Television Championship for the first time. On 23 December episode of Impact! Williams defended the Television Championship against Styles in a 20-minute Iron Man match. The match ended in a 1–1 draw, setting up another match at Genesis on 9 January 2011, where Styles would put his spot in Fortune on the line against Williams' title. However, Styles was forced to pull out of the match after suffering a legitimate hip injury and was replaced by his ally Abyss, who proceeded to defeat Williams for the Television Championship, after Styles ran in and hit Williams with the title belt.

British Invasion reunion and singles competition (2011–2012)
On 31 March episode of Impact!, Williams turned heel and reformed the British Invasion with Magnus by attacking the team of Eric Young and Orlando Jordan. In their first match back together, Williams and Magnus were defeated by Ink Inc. (Jesse Neal and Shannon Moore) on 12 April episode of Xplosion. On 17 April at Lockdown, Williams and Magnus were defeated again by Ink Inc. in a four tag team number one contenders' Steel Cage match. Despite not having won a single match since their reformation, it was announced on 2 June episode of Impact Wrestling that the British Invasion would receive a shot at the TNA World Tag Team Championship at Slammiversary IX. At the pay-per-view, Williams and Magnus were unsuccessful in their challenge for the title, losing to the team of James Storm and Alex Shelley, who filled in for Robert Roode, who was sidelined with a storyline injury. On 30 June episode of Impact Wrestling, Williams and Magnus turned face by responding to Mexican America's bad remarks by praising the United States. They were then attacked until Rob Terry returned to save them, effectively adding him back to the group. On 14 July episode of Impact Wrestling, Mexican America's Anarquia and Hernandez defeated Williams and Magnus in a TNA World Tag Team Championship number one contender's match, following outside interference from their stablemate Rosita. In December, Magnus formed a new tag team with Samoa Joe, which eventually led to the dissolution of the British Invasion.

On 29 December episode of Impact Wrestling, Williams was sidelined with a storyline injury, after Gunner dropped him onto the concrete floor with a DDT. Williams returned to Impact Wrestling on 9 February 2012, episode, taped in London, losing to Alex Shelley in a three-way match, which also inincluded TNA X Division Champion Austin Aries. On 8 July at Destination X, Williams entered a tournament to determine the new X Division Champion, but was defeated in his first round match by Kenny King.

Ohio Valley Wrestling (2012–2013)
During the summer of 2012, Williams began working as a trainer in TNA's developmental territory Ohio Valley Wrestling (OVW). In September, Williams also began making appearances for OVW's weekly television program. Despite his new job in OVW, Williams also continued making appearances for TNA, unsuccessfully challenging Zema Ion for the TNA X Division Championship on 4 October episode of Impact Wrestling. On 30 January 2013, Williams defeated Rob Terry to win the OVW Heavyweight Championship for the first time. He held the title until 10 April, when he lost it to Jamin Olivencia. On 5 June 2013, Durdle confirmed that he had been released from his TNA Wrestling contract, which also ended his run with OVW.

Return (2014)
On 1 February 2014, Williams returned to take part in TNA's Impact Wrestling tapings in London, he lost to Ethan Carter III in a singles match. The following day, Williams also took part in the tapings of the Joker's Wild II pay-per-view, where he and Magnus reunited to defeat Chris Sabin and Gunner in a tag team match.

Independent circuit (2014–2018)
After leaving TNA Wrestling, Williams began working for various independent promotions, most notably in United Kingdom. On 4 October, Williams faced AJ Styles in a 3 way bout for the IWGP heavyweight championship but failed to win the title. on 8 November, Doug Williams defeated Tiger Ali to win the 4FW heavyweight title. on 29 December, Williams defeated Josh Faulkner in the semi-finals then defeated Zack Sabre Jr. to win the 2014 Worthing Trophy. On 28 October 2015, Williams and Nick Aldis reunited as The British Invasion as part of Global Force Wrestling's GFW UK Invasion tour, defeating Marty Scurll and Rampage Brown. On 20 May 2018, Douglas defeated Joseph Conners, Rampage Brown and Rob Lynch to win the vacant Progress Atlas Championship. He lost the title on 30 September 2018 at the event Hello Wembley. He was scheduled for a match on 21 December 2018 against Yoshinari Ogawa for Pro Wrestling Noah, in what was considered to be his final advertised match.

Return (2019–2023)
In 2019, Williams began taking bookings in Japan and Germany before committing to a return. On 15 March 2021, Williams announced his return to professional wrestling with the following statement on his twitter; "The events of the past year has made me reassess my decision to retire as it made me understand that you should use your life to fullest doing what you enjoy. As such I am open to bookings again, email is in my bio...". After a 3-year hiatus from the ring, Williams once again returned to the ring, touring the UK Independent circuit under the moniker of "The Ambassador" with companies such as ICW, PROGRESS, 1PW and Rev Pro.

On 17 June 2021, Williams defeated Lookachu in a tournament final for the All Nations Championship at an event promoted jointly by Solent Wrestling Federation and Kreative Alternative Professional Organised Wrestling.

Williams began to wind down his in-ring career in mid 2023. He challenged Andy Roberts for the Union of European Wrestling Alliances European Heavyweight Championship in Scotland for British Championship Wrestling and Martin Steers for the Irish Junior Heavyweight Championship in Belgium.

Williams indicated he was retiring on 17 June 2023 before confirming it in July by saying: ''Doug Williams the Pro Wrestler is over. A new story awaits. Farewell for now. Thank you for the appreciation. 30 Years of incredible memories.''

National Wrestling Alliance (2021–2022)
On 7 December 2021, Williams made his NWA Power debut teaming with Nick Aldis reforming The British Invasion defeating Hawx Aerie (PJ and Luke Hawx). On March 22, Williams and Harry Smith were defeated by The Briscoe Brothers in the finals of the Crockett Cup. On June 11, Williams and Smith, now called the Commonwealth Connection defeated La Rebelión Amarilla to become the NWA World Tag Team Champions. The duo defended the titles against the Dirty Sexy Boys (JTG & Dirty Dango) before having to vacate the titles after 77 days.

Martial arts

 * Judo
 * British Judo Championship (72 kg weight class; 1992)

Professional wrestling

 * 3 Count Wrestling
 * 3CW Heavyweight Championship (1 time)
 * 4 Front Wrestling
 * 4FW Heavyweight Championship (1 time)
 * 4FW Inter-Trophy Championship (1 time)
 * All Star Wrestling
 * ASW British Heavyweight Championship (1 time)
 * ASW Middle Heavyweight Championship (1 time)
 * ASW People's Championship (1 time)
 * Athletik Club Wrestling
 * ACW German Championship (1 time)
 * ACW World Wrestling Championship (1 time)
 * Dynamic Over-The-Top Action Wrestling
 * DOA UK Heavyweight Championship (1 time)
 * European Wrestling Association
 * EWA Intercontinental Championship (1 time)
 * European Wrestling Promotion
 * EWP Intercontinental Championship (1 time)
 * EWP Submission Shootfight Championship (1 time)
 * Frontier Wrestling Alliance
 * FWA British Heavyweight Championship (2 times)
 * FWA British Heavyweight Championship Tournament (2001)
 * German Stampede Wrestling
 * GSW Tag Team Championship (1 times) - with Martin Stone
 * German Wrestling Promotion
 * GWP WrestlingCorner Championship (2 times)
 * Hammerlock Wrestling
 * Hammerlock Allcomers Championship (1 time)
 * Hammerlock Allcomers Championship Tournament (1995)
 * One Night Tournament (1997)
 * International Catch Wrestling Alliance
 * ICWA World Heavyweight Championship (1 time)
 * K-Star Promotions
 * One Night Tournament (1998)
 * Maximum Championship (1 time)
 * Maximum Rumble (2016)
 * NWA UK Hammerlock
 * King of the Ring (1998)
 * Survivor Series Tournament (1996, 1997)
 * National Wrestling Alliance
 * NWA World Tag Team Championship (1 time) – with Harry Smith
 * New Japan Pro-Wrestling
 * IWGP Tag Team Championship (1 time) – with Brutus Magnus
 * Ohio Valley Wrestling
 * OVW Heavyweight Championship (1 time)
 * Power of Wrestling
 * POW Intercontinental Championship (1 time)
 * POW Tag Team Championship (2 times) - with Dave Mastiff (1 time), Michael Kovac (1 time)
 * Premier Wrestling Federation
 * PWF Heavyweight Championship (1 time)
 * PWF International Championship (1 time)
 * PWF Mid-Heavyweight Championship (1 time)
 * Worthing Trophy (2002, 2003, 2004, 2006, 2014, 2017)
 * Wrestler of the Year (2002–2004, 2006, 2007)
 * Preston City Wrestling
 * PCW Heavyweight Championship (1 time)
 * Pro Wrestling Illustrated
 * Ranked No. 45 of the top 500 singles wrestlers in the PWI 500 in 2010
 * Pro Wrestling Noah
 * GHC Tag Team Championship (1 time) – with Scorpio
 * Pro Wrestling Pride
 * PWP Catch Division Championship (1 time)
 * Progress Wrestling
 * Progress Atlas Championship (1 time)
 * Revolution British Wrestling
 * One Night Tournament (2005)
 * Ring of Honor
 * ROH Pure Championship (1 time)
 * ROH Pure Championship Tournament (2004)
 * Rings of Europe
 * Thermencup (2005)
 * Scottish Wrestling Alliance
 * SWA Scottish Heavyweight Championship (1 time)
 * Solent Wrestling Federation
 * All Nations Championship (1 time)
 * All Nations Title Tournament (2021)
 * Total Nonstop Action Wrestling
 * TNA Television Championship (1 time)
 * TNA X Division Championship (2 times)
 * TNA World Tag Team Championship (1 time) – with Brutus Magnus
 * Feast or Fired (2009 – X Division Championship contract)
 * The Wrestling Alliance
 * Universal British Heavyweight Championship (1 time)
 * TWA British Heavyweight Championship (2 times)
 * TWA European Heavyweight Championship (1 time)
 * TWA British Tag Team Championship (1 time) – with Robbie Brookside
 * TWA British Heavyweight Championship Tournament (2001)
 * British Heavyweight Championship Tournament (2002)
 * TopCatch
 * TopCatch Europameisterschaft Championship (1 time)
 * Triple X Wrestling
 * TXW Crush Championship (1 time)
 * Ultimate Wrestling Alliance
 * UWA Championship (1 time)
 * Westside Xtreme Wrestling
 * wXw Tag Team Championship (1 time) – with Martin Stone
 * Trios Tournament (2006) – with Takashi Sugiura and Yoshinobu Kanemaru
 * wXw Hall of Fame (Class of 2018)
 * World Association of Wrestling
 * WAW European Heavyweight Championship (1 time)
 * WrestleZone
 * Battle of the Nations (2015)

title: Doug Williams (wrestler)


text:
Jeffersonville

Jeffersonville is the name of several places in the United States:

 * Jeffersonville, Georgia
 * Jeffersonville, Illinois
 * Jeffersonville, Indiana
 * Jeffersonville, Kentucky
 * Jeffersonville, New York
 * Jeffersonville, Ohio
 * Jeffersonville, Vermont

title: Jeffersonville


text:
Alicia Bridges

Alicia Bridges (born July 15, 1953) is an American singer and songwriter who co-wrote and performed her international hit "I Love the Nightlife (Disco 'Round)" in 1978.

Early years
Born in Charlotte, North Carolina and raised in the small Cleveland County town of Lawndale, Alicia Bridges sang from a very young age, learning to play guitar at the age of 10. At the age of 12, she had her own radio program: The Alicia Bridges Show, broadcast every Saturday on station WADA in Shelby, North Carolina. From age thirteen, Bridges fronted bands that performed in local clubs: "They were pretty rough places ... but these were the only places I could get ... performing experience ... so I had to adjust to that rough environment and learn to survive in it."

Discovery and fame (1970s)
Bridges first recorded for the Nashville-based Mega Records, with two 1973 single releases of her original material. Despite the lack of success of these releases, Bridges came to the attention of Bill Lowery, who became her manager, subsidizing Bridges over a roughly two-year period. During this time, he canvassed record labels on her behalf, which led to her signing with Polydor in 1977.

Bridges's self-titled debut album was the inaugural production by Steve Buckingham, who was assigned to produce Bridges after playing guitar on one of her earliest sessions. Bridges and her personal and songwriting partner Susan Hutcheson had written a song entitled "Disco 'Round", which, at Buckingham's suggestion, was recorded as "I Love the Nightlife", as Buckingham felt that the song was R&B rather than disco ("Disco 'Round" was retained as the song's subtitle): Bridges herself would later admit she had had hopes that the song would be received as a Memphis soul number, calling it "something Al Green might sing". However, club support was a key factor in the success of the track, which became a "sleeper" Top 40 hit, spending 27 weeks on the Billboard Hot 100 with a peak of number five, and also appearing on several international hit parades as evinced by the Billboard ' s "Hits of the World" chart. The song was also in the #5 position on the Record World and Cash Box charts. It was also a "crossover" hit, receiving play even on country music stations. The cover of the 1978 self-titled LP features an early skyline picture of Atlanta, Georgia.

In 1978, Bridges was a Grammy nominee and was the opening act, performing "I Love the Nightlife". Bridges' self-titled LP, Alicia Bridges, was also rising on the Billboard, Cash Box and Record World charts, and the LP remained on the charts for up to 35 weeks. Both the LP and the single release were international successes, and "I Love the Nightlife" went top-40 in the UK and Germany, Spain and other European countries. Bridges's music was especially popular in Japan. She received an RIAA Certified Gold Record for the sale of over one million copies of her hit single in the United States, as well as a gold record from Canada.

Contrary to the impression created by her success with "I Love the Nightlife", Bridges was primarily a singer of R&B, rock, blues and love songs. "I Love the Nightlife" has a definite back beat and organ similar to early 1970s R&B; the original intention was for "I Love the Nightlife" to be an R&B song, but when Bill Lowery first heard the song, he saw its disco potential: subsequently, a 12" single remix was created by Jim Burgess, ultimately becoming a worldwide club smash.

"I Love the Nightlife" has been featured in several movies, including Love at First Bite, The Last Days of Disco and The Adventures of Priscilla, Queen of the Desert.

Bridges's second single, "Body Heat", was a rock song with strong electric guitar rhythms. It was a "rock/dance" song released at a time when there was some rivalry between disco fans and rock fans; this time period began to see more songs that mixed the genres of rock and disco to create a "rock/disco" flavor.

In an interview, Bridges mentioned Elvis Presley as the singer who originally aroused her interest in rock music, but she has always considered herself an R&B singer. She cited Tina Turner, Aretha Franklin, Gladys Knight and Patti LaBelle as her 1970s influences. She said that she did not care for white female singers as a rule, with the exceptions being Joni Mitchell and Janis Joplin. She complained that white women sounded as though they were "sleepwalking" through their performances. "Yeah, I jump around a lot when I'm performing," she said.

Impostor
In 1983, a woman was arrested in Nashville, Tennessee for pretending to be Alicia Bridges during a country music convention. Bridges drove from her home in Atlanta to give evidence in the case. Bridges stated that she had been plagued by an impostor for five years.

Recent years
After the 1984 release of the second of two consecutive unsuccessful album releases, Bridges began working as a disc jockey at Atlanta dance clubs. She was working at the Ponce de Leon Avenue venue the Atlanta Eagle when her disco classic "I Love the Nightlife" had its 1994 revival due to its inclusion on the soundtrack of the movie The Adventures of Priscilla, Queen of the Desert. This revival spurred Bridges to perform at retro disco shows for a number of years, before retiring to Charlotte, North Carolina.

Alicia Bridges Music Publications was founded in 2006 and has produced three albums.

In November 2006, Bridges released a remixed and remastered compilation of her songs titled This Girl Don't Care. In 2007, she released Say It Sister. Her third CD, Faux Diva, was released in 2008; this included a remix of "I Love The Nightlife".

Bridges is currently active in the music community as a producer and DJ. Her works, digital versions of her recordings, a blog and videos can be found on her official web sites. In 2009, Bridges was inducted into the North Carolina Music Hall of Fame.

Personal life
Bridges, who is a lesbian, has stated that, as a teenager, she realized she was "different from the rest", as discussed during a 1998 interview for The Advocate.

Albums

 * 1978: Alicia Bridges (Polydor, 1978)
 * 1979: Play It as It Lays
 * 1984: Hocus Pocus
 * 2002: The Collection: I Love the Nightlife (compilation)
 * 2006: This Girl Don't Care (remixed and remastered compilation)
 * 2007: Say It Sister (remastered compilation)
 * 2008: Faux Diva (remixed and remastered compilation)

EP

 * 1978: The AOR Mini-Album

title: Alicia Bridges


text:
Joondanna, Western Australia

Joondanna is a suburb in Perth, Western Australia. Its local government area is the City of Stirling. There is a Catholic church, St Denis, and a small Catholic primary school, St Denis School, named after the church. Joondanna is named after a farm said to be owned by an early settler in the area.

Joondanna Heights was the first declared township in the Perth Road District (now the City of Stirling) - the Government Gazette of 17 November 1939 described its boundaries as Balcatta Beach Road (now North Beach Drive, Tuart Hill), Main Street, Wanneroo Road, and what was then the City of Perth boundary (Green Street, now the City of Vincent boundary). Joondanna's neighbouring suburbs are Yokine, Coolbinia, North Perth, Mount Hawthorn, Glendalough, Osborne Park, and Tuart Hill.

Footballer Chris Judd lived in Joondanna when he played for the West Coast Eagles, and Paralympian Louise Sauvage grew up in the suburb.

title: Joondanna, Western Australia


text:
Bristol Bulldog

The Bristol Bulldog is a British Royal Air Force single-seat biplane fighter designed during the 1920s by the Bristol Aeroplane Company. More than 400 Bulldogs were produced for the RAF and overseas customers, and it was one of the most famous aircraft used by the RAF during the inter-war period.

Background
The design of the Bulldog was the outcome of a series of design studies for fighters undertaken by Frank Barnwell during the 1920s. In 1924 Barnwell had started work on a fighter powered by the Rolls-Royce Falcon to meet the requirements of specification F.17/24. The project was shelved since Bristol preferred to use its own engine designs, but was revived in 1926 when Barnwell started work on a design, designated the Bristol 102, to meet either F.9/26 for a day-and-night fighter or N.21/26 for a shipborne fighter.

The Type 105 designation was first applied to a subsequent proposal for another aircraft to meet F.9/26 powered by the Mercury engine then under development at Bristol. These proposals looked promising enough for a pair of mockups to be constructed for inspection by the Air Ministry in February 1927. The two aircraft were similar in design, the interceptor to specification F.17/24 design being slightly smaller and lighter and not equipped with radio. As a result, Bristol was asked to revise the design so that it met a later interceptor specification, F.20/27. Subsequently, a prototype aircraft, now designated the Type 107 Bullpup was ordered for evaluation, but the other design did not gain official backing. Nevertheless, Bristol considered it promising enough to build a prototype to be entered for the F.9/26 trials as a private venture, powered by a Bristol Jupiter because the supply of Mercurys was expected to be limited.

Design and development
The Type 105 was an unequal span single bay biplane powered by a supercharged Bristol Jupiter VII air-cooled radial engine driving a two-bladed propeller. The structure was all-metal with a fabric covering, using members built up from rolled high-tensile steel strips riveted together. In order to ensure the maximum field of view there was a large semi-circular cut-out in the trailing edge of the upper wing and the inboard section of the lower was of reduced chord. Frise ailerons were fitted to the top wing only. It was armed with a pair of 0.303 in Vickers machine guns mounted one either side of the cockpit.

The prototype Bulldog first flew on 17 May 1927. Initial testing was entirely satisfactory and it was delivered to RAF Martlesham Heath in June. After consideration of all the types entered to meet the specification, the Bulldog and the Hawker Hawfinch were selected for more detailed evaluation. The manoeuvrability and strength of the Bulldog were praised by the RAF. It had poor spin recovery characteristics, which were remedied by fitting an enlarged fin and rudder but this led to difficulties in taxiing in a crosswind.

A second prototype with a lengthened rear fuselage was ordered for further evaluation in comparison with the Hawfinch. In this form, the Type 105A or Bulldog Mk. II, it was first flown by Cyril Uwins on 21 January 1928 and shortly afterwards delivered to Martlesham Heath. Performance was so close to that of the Hawfinch that a decision was deferred until the aircraft had been evaluated by service pilots; the eventual choice of the Bulldog was made largely because it was easier to maintain. An initial contract for 25 aircraft was placed: Bristol accordingly laid down 26 airframes, the extra one being intended as a company demonstration aircraft. The first of these were delivered on 8 May 1929 and deliveries were complete by 10 October.

Later production aircraft were of a refined version, the '''Mk. IIA''' with revised wing spars and a stronger fuselage, powered by the uprated Jupiter VII F. One production aircraft was modified for use as an advanced trainer: after evaluation by the Central Flying School at Upavon this was ordered by the RAF, the production aircraft differing from the prototype in having slightly swept wings and an enlarged fin to improve spin recovery.

Operational history
The Bulldog never saw combat with the RAF, although during the Abyssinia Crisis of 1935–36, Bristol Bulldogs were sent to the Sudan to reinforce Middle East Command. Douglas Bader, better known for his Second World War actions, lost both of his legs when he crashed his Bristol Bulldog while he was performing unauthorised aerobatics at Woodley airfield near Reading. The Bulldog was withdrawn from RAF Fighter Command in July 1937, being primarily replaced by the Gloster Gauntlet. but continued to serve the RAF for a few years with Service Flying Training Schools. The Bulldog was exported to foreign air forces, seeing service with Australia, Denmark, Estonia, Finland, Japan, Latvia, Siam and Sweden.

In 1936, Latvia, intent on replacing its Bulldogs with more modern aircraft, sold 11 Bulldogs to Basque nationalist forces. These became part of the Spanish Republican Air Force in the Spanish Civil War; remaining in use until the Battle of Santander. Ten Bulldogs saw combat as part of the Finnish Air Force during the Winter War against the Soviet Union, which began in 1939. The Bulldogs fought against their Soviet opponent, gaining two kills by two pilots for the loss of one of their own, the types shot down being one Polikarpov I-16 and one Tupolev SB, both of which were superior in terms of technology compared to the Bulldog. The first aerial victory of the Finnish Air Force was achieved by a Bulldog piloted by SSgt Toivo Uuttu on 1 December 1939, over an I-16. The Bulldogs were used in advanced training during the subsequent Continuation War against the Soviet Union.

Variants

 * Bulldog Mk. I
 * Single-seat day-and-night fighter prototype; two built.

 * High-altitude Bulldog
 * Modification of first prototype with enlarged wings intended for an attempt on the world altitude record.

 * Bulldog Mk. II
 * Second prototype and initial production version. Powered by a 440 hp (330 kW) Bristol Jupiter VII radial piston engine; 92 built by Bristol.

 * Bulldog Mk. IIA
 * Powered by a 490 hp (370 kW) Bristol Jupiter VIIF radial piston engine and revised detail design; 268 built by Bristol.

 * Bulldog Mk. IIIA
 * Development powered by a Bristol Mercury IV enclosed within a Townend ring with a revised wings and stronger fuselage. Only two built, one of which was converted to become the prototype Mk. IV.

 * Bulldog Mk. IVA
 * Development of the Mk.III to meet specification F.7/30 for a four-gun day-and-night fighter. Not ordered by the RAF but 17 sold to Finland, armed with two 7.7 mm Vickers guns; 18 built by Bristol.

 * Bulldog TM (Type 124)
 * Two-seat training version; 59 built. Powered by an Armstrong Siddeley Cheetah IX of 310 hp.

 * "J.S.S.F." (Japanese Single-Seat Fighter)
 * Two aircraft license-built by Nakajima Aircraft Works, Japan.

Operators

 * Royal Australian Air Force
 * No. 1 Squadron RAAF – Bulldog Mk. IIA
 * No. 2 Squadron RAAF – Bulldog Mk. IIA
 * Royal Danish Air Force
 * Estonian Air Force
 * Finnish Air Force – Bulldog Mk. IVA
 * Imperial Japanese Navy Air Service
 * Latvian Air Force
 * 🇹🇭 Thailand (now Thailand)
 * Royal Siamese Air Force Two bought for comparison purposes
 * Spanish Republic
 * Spanish Republican Air Force
 * Royal Swedish Air Force Two remaining Bulldog Mk. IIA donated to Finland 1939
 * Royal Air Force
 * 🇹🇭 Thailand (now Thailand)
 * Royal Siamese Air Force Two bought for comparison purposes
 * Spanish Republic
 * Spanish Republican Air Force
 * Royal Swedish Air Force Two remaining Bulldog Mk. IIA donated to Finland 1939
 * Royal Air Force
 * Royal Air Force
 * Royal Air Force

 * No. 3 Squadron RAF – Bulldog Mk. II and Bulldog Mk. IIA
 * No. 17 Squadron RAF – Bulldog Mk. II and Bulldog Mk. IIA
 * No. 19 Squadron RAF – Bulldog Mk. IIA
 * No. 23 Squadron RAF – Bulldog Mk. IIA
 * No. 24 Squadron RAF – One two-seat Bulldog was used by No. 24 (Communications) Squadron.
 * No. 29 Squadron RAF – Bulldog Mk. IIA
 * No. 32 Squadron RAF – Bulldog Mk. IIA
 * No. 41 Squadron RAF – Bulldog Mk. IIA
 * No. 54 Squadron RAF – Bulldog Mk. IIA
 * No. 56 Squadron RAF – Bulldog Mk. IIA
 * No. 111 Squadron RAF – Bulldog Mk. IIA
 * No. 3 Flying Training School
 * No. 5 Flying Training School
 * Central Flying School RAF
 * RAF College, Cranwell

 * United States Navy – The U.S. Navy purchased two aircraft between 1929–1930. The aircraft were used for test and trials. During trial on 25 November 1929, one crashed near Bolling Field, killing its pilot, Lieutenant George T. Cuddihy.
 * United States Navy – The U.S. Navy purchased two aircraft between 1929–1930. The aircraft were used for test and trials. During trial on 25 November 1929, one crashed near Bolling Field, killing its pilot, Lieutenant George T. Cuddihy.

Surviving aircraft

 * Finland
 * BU-59 – Bulldog IVA on static display at the Hallinportti Aviation Museum in Kuorevesi, Jämsä.

 * United Kingdom
 * G-ABBB – Bulldog IIA on static display at the Royal Air Force Museum London in London. This aircraft was the civilian demonstrator and is painted as K2227. It was severely damaged in a crash in 1964 at the Farnborough Airshow and repaired in the late 1990s.

In addition an airworthy replica was constructed by Ed Storo of Tillamook, Oregon between 2000 and 2022. It is powered by a Pratt & Whitney R-1340 Wasp due to the scarcity of the original Bristol Jupiter engine.

title: Bristol Bulldog


text:
Tim Curtis

Timothy Stephen Curtis (born 15 January 1960) is a former England cricketer, English teacher and Director of Sport at RGS Worcester. He retired from teaching in 2016.

A right-handed batsman born at Chislehurst in Kent, Curtis was a prolific scorer for Worcestershire and county captain between 1992 and 1995. He is one of only two players to score 10,000 one day runs for the county, and during the late 1980s he had a brief international career.

As cricket correspondent, Colin Bateman, stated Curtis was "a diligent, determined opener... brought in to stop the rot against the 1988 West Indians with about as much success as those who had gone before him".

Curtis was perhaps unfortunate to play in an era of very poor England performances and much was made of Curtis's batting stance and technique during his England career. In 1989, Terry Alderman frequently had Curtis lbw playing across the line and much of the TV media blamed this on his stance and backlift. This despite the fact that his opening partner Graham Gooch had a very similar set-up at the crease. While Curtis showed considerable patience and courage during his encounters with the West Indies, he looked highly suspect against Australia and at age 29 was not considered for the rebuilding process which England began under Graham Gooch's captaincy in 1990.

Life and career
Curtis moved to Worcestershire in the 1960s to live in Malvern, where he attended The Grove county primary school. He entered the Royal Grammar School Worcester, where he became captain of rugby and cricket and Head Boy of the school. He moved on to Durham University (Hatfield College) in 1979, where he captained the university cricket team. That year also saw his debut in first-class cricket. Curtis also played rugby alongside cricket at university and represented Durham University RFC as a fullback in the University Athletic Union (UAU) final against Exeter University, which Durham won 6–3. In most rugby matches he played at university his name was listed in the programme as 'A. N. Other' – this subterfuge was the result of Curtis being contracted to play county cricket for Worcestershire, and he did not think they would approve of him risking injuries by playing rugby in the closed season.

In 1983, Curtis went to Magdalene College, Cambridge and gained a blue for cricket. After returning to Worcester to teach English at the Royal Grammar School, he played cricket for Worcestershire. Curtis is the only man, other than Graeme Hick, to have scored more than 10,000 runs in List A cricket for Worcestershire. In 1988, he was selected for England and played in a total of five Test matches against Australia and West Indies as an opening batsman. However, his international career was unsuccessful, and he never reached fifty in a Test innings. He had more success at this time with Worcestershire, helping them to win the County Championship in 1988 and 1989, the Refuge Assurance League in 1987 and 1988, and the Benson & Hedges Cup and Refuge Assurance Cup in 1991. In 1992, Curtis was appointed captain of Worcestershire, leading the county to success in the 1994 NatWest Trophy. He retired from first-class cricket in 1996 and continues to write about, and coach, cricket.

In 1994, he along with Tom Moody set the record for the highest ever partnership for the 3rd wicket in List A cricket history (309*).

Curtis had a spell as chairman of the Professional Cricketers' Association.

title: Tim Curtis


text:
Karrinyup, Western Australia

Karrinyup is a suburb of Perth, the capital city of Western Australia, and is located 12 km north of Perth's central business district. Its local government area is the City of Stirling.

History
The name Karrinyup was originally derived from the word Careniup, a Noongar name for a nearby swamp, an Aboriginal word apparently meaning "the place where bush kangaroos graze". It may also mean "the place where spiders are". In the 1840s, Samuel Moore took up a grant of 780 acre in the northern part of the suburb. Moore's grant, Swan Location 92 was surveyed by P Chauncey in 1844 and Chauncey recorded a large swamp just to the east of Karrinyup as Careniup Swamp.

In 1929, the foundation committee developing the Lake Karrinyup Country Club golf course opted to change the spelling.

While the area had been subdivided by Charles Stoneman in 1904 and roads built, the country club remained the only significant feature in the area, and rapid growth did not begin until 1957, with the part south of Karrinyup Road developing first. At this stage, the only access to the area from Perth was via Wanneroo Road and Balcatta Beach Road. The building of the Mitchell Freeway to Karrinyup Road in 1983-84 facilitated the growth of Karrinyup and nearby Stirling as a regional hub.

Geography
Karrinyup is bounded by North Beach Road to the north (Reid Highway is located just a few hundred metres further north), Marmion Avenue to the west, Newborough Street to the south and Huntriss Road and the country club to the east. About one-third of Karrinyup's land area is reserve or bushland, or part of the suburb's two golf courses. Karrinyup Road links Marmion Avenue and West Coast Highway to Mitchell Freeway through the suburb.

At the 2016 Australian census, Karrinyup had a population of 9,283. Most of the houses in Karrinyup are relatively modern, though the prolonged period of development has resulted in a range of styles from various eras. Many of the homes within the suburb are of two storeys and the vast majority are of brick and tile construction.

Facilities
The Karrinyup Shopping Centre contains a bus station, community centre and library as well as two major department stores. It was built in 1973 and has since been extended to offer 54,587 m2 of retail accommodation with undercover and open-air parking. A major renovation is expected to be finished in 2021, with plans for apartment towers in the north-east of the complex. It's owned by the superannuation fund Unisuper and managed by AMP.

Karrinyup has two golf courses, Hamersley (public) and Lake Karrinyup (private). Open spaces exist at Lake Karrinyup and at the south-west of the suburb. Karrinyup contains three state primary schools (Karrinyup, Deanmore and Newborough) and a private college, St Mary's Anglican Girls' School, founded in 1921 at West Perth and relocated to Karrinyup in 1961.

Transport
Karrinyup is served by the Karrinyup bus station, located at the shopping centre, with Transperth bus routes 422, 423, 424 and 425 providing a link to Stirling train station. Further west is a bus depot operated by Swan Transit. All services are operated by Swan Transit.

Bus Stations

 * Karrinyup Bus Station

Bus Routes

 * 422 Stirling Station to Scarborough Beach Bus Station – serves Huntriss Road, Finnerty Street, Ramsay Street, Dean Street, Francis Avenue, Pascoe Street, Burroughs Road, Karrinyup Bus Station, Jeanes Road, Deanmore Road, Rinaldi Crescent, Rainer Street and Newborough Street
 * 423 Stirling Station to Warwick Station – serves Huntriss Road, Karrinyup Road and Karrinyup Bus Station
 * 424 Stirling Station to Karrinyup Bus Station – serves North Beach Road and Karrinyup Road
 * 425 Stirling Station to Warwick Station – serves Karrinyup Road, Karrinyup Bus Station, Milverton Avenue and North Beach Road

Politics
Karrinyup is a reasonably affluent suburb with many "mortgage belt" families and socially liberal voters. It consistently supports the Liberal Party at both federal and state elections, although the part south of Karrinyup Road leans more towards the Australian Labor Party.

Notable residents

 * Peter Dowding, former Premier of Western Australia (1988 to 1990)
 * Dennis Lillee, former Australian cricketer
 * Emma Matthews, soprano
 * Jeff Newman, former television presenter
 * Tim Winton, novelist and National Living Treasure, grew up in Gwelup Street

title: Karrinyup, Western Australia


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User:Montanean

Inquiring about deletion of the article "Task Force to Overcome Racism in Topeka"

I'm not seeing a lot of advocates making themselves available in regard to the article referenced above. It is possible to find secondary sources on this topic but it will require research in the newspaper library of the Topeka Capitol Journal. The organization did exist, albeit for a short time. It did have an impact on the community of Topeka which would be evidenced by a greater awareness of the impact racism was having on life in that community. There are no other references to this organization on the web in part because the web did not exist during the time this organization was in existence. There is another organization that was a collaborator with TFORT called Concerned Citizens for Equal Justice. A short description of this group also was written but it has been deleted as well.

Wikipedia can try to be like the Encyclopedia Britannica or other authorized compendium of history. Or it can be an alternative that allows a number of voices to be heard. My impression is that the collective wisdom is tilted toward the former. TFORT was an important organization. The story about it needs to be told.

title: User:Montanean


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Jemison

Jemison is an English surname. Notable people with the surname include:

 * Alice Lee Jemison (1901–1964), American political activist
 * Dick Jemison (1886–1965), American sportswriter
 * Eddie Jemison (born 1963), American actor
 * Edwin Francis Jemison (1844–1862), American Civil War soldier
 * Julio Jemison (born 1994), Bahamian footballer
 * Mae Jemison (born 1956), American astronaut, engineer and physician
 * Marty Jemison (born 1965), American cyclist
 * Mary Jemison (1743–1833), Scots-Irish American frontierswoman
 * Mike Jemison (born 1983), American football player
 * Robert Jemison Jr. (1802–1871), American politician and entrepreneur
 * Steffani Jemison (born 1981), American artist
 * T. J. Jemison (1918–2013), American clergyman
 * Trey Jemison (born 1999), American basketball player

title: Jemison


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User talk:Andrew Dunn

Welcome
Hello Andrew, welcome to Wikipedia.

You might find these links helpful in editing pages or creating new ones: How to edit a page, Tutorial, Naming conventions, Manual of Style. You should probably read our policies at some point too.

But don't feel you have to read every policy document before you do anything. Dive in, be bold in editing, and if you do anything wrong, someone will be quick to correct it and let you know (hopefully, politely!)

If you have any questions, see the help pages, add a question to the village pump or ask me on my talk page. I hope you enjoy editing here and being a Wikipedian!

You may have already noticed that you can sign your name using three tildes, like this:. If you use four, you can add a datestamp too, as I've done below. This is mostly useful, and is considered good wikiquette, on talk pages. But be aware that the actual wikipedia articles should never be signed - they belong to the community, rather than the individual who created them.

Again, welcome! -- Finlay McWalter | Talk 15:59, 22 May 2004 (UTC)

authors: Template namespace initialisation script

title: User talk:Andrew Dunn


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Menora, Western Australia

Menora is a suburb of Perth, Western Australia. Its local government area is the City of Stirling.

In 1954, with the establishment of official suburban boundaries, the new offshoot from Mount Lawley was named Menora in honour of an old theatre of that name which was located within its borders. Further support for the name was provided by the local Jewish community for whom the Menorah holds special significance. From an aerial point of view, a section of the roads are shaped like the Menorah.

Menora is home to a large Jewish community (8.1%), the heart of which is the Perth Hebrew Congregation. It is close to the Carmel Jewish School and other Kosher facilities like the Kosher Food Centre in Menora and the Kosher Besh Fresh cafe.

In the, Menora had a median age of 53 (as compared with 38 for Australia), children aged 0–14 years made up 13.8% and 37.6% of its population were over 65 years of age. This is likely attributable to a high concentration of retirement villages in the district.

title: Menora, Western Australia


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Batalin–Vilkovisky formalism

In theoretical physics, the Batalin–Vilkovisky (BV) formalism (named for Igor Batalin and Grigori Vilkovisky) was developed as a method for determining the ghost structure for Lagrangian gauge theories, such as gravity and supergravity, whose corresponding Hamiltonian formulation has constraints not related to a Lie algebra (i.e., the role of Lie algebra structure constants are played by more general structure functions). The BV formalism, based on an action that contains both fields and "antifields", can be thought of as a vast generalization of the original BRST formalism for pure Yang–Mills theory to an arbitrary Lagrangian gauge theory. Other names for the Batalin–Vilkovisky formalism are field-antifield formalism, Lagrangian BRST formalism, or BV–BRST formalism. It should not be confused with the Batalin–Fradkin–Vilkovisky (BFV) formalism, which is the Hamiltonian counterpart.

Batalin–Vilkovisky algebras
In mathematics, a Batalin–Vilkovisky algebra is a graded supercommutative algebra (with a unit 1) with a second-order nilpotent operator Δ of degree −1. More precisely, it satisfies the identities
 * $$(ab)c = a(bc)    $$  (The product is associative)
 * $$ab = (-1)^{|a||b|}ba  $$    (The product is (super-)commutative)
 * $$|ab| = |a| + |b| $$  (The product has degree 0)
 * $$|\Delta(a)| = |a| - 1   $$  (Δ has degree −1)
 * $$\Delta^2 = 0     $$ (Nilpotency (of order 2))
 * The Δ operator is of second order:
 * $$\begin{align}

0 = & \Delta(abc) \\ &-\Delta(ab)c-(-1)^{|a|}a\Delta(bc)-(-1)^{(|a|+1)|b|}b\Delta(ac)\\ &+\Delta(a)bc+(-1)^{|a|}a\Delta(b)c+(-1)^{|a|+|b|}ab\Delta(c)\\ &-\Delta(1)abc \end{align}$$

One often also requires normalization:

 * $$\Delta(1)=0 $$ (normalization)

Antibracket
A Batalin–Vilkovisky algebra becomes a Gerstenhaber algebra if one defines the Gerstenhaber bracket by
 * $$(a,b) := (-1)^{\left|a\right|}\Delta(ab) - (-1)^{\left|a\right|}\Delta(a)b - a\Delta(b)+a\Delta(1)b .$$

Other names for the Gerstenhaber bracket are Buttin bracket, antibracket, or odd Poisson bracket. The antibracket satisfies
 * $$|(a,b)| = |a|+|b| - 1 $$ (The antibracket has degree −1)
 * $$ (a,b) = -(-1)^{(|a|+1)(|b|+1)}(b,a) $$    (Skewsymmetry)
 * $$ (-1)^{(|a|+1)(|c|+1)}(a,(b,c)) +  (-1)^{(|b|+1)(|a|+1)}(b,(c,a)) +  (-1)^{(|c|+1)(|b|+1)}(c,(a,b)) = 0 $$ (The Jacobi identity)
 * $$ (ab,c) = a(b,c) + (-1)^{|a||b|}b(a,c)$$    (The Poisson property; the Leibniz rule)

Odd Laplacian
The normalized operator is defined as
 * $$ {\Delta}_{\rho} := \Delta-\Delta(1) . $$

It is often called the odd Laplacian, in particular in the context of odd Poisson geometry. It "differentiates" the antibracket The square $${\Delta}_{\rho}^{2}=(\Delta(1),\cdot)$$ of the normalized $${\Delta}_{\rho}$$ operator is a Hamiltonian vector field with odd Hamiltonian Δ(1) which is also known as the modular vector field. Assuming normalization Δ(1)=0, the odd Laplacian $$ {\Delta}_{\rho} $$ is just the Δ operator, and the modular vector field $$ {\Delta}_{\rho}^{2} $$ vanishes.
 * $$ {\Delta}_{\rho}(a,b) = ({\Delta}_{\rho}(a),b) - (-1)^{\left|a\right|}(a,{\Delta}_{\rho}(b)) $$        (The $${\Delta}_{\rho}$$ operator  differentiates )
 * $$ {\Delta}_{\rho}^{2}(ab) = {\Delta}_{\rho}^{2}(a)b+ a{\Delta}_{\rho}^{2}(b) $$      (The Leibniz rule)

Compact formulation in terms of nested commutators
If one introduces the left multiplication operator $$L_{a}$$ as
 * $$ L_{a}(b) := ab,  $$

and the supercommutator [,] as
 * $$[S,T]:=ST - (-1)^{\left|S\right|\left|T\right|}TS $$

for two arbitrary operators S and T, then the definition of the antibracket may be written compactly as
 * $$ (a,b) := (-1)^{\left|a\right|} [[\Delta,L_{a}],L_{b}]1, $$

and the second order condition for Δ may be written compactly as
 * $$ [[[\Delta,L_{a}],L_{b}],L_{c}]1 = 0  $$    (The Δ operator is of second order)

where it is understood that the pertinent operator acts on the unit element 1. In other words, $$ [\Delta,L_{a}] $$ is a first-order (affine) operator, and $$ [[\Delta,L_{a}],L_{b}] $$ is a zeroth-order operator.

Master equation
The classical master equation for an even degree element S (called the action) of a Batalin–Vilkovisky algebra is the equation
 * $$(S,S) = 0  . $$

The quantum master equation for an even degree element W of a Batalin–Vilkovisky algebra is the equation
 * $$ \Delta\exp \left[\frac{i}{\hbar}W\right] = 0 ,$$

or equivalently,
 * $$\frac{1}{2}(W,W) = i\hbar{\Delta}_{\rho}(W)+\hbar^{2}\Delta(1) . $$

Assuming normalization Δ(1) = 0, the quantum master equation reads
 * $$\frac{1}{2}(W,W) = i\hbar\Delta(W) . $$

Generalized BV algebras
In the definition of a generalized BV algebra, one drops the second-order assumption for Δ. One may then define an infinite hierarchy of higher brackets of degree −1
 * $$ \Phi^{n}(a_{1},\ldots,a_{n}) := \underbrace{[[\ldots[\Delta,L_{a_{1}}],\ldots],L_{a_{n}}]}_{n~{\rm nested~commutators}}1  .   $$

The brackets are (graded) symmetric
 * $$ \Phi^{n}(a_{\pi(1)},\ldots,a_{\pi(n)}) = (-1)^{\left|a_{\pi}\right|}\Phi^{n}(a_{1},\ldots, a_{n})   $$  (Symmetric brackets)

where $$\pi\in S_{n}$$ is a permutation, and $$(-1)^{\left|a_{\pi}\right|}$$ is the Koszul sign of the permutation
 * $$a_{\pi(1)}\ldots a_{\pi(n)} = (-1)^{\left|a_{\pi}\right|}a_{1}\ldots a_{n}$$.

The brackets constitute a homotopy Lie algebra, also known as an $$L_{\infty}$$ algebra, which satisfies generalized Jacobi identities
 * $$ \sum_{k=0}^n \frac{1}{k!(n\!-\!k)!}\sum_{\pi\in S_{n}}(-1)^{\left|a_{\pi}\right|}\Phi^{n-k+1}\left(\Phi^{k}(a_{\pi(1)}, \ldots, a_{\pi(k)}), a_{\pi(k+1)}, \ldots, a_{\pi(n)}\right) = 0. $$  (Generalized Jacobi identities)

The first few brackets are: In particular, the one-bracket $$ \Phi^{1}={\Delta}_{\rho}$$ is the odd Laplacian, and the two-bracket $$ \Phi^{2}$$ is the antibracket up to a sign. The first few generalized Jacobi identities are: where the Jacobiator for the two-bracket $$\Phi^{2}$$ is defined as
 * $$ \Phi^{0} := \Delta(1) $$     (The zero-bracket)
 * $$ \Phi^{1}(a) := [\Delta,L_{a}]1 = \Delta(a) - \Delta(1)a =: {\Delta}_{\rho}(a) $$   (The one-bracket)
 * $$ \Phi^{2}(a,b) := [[\Delta,L_{a}],L_{b}]1 =: (-1)^{\left|a\right|}(a,b) $$   (The two-bracket)
 * $$ \Phi^{3}(a,b,c) := [[[\Delta,L_{a}],L_{b}],L_{c}]1 $$   (The three-bracket)
 * $$ \vdots $$
 * $$ \Phi^{1}(\Phi^0) = 0 $$ ($$\Delta(1)$$  is $$\Delta_\rho$$-closed)
 * $$ \Phi^{2}(\Phi^{0},a)+\Phi^{1}\left(\Phi^{1}(a)\right)$$ ($$\Delta(1)$$  is the Hamiltonian for the modular vector field $${\Delta}_{\rho}^{2}$$)
 * $$ \Phi^{3}(\Phi^{0},a,b) + \Phi^{2}\left(\Phi^{1}(a),b\right)+(-1)^{|a|}\Phi^{2}\left(a,\Phi^{1}(b)\right) +\Phi^{1}\left(\Phi^{2}(a,b)\right) = 0 $$ (The $$ {\Delta}_{\rho} $$ operator differentiates  generalized)
 * $$ \Phi^{4}(\Phi^{0},a,b,c) + {\rm Jac}(a,b,c)+ \Phi^{1}\left(\Phi^{3}(a,b,c)\right) + \Phi^{3}\left(\Phi^{1}(a),b,c\right) + (-1)^{\left|a\right|}\Phi^{3}\left(a,\Phi^{1}(b),c\right) +(-1)^{\left|a\right|+\left|b\right|}\Phi^{3}\left(a,b,\Phi^{1}(c)\right) = 0 $$ (The generalized Jacobi identity)
 * $$ \vdots $$
 * $$ {\rm Jac}(a_{1},a_{2},a_{3}) :=

\frac{1}{2} \sum_{\pi\in S_{3}}(-1)^{\left|a_{\pi}\right|} \Phi^{2}\left(\Phi^{2}(a_{\pi(1)},a_{\pi(2)}),a_{\pi(3)}\right). $$

BV n-algebras
The Δ operator is by definition of n'th order if and only if the (n + 1)-bracket $$ \Phi^{n+1} $$ vanishes. In that case, one speaks of a BV n-algebra. Thus a BV 2-algebra is by definition just a BV algebra. The Jacobiator $$ {\rm Jac}(a,b,c)=0 $$ vanishes within a BV algebra, which means that the antibracket here satisfies the Jacobi identity. A BV 1-algebra that satisfies normalization Δ(1) = 0 is the same as a differential graded algebra (DGA) with differential Δ. A BV 1-algebra has vanishing antibracket.

Odd Poisson manifold with volume density
Let there be given an (n|n) supermanifold with an odd Poisson bi-vector $$ \pi^{ij}$$ and a Berezin volume density $$\rho$$, also known as a P-structure and an S-structure, respectively. Let the local coordinates be called $$x^{i}$$. Let the derivatives $$ \partial_{i}f $$ and
 * $$ f\stackrel{\leftarrow}{\partial}_{i}:=(-1)^{\left|x^{i}\right|(|f|+1)}\partial_{i}f $$

denote the left and right derivative of a function f wrt. $$x^{i}$$, respectively. The odd Poisson bi-vector $$ \pi^{ij}$$ satisfies more precisely Under change of coordinates $$x^{i} \to x^{\prime i} $$ the odd Poisson bi-vector $$ \pi^{ij}$$ and Berezin volume density $$\rho$$ transform as where sdet denotes the superdeterminant, also known as the Berezinian. Then the odd Poisson bracket is defined as
 * $$ \left|\pi^{ij}\right| = \left|x^{i}\right| + \left|x^{j}\right| -1 $$   (The odd Poisson structure has degree –1)
 * $$ \pi^{ji} = -(-1)^{(\left|x^{i}\right|+1)(\left|x^{j}\right|+1)} \pi^{ij} $$      (Skewsymmetry)
 * $$ (-1)^{(\left|x^{i}\right|+1)(\left|x^{k}\right|+1)}\pi^{i\ell}\partial_{\ell}\pi^{jk} + {\rm cyclic}(i,j,k) = 0 $$   (The Jacobi identity)
 * $$ \pi^{\prime k\ell} = x^{\prime k}\stackrel{\leftarrow}{\partial}_{i} \pi^{ij} \partial_{j}x^{\prime \ell} $$
 * $$\rho^{\prime} = \rho/{\rm sdet}(\partial_{i}x^{\prime j}) $$
 * $$ (f,g) := f\stackrel{\leftarrow}{\partial}_{i}\pi^{ij}\partial_{j}g . $$

A Hamiltonian vector field $$ X_{f}$$ with Hamiltonian f can be defined as
 * $$ X_{f}[g] := (f,g) .$$

The (super-)divergence of a vector field $$ X=X^{i}\partial_{i} $$ is defined as
 * $$ {\rm div}_{\rho} X := \frac{(-1)^{\left|x^{i}\right|(|X|+1)}}{\rho} \partial_{i}(\rho X^{i}) $$

Recall that Hamiltonian vector fields are divergencefree in even Poisson geometry because of Liouville's Theorem. In odd Poisson geometry the corresponding statement does not hold. The odd Laplacian $$ {\Delta}_{\rho}$$ measures the failure of Liouville's Theorem. Up to a sign factor, it is defined as one half the divergence of the corresponding Hamiltonian vector field,
 * $$ {\Delta}_{\rho}(f) := \frac{(-1)^{\left|f\right|}}{2}{\rm div}_{\rho} X_{f} = \frac{(-1)^{\left|x^{i}\right|}}{2\rho}\partial_{i}\rho \pi^{ij}\partial_{j}f.$$

The odd Poisson structure $$ \pi^{ij}$$ and Berezin volume density $$\rho$$ are said to be compatible if the modular vector field $$ {\Delta}_{\rho}^{2} $$ vanishes. In that case the odd Laplacian $$ {\Delta}_{\rho}$$ is a BV Δ operator with normalization Δ(1)=0. The corresponding BV algebra is the algebra of functions.

Odd symplectic manifold
If the odd Poisson bi-vector $$ \pi^{ij}$$ is invertible, one has an odd symplectic manifold. In that case, there exists an odd Darboux Theorem. That is, there exist local Darboux coordinates, i.e., coordinates $$ q^{1}, \ldots, q^{n} $$, and momenta $$ p_{1},\ldots, p_{n} $$, of degree
 * $$ \left|q^{i}\right|+\left|p_{i}\right|=1, $$

such that the odd Poisson bracket is on Darboux form
 * $$ (q^{i},p_{j}) = \delta^{i}_{j} . $$

In theoretical physics, the coordinates $$q^{i} $$ and momenta $$p_{j} $$ are called fields and antifields, and are typically denoted $$\phi^{i} $$ and $$\phi^{*}_{j} $$, respectively.
 * $$\Delta_{\pi} := (-1)^{\left|q^{i}\right|}\frac{\partial}{\partial q^{i}}\frac{\partial}{\partial p_{i}} $$

acts on the vector space of semidensities, and is a globally well-defined operator on the atlas of Darboux neighborhoods. Khudaverdian's $$\Delta_{\pi}$$ operator depends only on the P-structure. It is manifestly nilpotent $$\Delta_{\pi}^{2}=0$$, and of degree −1. Nevertheless, it is technically not a BV Δ operator as the vector space of semidensities has no multiplication. (The product of two semidensities is a density rather than a semidensity.) Given a fixed density $$\rho$$, one may construct a nilpotent BV Δ operator as
 * $$ \Delta(f) :=\frac{1}{\sqrt{\rho}}\Delta_{\pi}(\sqrt{\rho}f),$$

whose corresponding BV algebra is the algebra of functions, or equivalently, scalars. The odd symplectic structure $$ \pi^{ij}$$ and density $$\rho$$ are compatible if and only if Δ(1) is an odd constant.

Examples

 * The Schouten–Nijenhuis bracket for multi-vector fields is an example of an antibracket.
 * If L is a Lie superalgebra, and Π is the operator exchanging the even and odd parts of a super space, then the symmetric algebra of Π(L) (the "exterior algebra" of L) is a Batalin–Vilkovisky algebra with Δ given by the usual differential used to compute Lie algebra cohomology.

Pedagogical

 * Costello, K. (2011). "Renormalization and Effective Field Theory". ISBN 978-0-8218-5288-0 (Explains perturbative quantum field theory and the rigorous aspects, such as quantizing Chern-Simons theory and Yang-Mills theory using BV-formalism)

title: Batalin–Vilkovisky formalism


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File:GL Lightfoot!.jpg

Summary
Gordon Lightfoot - Lightfoot! album cover

title: File:GL Lightfoot!.jpg