text:
Talk:Jesus Christ Vampire Hunter

Categorized as a Cult Film
I didn't know of this movie until I saw it listed in the category of cult films.

This sounds totally hilarious. I hope to see it now. See, MPAA, the internet actually ENCOURAGES paying to see movies! --TIB 04:56, Aug 7, 2004 (UTC)

And I thought Ed Wood was bad...Vincent 05:52, 5 Dec 2004 (UTC)

 * Oh man. This movie ruled.  A Kung Fu musical about Our Lord and Savior Jesus The Christ saving lesbian Canadian vampires.  I mean, the only thing it was lacking were ninjas and pirates.  Heh heh. Stargoat 20:35, 6 Dec 2004 (UTC)

Waste of Time
I just watched this. I WANT MY 80 MINUTES OF LIFE BACK!! wa wa wa

 * It's all good, it's all right... everybody waste some time TONIGHT! Erikeltic (talk) 17:43, 7 April 2009 (UTC)

Good Premise but Something went Wrong
And the premise sounds good. Too bad they took 3 times more money than "Clerks" and made this monstrosity. Mcpusc 07:09, 22 Jan 2005 (UTC)
 * It wasn't really 3 times the budget of Clerks. That film was shot for around 28,000 USD, sure, but by the time it got through post-production it ended up costing well over $200,000 USD.  Since this film cost a total of $100,000 Canadian--with 8 years of inflation separating the two films--you could basically say this cost a THIRD of what Clerks did.  God, I just realized I'm responding to a comment posted nearly a year ago.

For the record, they DELIBERATELY set out to make a bad B-movie, complete with jive hackneyed dialogue, Mexican wrestlers, B-movie Russ Meyer fake lesbian titillation, and inappropriate musical numbers. If you look at it through that angle, the movie is genius.

Unfortuantly, most people can't understand or comprehend the tongue-in-cheek way this film was made, and therefore can't understand how such a horrible movie is actually kick ass. —Preceding unsigned comment added by 68.11.215.101 (talk) 18:57, 29 March 2008 (UTC)

This was the worst movie ever made. The 100,000 dollars was actually spent on cocaine and cocaine accessories. —Preceding unsigned comment added by 99.236.29.5 (talk) 17:49, 23 October 2009 (UTC)

What about DVD?
Does the DVD contain a director's commentary?

At least Pi had a director's commentary.

 * Yes. That's where the 100000CAD figure comes from, I believe. --Ted 01:28, 16 May 2006 (UTC)

Fair use rationale for Image:Jesus Christ Vampire Hunter.jpg
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Sean Secor is not a really in this movie
Sean Secor is not in this movie http://www.imdb.com/title/tt0311361/fullcredits#cast

Phil Caracas	 ... Jesus Christ Murielle Varhelyi	 ... Maxine Schreck Ian Driscoll	 ... Johnny Golgotha Jeff Moffet	 ... Santo Enmascarado de Plata Josh Grace	 ... Dr. Pretorious Jason McMaster	 ... Atheist 1

Johnny Vegas	 ... Himself rest of cast listed alphabetically: Thetis Bernard	 ... Lesbian Vampire Hottie #3 Christopher R. Bugg	 ... Atheist #1 Mark Cayer	 ... Worshipper Marc Charron	 ... The cop Olga Chrzanowska	 ... Worshipper Graham Collins	 ... Store Clerk Christine Demarbre	 ... The screaming lesbian Eric Demarbre	 ... Hockey Jersey Atheist Tim Devries	 ... Father Eustace Rayna Farr-Dutchin	 ... The little blind girl Miles Finlayson	 ... Atheist Ivan Freud	 ... Narrator Mike Funk	 ... Danny Sabbath Alex Godin	 ... Father Avellino Stephanie Halin	 ... Lesbian Vampire Stu Hutchings	 ... Danny Sabbath's bodyguard Glen Jones	 ... Father Alban

Claudia Jurt	 ... Misty Winters Tracy Lance	 ... Gloria Oddbottom Yves Macra	 ... The limo driver Maria Moulton	 ... Mary Magnum Ean Murphy	 ... Female Lead Atheist Erica Murton	 ... Maggie Karm Parker	 ... Atheist #6 Shelly Pederson	 ... Woman blessed on the beach Mark Pollesel	 ... Dancer Elinor Quesnel	 ... Praetorius' patient #1/Wrestling girl #1 Ranita Raggo	 ... The Straight Shooters Waitress Nancy Riehle	 ... Pedestrian, Nurse, Atheist, voice of Atheist Woman, Cadaver #2, Wresting Girl Lucky Ron	 ... Blind Jimmy Leper Jose Sanchez	 ... The Transvestite Steve Sauve	 ... Pedestrian, Homeless Person Casey Tourangeau	 ... Singer, Musical Scene Xcrunner5k 20:26, 23 September 2007 (UTC)
 * And Nicholas Edwards is not credited there, but he is credited in the article.

%ce%a0_%28film%29
In the article someone had written about traditional vampire-killing; until a myth is proven as actual fact, the elements are, as well, mythical.

Above, someone had written about pi,... vam pi re?? Spanish pies, food of defea[e]ted vampires.

[[ hopiakuta Please do  sign  your  signature  on your  message. %7e%7e  Thank You. -]] 11:35, 8 October 2007 (UTC)

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title: Talk:Jesus Christ Vampire Hunter


text:
Fujiwara no Kamatari

Fujiwara no Kamatari (藤原 鎌足, 614 – November 14, 669), also known as Nakatomi no Kamatari (中臣 鎌足), was a Japanese politician and aristocrat who, together with Prince Naka no Ōe (later Emperor Tenji), carried out the Taika Reform. He was the founder of the Fujiwara clan, the most powerful aristocratic family in Japan during the Nara and Heian periods. He, along with the Mononobe clan, was a supporter of Shinto and fought the introduction of Buddhism to Japan. The Soga clan, defenders of Buddhism in the Asuka period, defeated Kamatari and the Mononobe clan, and Buddhism became the dominant religion of the Imperial Court. Kamatari was appointed Inner Minister, and, along with Prince Naka no Ōe, later Emperor Tenji (626–672), launched the Taika Reform of 645, which centralized and strengthened the central government. Just before his death, he received the surname Fujiwara and the rank Taishokan from Emperor Tenji, thus establishing the Fujiwara clan.

Biography
Kamatari was born to the Nakatomi clan, an aristocratic kin group claiming descent from their ancestral god Ame-no-Koyane. He was the son of Nakatomi no Mikeko, and named Nakatomi no Kamatari (中臣 鎌足) at birth. His early life and exploits are described in the 8th century clan history Tōshi Kaden (藤氏家伝).

He was a friend and supporter of the Prince Naka no Ōe, later Emperor Tenji. Kamatari was the head of the Jingi no Haku, or Shinto ritualists; as such, he was one of the chief opponents of the increasing power and prevalence of Buddhism in the court, and in the nation. As a result, in 645, Prince Naka no Ōe and Kamatari made a coup d'état in the court. They slew Soga no Iruka who had a strong influence over Empress Kōgyoku; thereafter, Iruka's father, Soga no Emishi, committed suicide.

Empress Kōgyoku was forced to abdicate in favor of her younger brother, who became Emperor Kōtoku; Kōtoku then appointed Kamatari naidaijin (Inner Minister).

Kamatari was a leader in the development of what became known as the Taika Reforms, a major set of reforms based on Chinese models and aimed at strengthening Imperial power. He acted as one of the principal editors responsible for the development of the Japanese legal code known as Sandai-kyaku-shiki, sometimes referred to as the Rules and Regulations of the Three Generations.

During his life Kamatari continued to support Prince Naka no Ōe, who became Emperor Tenji in 661. Tenji granted him the highest rank Taishōkan (or Daishokukan) (大織冠) and a new clan name, Fujiwara (藤原), as honors.

Legacy

Kamatari's son was Fujiwara no Fuhito. Kamatari's nephew, Nakatomi no Omimaro became head of Ise Shrine, and passed down the Nakatomi name.

In the 13th century, the main line of the Fujiwara family split into five houses: Konoe, Takatsukasa, Kujō, Nijō and Ichijō. These five families in turn provided regents for the Emperors, and were thus known as the Five Regent Houses. The Tachibana clan (samurai) also claimed descent from the Fujiwara. Emperor Montoku of the Taira clan was descended through his mother to the Fujiwara.

Until the marriage of the Crown Prince Hirohito (posthumously Emperor Shōwa) to Princess Kuni Nagako (posthumously Empress Kōjun) in January 1924, the principal consorts of emperors and crown princes had always been recruited from one of the Sekke Fujiwara. Imperial princesses were often married to Fujiwara lords - throughout a millennium at least. As recently as Emperor Shōwa's third daughter, the late former Princess Takanomiya (Kazoku), and Prince Mikasa's elder daughter, the former Princess Yasuko, married into Takatsukasa and Konoe families, respectively. Empress Shōken was a descendant of the Fujiwara clan and through Hosokawa Gracia of the Minamoto clan. Likewise a daughter of the last Tokugawa Shōgun married a second cousin of Emperor Shōwa.

Among Kamatari's descendants are Fumimaro Konoe the 34th/38th/39th Prime Minister of Japan and Konoe's grandson Morihiro Hosokawa the 79th Prime Minister of Japan (who is also a descendant of the Hosokawa clan via the Ashikaga clan of the Minamoto clan).

Abuyama Kofun
Abuyama Kofun, a megalithic tomb in Takatsuki and Ibaraki, Osaka has been identified as Fujiwara no Kamatari's tomb. The tomb and a mummy buried inside a coffin were first discovered in 1934. 50 years later, radiographic images and samples taken at the time were examined uncovering a mummy wrapped in gold thread. The kanmuri headwear found in the tomb indicates that the person buried was a noble of the highest rank Taishokkan. It was concluded that it is highly likely that the tomb was dedicated to Kamatari. According to the analysis, the mummified person had a strong bone structure and an athletic body, with the so-called pitcher's elbow. The cause of death was complications from injuries to the vertebral column and lumbar vertebrae sustained from a fall from horseback or a high ground. The injury is thought to have left the lower body paralyzed and caused secondary complications such as pneumonia or urinary tract infection. The cause of death matches with that of Kamatari's, whom is recorded to have died from a fall from horseback.

Higashinara site
In 2014, the Ibaraki City Education Committee announced that ancient sen bricks discovered at Higashinara site in Ibaraki, Osaka match with the bricks found in Abuyama Kofun. The site is believed to have been the location of Mishima Betsugyō, a villa where Kamatari stayed before the Isshi Incident which triggered the Taika Reform (645).

Family

 * Father: Nakatomi no Mikeko (中臣御食子)
 * Mother: Ōtomo no Chisen-no-iratsume (大伴智仙娘), daughter of Otomo no Kuiko (大伴囓子). Also known as "Ōtomo-bunin" (大伴夫人).
 * Main wife: Kagami no Ōkimi (鏡王女, ?-683)
 * Wife: Kurumamochi no Yoshiko-no-iratsume (車持与志古娘), daughter of Kurumamochi no Kuniko (車持国子).
 * 1st son: Jo'e (定恵, 643–666), buddhist monk who traveled to China.
 * 2nd son: Fujiwara no Fuhito (藤原不比等, 659–720)
 * Children with unknown mother:'''
 * Daughter: Fujiwara no Hikami-no-iratsume (藤原氷上娘, ?–682), Bunin of Emperor Tenmu, mother of Princess Tajima.
 * Daughter: Fujiwara no Ioe-no-iratsume (藤原五百重娘), Bunin of Emperor Tenmu, wife of Fujiwara no Fuhito and mother of Prince Niitabe and Fujiwara no Maro.
 * Daughter: Fujiwara no Mimimotoji (藤原耳面刀自), Bunin of Emperor Kōbun, mother of Princess Ichishi-hime (壱志姫王).
 * Daughter: Fujiwara no Tome/Tone-no-iratsume (藤原斗売娘), wife of Nakatomi no Omimaro (中臣意美麻呂), mother of Nakatomi no Azumahito (中臣東人).

Popular culture

 * Portrayed by Noh Seung-jin in the 2012–2013 KBS1 TV series Dream of the Emperor.
 * Portrayed by Junichi Okada in 2005 NHK special drama Taika Kaishin.

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

title: Fujiwara no Kamatari


text:
Martin Newell (computer scientist)

Martin Edward Newell is a British-born computer scientist specializing in computer graphics who is perhaps best known as the creator of the Utah teapot computer model.

Career
Before emigrating to the US, he worked at what was then the Computer-Aided Design Centre (CADCentre) in Cambridge, UK, along with his brother Dick Newell (who went on to co-found two of the most important UK graphics software companies – Cambridge Interactive Systems (CIS) in 1977 and Smallworld in 1987). At CADCentre, the two Newells and Tom Sancha developed Newell's algorithm, a technique for eliminating cyclic dependencies when ordering polygons to be drawn by a computer graphics system.

Newell developed the Utah teapot while working on a Ph.D. at the University of Utah, where he also helped develop a version of the painter's algorithm for rendering. He graduated in 1975, and was on the Utah faculty from 1977 to 1979. Later he worked at Xerox PARC, where he worked on JaM, a predecessor of PostScript. JaM stood for "John and Martin" – the John was John Warnock, co-founder of Adobe Systems.

Newell departed Xerox PARC to join CADLINC Inc., a factory automation startup, as VP of Advanced Development. There he led the development of a variety of CAD/CAM software applications, such as CimCAD (a 3-D drafting program) and Intelligent Documentation (an early electronic document editor integrating text, graphics, and information from relational databases).

He departed CADLINC to found the computer-aided design software company Ashlar in 1988. In 2007, Newell was elected a member of the National Academy of Engineering for contributions to computer-graphics modeling, rendering, and printing. He recently retired as an Adobe Fellow at Adobe Systems.

title: Martin Newell (computer scientist)


text:
Poisson manifold

In differential geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics.

A Poisson structure (or Poisson bracket) on a smooth manifold $$ M $$ is a function$$ \{ \cdot,\cdot \}: \mathcal{C}^\infty(M) \times \mathcal{C}^\infty(M) \to \mathcal{C}^\infty(M) $$on the vector space $$ {C^{\infty}}(M) $$ of smooth functions on $$ M $$, making it into a Lie algebra subject to a Leibniz rule (also known as a Poisson algebra). Poisson structures on manifolds were introduced by André Lichnerowicz in 1977 and are named after the French mathematician Siméon Denis Poisson, due to their early appearance in his works on analytical mechanics.

A Poisson structure on a manifold $$M$$ gives a way of deforming the product of functions on $$M$$ to a new product that is typically not commutative. This process is known as deformation quantization, since classical mechanics can be based on Poisson structures, while quantum mechanics involves non-commutative rings.

From phase spaces of classical mechanics to symplectic and Poisson manifolds
In classical mechanics, the phase space of a physical system consists of all the possible values of the position and of the momentum variables allowed by the system. It is naturally endowed with a Poisson bracket/symplectic form (see below), which allows one to formulate the Hamilton equations and describe the dynamics of the system through the phase space in time.

For instance, a single particle freely moving in the $$ n $$-dimensional Euclidean space (i.e. having $$ \mathbb{R}^n $$ as configuration space) has phase space $$ \mathbb{R}^{2n} $$. The coordinates $$ (q^1,...,q^n,p_1,...,p_n) $$ describe respectively the positions and the generalised momenta. The space of observables, i.e. the smooth functions on $$ \mathbb{R}^{2n} $$, is naturally endowed with a binary operation called the Poisson bracket, defined as

 * $$ \{ f,g \} := \sum_{i=1}^n \left( \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} - \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} \right) .$$

Such a bracket satisfies the standard properties of a Lie bracket, plus a further compatibility with the product of functions, namely the Leibniz identity $$ \{f,g \cdot h\} = g \cdot \{f,h\} + \{f,g\} \cdot h $$. Equivalently, the Poisson bracket on $$ \mathbb{R}^{2n} $$ can be reformulated using the symplectic form

 * $$ \omega := \sum_{i=1}^n dp_i \wedge dq^i .$$

Indeed, if one considers the Hamiltonian vector field

 * $$ X_f := \sum_{i=1}^n \frac{\partial f}{\partial p_i} \partial_{q_i} - \frac{\partial f}{\partial q_i} \partial_{p_i} $$

associated to a function $$ f $$, then the Poisson bracket can be rewritten as $$ \{f,g\} = \omega (X_f,X_g). $$

A standard example of a symplectic manifold, and thus of a Poisson manifold, is the cotangent bundle $$ T^*Q $$ of any finite-dimensional smooth manifold $$ Q .$$ The coordinates on $$ Q $$ are interpreted as particle positions; the space of tangents at each point forming the space of (canonically) conjugate momenta. If $$ Q $$ is $$ n $$-dimensional, $$ T^*Q $$ is a smooth manifold of dimension $$ 2n ;$$ it can be regarded as the associated phase space. The cotangent bundle is naturally equipped with a canonical symplectic form, which, in canonical coordinates, coincides with the one described above. In general, by Darboux theorem, any arbitrary symplectic manifold $$ (M,\omega) $$ admits special coordinates where the form $$ \omega $$ and the bracket $$ \{f,g\} = \omega (X_f,X_g) $$ are equivalent with, respectively, the symplectic form and the Poisson bracket of $$ \mathbb{R}^{2n} $$. Symplectic geometry is therefore the natural mathematical setting to describe classical Hamiltonian mechanics.

Poisson manifolds are further generalisations of symplectic manifolds, which arise by axiomatising the properties satisfied by the Poisson bracket on $$\mathbb{R}^{2n}$$. More precisely, a Poisson manifold consists of a smooth manifold $$M$$ (not necessarily of even dimension) together with an abstract bracket $$\{\cdot,\cdot\}: \mathcal{C}^\infty(M) \times \mathcal{C}^\infty(M) \to \mathcal{C}^\infty(M) $$, still called Poisson bracket, which does not necessarily arise from a symplectic form $$\omega$$, but satisfies the same algebraic properties.

Poisson geometry is closely related to symplectic geometry: for instance every Poisson bracket determines a foliation of the manifold into symplectic submanifolds. However, the study of Poisson geometry requires techniques that are usually not employed in symplectic geometry, such as the theory of Lie groupoids and algebroids.

Moreover, there are natural examples of structures which should be "morally" symplectic, but exhibit singularities, i.e. their "symplectic form" should be allowed to be degenerate. For example, the smooth quotient of a symplectic manifold by a group acting by symplectomorphisms is a Poisson manifold, which in general is not symplectic. This situation models the case of a physical system which is invariant under symmetries: the "reduced" phase space, obtained quotienting the original phase space by the symmetries, in general is no longer symplectic, but is Poisson.

History
Although the modern definition of Poisson manifold appeared only in the 70's–80's, its origin dates back to the nineteenth century. Alan Weinstein summarized the early history of Poisson geometry as follows:"'Poisson invented his brackets as a tool for classical dynamics. Jacobi realized the importance of these brackets and elucidated their algebraic properties, and Lie began the study of their geometry.'"

Indeed, Siméon Denis Poisson introduced in 1809 what we now call Poisson bracket in order to obtain new integrals of motion, i.e. quantities which are preserved throughout the motion. More precisely, he proved that, if two functions $$ f $$ and $$ g $$ are integrals of motion, then there is a third function, denoted by $$ \{ f,g \} $$, which is an integral of motion as well. In the Hamiltonian formulation of mechanics, where the dynamics of a physical system is described by a given function $$ h $$ (usually the energy of the system), an integral of motion is simply a function $$ f $$ which Poisson-commutes with $$ h $$, i.e. such that $$ \{f,h\} = 0 $$. What will become known as Poisson's theorem can then be formulated as

 * $$ \{f,h\} = 0, \{g,h\} = 0 \Rightarrow \{\{f,g\},h\} = 0.$$

Poisson's computations occupied many pages, and his results were rediscovered and simplified two decades later by Carl Gustav Jacob Jacobi. Jacobi was the first to identify the general properties of the Poisson bracket as a binary operation. Moreover, he established the relation between the (Poisson) bracket of two functions and the (Lie) bracket of their associated Hamiltonian vector fields, i.e.$$ X_{\{f,g\}} = [X_f,X_g],$$in order to reformulate (and give a much shorter proof of) Poisson's theorem on integrals of motion. Jacobi's work on Poisson brackets influenced the pioneering studies of Sophus Lie on symmetries of differential equations, which led to the discovery of Lie groups and Lie algebras. For instance, what are now called linear Poisson structures (i.e. Poisson brackets on a vector space which send linear functions to linear functions) correspond precisely to Lie algebra structures. Moreover, the integrability of a linear Poisson structure (see below) is closely related to the integrability of its associated Lie algebra to a Lie group.

The twentieth century saw the development of modern differential geometry, but only in 1977 did André Lichnerowicz introduce Poisson structures as geometric objects on smooth manifolds. Poisson manifolds were further studied in the foundational 1983 paper of Alan Weinstein, where many basic structure theorems were first proved.

These works exerted a huge influence in the subsequent decades on the development of Poisson geometry, which today is a field of its own, and at the same time is deeply entangled with many others, including non-commutative geometry, integrable systems, topological field theories and representation theory.

Formal definition
There are two main points of view to define Poisson structures: it is customary and convenient to switch between them.

As bracket
Let $$ M $$ be a smooth manifold and let $$ {C^{\infty}}(M) $$ denote the real algebra of smooth real-valued functions on $$ M $$, where the multiplication is defined pointwise. A Poisson bracket (or Poisson structure) on $$ M $$ is an $$ \mathbb{R} $$-bilinear map

 * $$ \{ \cdot,\cdot \}: {C^{\infty}}(M) \times {C^{\infty}}(M) \to {C^{\infty}}(M) $$

defining a structure of Poisson algebra on $$ {C^{\infty}}(M) $$, i.e. satisfying the following three conditions:
 * Skew symmetry: $$ \{ f,g \} = - \{ g,f \} $$.
 * Jacobi identity: $$ \{ f,\{ g,h \} \} + \{ g,\{ h,f \} \} + \{ h,\{ f,g \} \} = 0 $$.
 * Leibniz's Rule: $$ \{ f g,h \} = f \{ g,h \} + g \{ f,h \} $$.

The first two conditions ensure that $$ \{ \cdot,\cdot \} $$ defines a Lie-algebra structure on $$ {C^{\infty}}(M) $$, while the third guarantees that, for each $$ f \in {C^{\infty}}(M) $$, the linear map $$ X_f := \{ f,\cdot \}: {C^{\infty}}(M) \to {C^{\infty}}(M) $$ is a derivation of the algebra $$ {C^{\infty}}(M) $$, i.e., it defines a vector field $$ X_{f} \in \mathfrak{X}(M) $$ called the Hamiltonian vector field associated to $$ f $$.

Choosing local coordinates $$ (U, x^i) $$, any Poisson bracket is given by$$ \{f, g\}_{\mid U} = \sum_{i,j} \pi^{ij} \frac{\partial f}{\partial x^i} \frac{\partial g}{\partial x^j}, $$for $$ \pi^{ij} = \{ x^i, x^j \} $$ the Poisson bracket of the coordinate functions.

As bivector
A Poisson bivector on a smooth manifold $$ M $$ is a bivector field $$ \pi \in \mathfrak{X}^2(M) := \Gamma \big( \wedge^{2} T M \big) $$ satisfying the non-linear partial differential equation $$ [\pi,\pi] = 0 $$, where

 * $$ [\cdot,\cdot]: {\mathfrak{X}^{p}}(M) \times {\mathfrak{X}^{q}}(M) \to {\mathfrak{X}^{p + q - 1}}(M) $$

denotes the Schouten–Nijenhuis bracket on multivector fields. Choosing local coordinates $$ (U, x^i) $$, any Poisson bivector is given by$$ \pi_{\mid U} = \sum_{i,j} \pi^{ij} \frac{\partial}{\partial x^i} \frac{\partial}{\partial x^j}, $$for $$ \pi^{ij} $$ skew-symmetric smooth functions on $$ U $$.

Equivalence of the definitions
Let $$ \{ \cdot,\cdot \} $$ be a bilinear skew-symmetric bracket (called an "almost Lie bracket") satisfying Leibniz's rule; then the function $$ \{ f,g \} $$ can be described a$$ \{ f,g \} = \pi(df \wedge dg), $$for a unique smooth bivector field $$ \pi \in \mathfrak{X}^2(M) $$. Conversely, given any smooth bivector field $$ \pi $$ on $$ M $$, the same formula $$ \{ f,g \} = \pi(df \wedge dg) $$ defines an almost Lie bracket $$ \{ \cdot,\cdot \} $$ that automatically obeys Leibniz's rule.

Then the following integrability conditions are equivalent:

A Poisson structure without any of the four requirements above is also called an almost Poisson structure.
 * $$ \{ \cdot,\cdot \} $$ satisfies the Jacobi identity (hence it is a Poisson bracket);
 * $$ \pi $$ satisfies $$ [\pi,\pi] = 0 $$ (hence it a Poisson bivector);
 * the map $$ {C^{\infty}}(M) \to \mathfrak{X}(M), f \mapsto X_f $$ is a Lie algebra homomorphism, i.e. the Hamiltonian vector fields satisfy $$ [X_f, X_g] = X_{\{f,g\}} $$;
 * the graph $$ {\rm Graph}(\pi) \subset TM \oplus T^*M $$ defines a Dirac structure, i.e. a Lagrangian subbundle $$ D \subset TM \oplus T^*M $$ which is closed under the standard Courant bracket.

Holomorphic Poisson structures
The definition of Poisson structure for real smooth manifolds can be also adapted to the complex case.

A holomorphic Poisson manifold is a complex manifold $$M$$ whose sheaf of holomorphic functions $$ \mathcal{O}_M $$ is a sheaf of Poisson algebras. Equivalently, recall that a holomorphic bivector field $$\pi$$ on a complex manifold $$M$$ is a section $$ \pi \in \Gamma (\wedge^2 T^{1,0}M)$$ such that $$ \bar{\partial} \pi = 0$$. Then a holomorphic Poisson structure on $$M $$ is a holomorphic bivector field satisfying the equation $$[\pi,\pi]=0$$. Holomorphic Poisson manifolds can be characterised also in terms of Poisson-Nijenhuis structures.

Many results for real Poisson structures, e.g. regarding their integrability, extend also to holomorphic ones.

Holomorphic Poisson structures appear naturally in the context of generalised complex structures: locally, any generalised complex manifold is the product of a symplectic manifold and a holomorphic Poisson manifold.

Deformation quantization
The notion of a Poisson manifold arises naturally from the deformation theory of associative algebras. For a smooth manifold $$M$$, the smooth functions $$C^{\infty}(M)$$ form a commutative algebra over the real numbers $$\mathbf{R}$$, using pointwise addition and multiplication (meaning that $$(fg)(x) = f(x)g(x)$$ for points $$x$$ in $$M$$). An $$n$$th-order deformation of this algebra is given by a formula
 * $$ f*g = fg + \epsilon B_1(f,g) + \cdots + \epsilon^n B_n(f,g) \pmod{\epsilon^{n+1}}$$

for $$f,g\in C^{\infty}(M)$$ such that the star-product is associative (modulo $$\epsilon^{n+1}$$), but not necessarily commutative.

A first-order deformation of $$C^{\infty}(M)$$ is equivalent to an almost Poisson structure as defined above, that is, a bilinear "bracket" map
 * $$ \{ \cdot,\cdot \}: {C^{\infty}}(M) \times {C^{\infty}}(M) \to {C^{\infty}}(M) $$

that is skew-symmetric and satisfies Leibniz's Rule. Explicitly, one can go from the deformation to the bracket by
 * $$f*g-g*f=\epsilon \{ f,g \} \pmod{\epsilon^2}.$$

A first-order deformation is also equivalent to a bivector field, that is, a smooth section of $$\wedge^2 TM$$.

A bracket satisfies the Jacobi identity (that is, it is a Poisson structure) if and only if the corresponding first-order deformation of $$C^{\infty}(M)$$ can be extended to a second-order deformation. Remarkably, the Kontsevich quantization formula shows that every Poisson manifold has a deformation quantization. That is, if a first-order deformation of $$C^{\infty}(M)$$ can be extended to second order, then it can be extended to infinite order.

Example: For any smooth manifold $$M$$, the cotangent bundle $$T^*M$$ is a symplectic manifold, and hence a Poisson manifold. The corresponding non-commutative deformation of $$C^{\infty}(T^*M)$$ is related to the algebra of linear differential operators on $$M$$. When $$M$$ is the real line $$\mathbf{R}$$, the non-commutativity of the algebra of differential operators (known as the Weyl algebra) follows from the calculation that
 * $$\bigg[ \frac{\partial}{\partial x},x \bigg] =1.$$

Symplectic leaves
A Poisson manifold is naturally partitioned into regularly immersed symplectic manifolds of possibly different dimensions, called its symplectic leaves. These arise as the maximal integral submanifolds of the completely integrable singular foliation spanned by the Hamiltonian vector fields.

Rank of a Poisson structure
Recall that any bivector field can be regarded as a skew homomorphism, the musical morphism $$ \pi^{\sharp}: T^{*} M \to T M, \alpha \mapsto \pi(\alpha,\cdot) $$. The image $$ {\pi^{\sharp}}(T^{*} M) \subset TM $$ consists therefore of the values $$ {X_{f}}(x) $$ of all Hamiltonian vector fields evaluated at every $$ x \in M $$.

The rank of $$ \pi $$ at a point $$ x \in M $$ is the rank of the induced linear mapping $$ \pi^{\sharp}_{x} $$. A point $$ x \in M $$ is called regular for a Poisson structure $$ \pi $$ on $$ M $$ if and only if the rank of $$ \pi $$ is constant on an open neighborhood of $$ x \in M $$; otherwise, it is called a singular point. Regular points form an open dense subspace $$ M_{\mathrm{reg}} \subseteq M $$; when $$ M_{\mathrm{reg}} = M $$, i.e. the map $$ \pi^\sharp $$ is of constant rank, the Poisson structure $$ \pi $$ is called regular. Examples of regular Poisson structures include trivial and nondegenerate structures (see below).

The regular case
For a regular Poisson manifold, the image $$ {\pi^{\sharp}}(T^{*} M) \subset TM $$ is a regular distribution; it is easy to check that it is involutive, therefore, by the Frobenius theorem, $$ M $$ admits a partition into leaves. Moreover, the Poisson bivector restricts nicely to each leaf, which therefore become symplectic manifolds.

The non-regular case
For a non-regular Poisson manifold the situation is more complicated, since the distribution $$ {\pi^{\sharp}}(T^{*} M) \subset TM $$ is singular, i.e. the vector subspaces $$ {\pi^{\sharp}}(T^{*}_x M) \subset T_xM $$ have different dimensions.

An integral submanifold for $$ {\pi^{\sharp}}(T^{*} M) $$ is a path-connected submanifold $$ S \subseteq M $$ satisfying $$ T_{x} S = {\pi^{\sharp}}(T^{\ast}_{x} M) $$ for all $$ x \in S $$. Integral submanifolds of $$ \pi $$ are automatically regularly immersed manifolds, and maximal integral submanifolds of $$ \pi $$ are called the leaves of $$ \pi $$.

Moreover, each leaf $$ S $$ carries a natural symplectic form $$ \omega_{S} \in {\Omega^{2}}(S) $$ determined by the condition $$ [{\omega_{S}}(X_{f},X_{g})](x) = - \{ f,g \}(x) $$ for all $$ f,g \in {C^{\infty}}(M) $$ and $$ x \in S $$. Correspondingly, one speaks of the symplectic leaves of $$ \pi $$. Moreover, both the space $$ M_{\mathrm{reg}} $$ of regular points and its complement are saturated by symplectic leaves, so symplectic leaves may be either regular or singular.

Weinstein splitting theorem
To show the existence of symplectic leaves in the non-regular case, one can use Weinstein splitting theorem (or Darboux-Weinstein theorem). It states that any Poisson manifold $$ (M^n, \pi) $$ splits locally around a point $$ x_0 \in M $$ as the product of a symplectic manifold $$ (S^{2k}, \omega) $$ and a transverse Poisson submanifold $$ (T^{n-2k}, \pi_T) $$ vanishing at $$ x_0 $$. More precisely, if $$ \mathrm{rank}(\pi_{x_0}) = 2k $$, there are local coordinates $$ (U, p_1,\ldots,p_k,q^1,\ldots, q^k,x^1,\ldots,x^{n-2k}) $$ such that the Poisson bivector $$ \pi $$ splits as the sum
 * $$ \pi_{\mid U} = \sum_{i=1}^{k} \frac{\partial}{\partial q^i} \frac{\partial}{\partial p_i} + \frac{1}{2} \sum_{i,j=1}^{n-2k} \phi^{ij}(x) \frac{\partial}{\partial x^i} \frac{\partial}{\partial x^j}, $$

where $$ \phi^{ij}(x_0) = 0 .$$ Notice that, when the rank of $$ \pi $$ is maximal (e.g. the Poisson structure is nondegenerate, so that $$n=2k$$), one recovers the classical Darboux theorem for symplectic structures.

Trivial Poisson structures
Every manifold $$ M $$ carries the trivial Poisson structure $$ \{ f,g \} = 0 $$, equivalently described by the bivector $$ \pi=0 $$. Every point of $$ M $$ is therefore a zero-dimensional symplectic leaf.

Nondegenerate Poisson structures
A bivector field $$ \pi $$ is called nondegenerate if $$ \pi^{\sharp}: T^{*} M \to T M $$ is a vector bundle isomorphism. Nondegenerate Poisson bivector fields are actually the same thing as symplectic manifolds $$ (M,\omega) $$.

Indeed, there is a bijective correspondence between nondegenerate bivector fields $$ \pi $$ and nondegenerate 2-forms $$ \omega $$, given by the musical isomorphism
 * $$ \pi^\sharp = (\omega^{\flat})^{-1}, $$

where $$ \omega $$ is encoded by $$ \omega^{\flat}: TM \to T^*M, \quad v \mapsto \omega(v,\cdot) $$. Furthermore, $$ \pi $$ is Poisson precisely if and only if $$ \omega $$ is closed; in such case, the bracket becomes the canonical Poisson bracket from Hamiltonian mechanics:
 * $$ \{ f,g \} := \omega (X_f,X_g). $$

Non-degenerate Poisson structures have only one symplectic leaf, namely $$ M $$ itself, and their Poisson algebra $$ (\mathcal{C}^{\infty}(M), \{\cdot, \cdot \}) $$ become a Poisson ring.

Linear Poisson structures
A Poisson structure $$ \{ \cdot, \cdot \} $$ on a vector space $$ V $$ is called linear when the bracket of two linear functions is still linear.

The class of vector spaces with linear Poisson structures coincides with that of the duals of Lie algebras. The dual $$ \mathfrak{g}^{*} $$ of any finite-dimensional Lie algebra $$ (\mathfrak{g},[\cdot,\cdot]) $$ carries a linear Poisson bracket, known in the literature under the names of Lie-Poisson, Kirillov-Poisson or KKS (Kostant-Kirillov-Souriau) structure:$$ \{ f, g \} (\xi) := \xi ([d_\xi f,d_\xi g]_{\mathfrak{g}}), $$where $$ f,g \in \mathcal{C}^{\infty}(\mathfrak{g}^*), \xi \in \mathfrak{g}^* $$ and the derivatives $$ d_\xi f, d_\xi g: T_{\xi} \mathfrak{g}^* \to \mathbb{R} $$ are interpreted as elements of the bidual $$ \mathfrak{g}^{**} \cong \mathfrak{g} $$. Equivalently, the Poisson bivector can be locally expressed as$$ \pi = \sum_{i,j,k} c^{ij}_k x^k \frac{\partial}{\partial x^i} \frac{\partial}{\partial x^j}, $$where $$ x^i $$ are coordinates on $$ \mathfrak{g}^{*} $$ and $$ c_k^{ij} $$ are the associated structure constants of $$ \mathfrak{g} $$,

Conversely, any linear Poisson structure $$ \{ \cdot, \cdot \} $$ on $$ V $$ must be of this form, i.e. there exists a natural Lie algebra structure induced on $$ \mathfrak{g}:=V^* $$ whose Lie-Poisson bracket recovers $$ \{ \cdot, \cdot \} $$.

The symplectic leaves of the Lie-Poisson structure on $$ \mathfrak{g}^* $$ are the orbits of the coadjoint action of $$ G $$ on $$ \mathfrak{g}^* $$.

Fibrewise linear Poisson structures
The previous example can be generalised as follows. A Poisson structure on the total space of a vector bundle $$ E \to M $$ is called fibrewise linear when the bracket of two smooth functions $$ E \to \mathbb{R} $$, whose restrictions to the fibres are linear, results in a bracket that is linear when restricted to the fibres. Equivalently, the Poisson bivector field $$ \pi $$ is asked to satisfy $$ (m_t)^*\pi = t \pi $$ for any $$ t >0 $$, where $$ m_t: E \to E $$ is the scalar multiplication $$ v \mapsto tv $$.

The class of vector bundles with linear Poisson structures coincides with that of the duals of Lie algebroids. The dual $$ A^* $$ of any Lie algebroid $$ (A, [\cdot, \cdot]) $$ carries a fibrewise linear Poisson bracket, uniquely defined by$$ \{ \mathrm{ev}_\alpha, \mathrm{ev}_\beta \}:= ev_{[\alpha,\beta]} \quad \quad \forall \alpha, \beta \in \Gamma(A), $$where $$ \mathrm{ev}_\alpha: A^* \to \mathbb{R}, \phi \mapsto \phi(\alpha) $$ is the evaluation by $$ \alpha $$. Equivalently, the Poisson bivector can be locally expressed as$$ \pi = \sum_{i,a} B^i_a(x) \frac{\partial}{\partial y_a} \frac{\partial}{\partial x^i} + \sum_{a < b,c} C_{ab}^c(x) y_c \frac{\partial}{\partial y_a} \frac{\partial}{\partial y_b}, $$where $$ x^i $$ are coordinates around a point $$ x \in M $$, $$ y_a $$ are fibre coordinates on $$ A^* $$, dual to a local frame $$ e_a $$ of $$ A $$, and $$ B^i_a $$ and $$ C^c_{ab} $$ are the structure function of $$ A $$, i.e. the unique smooth functions satisfying$$ \rho(e_a) = \sum_i B^i_a (x) \frac{\partial}{\partial x^i}, \quad \quad [e_a, e_b] = \sum_c C^c_{ab} (x) e_c. $$Conversely, any fibrewise linear Poisson structure $$ \{ \cdot, \cdot \} $$ on $$ E $$ must be of this form, i.e. there exists a natural Lie algebroid structure induced on $$ A:=E^* $$ whose Lie-Poisson backet recovers $$ \{ \cdot, \cdot \} $$.

The symplectic leaves of $$ A^* $$ are the cotangent bundles of the algebroid orbits $$ \mathcal{O} \subseteq A $$; equivalently, if $$ A $$ is integrable to a Lie groupoid $$ \mathcal{G} \rightrightarrows M $$, they are the connected components of the orbits of the cotangent groupoid $$ T^* \mathcal{G} \rightrightarrows A^* $$.

For $$ M = \{*\} $$ one recovers linear Poisson structures, while for $$ A = TM $$ the fibrewise linear Poisson structure is the nondegenerate one given by the canonical symplectic structure of the cotangent bundle $$ T^*M $$.

Other examples and constructions

 * Any constant bivector field on a vector space is automatically a Poisson structure; indeed, all three terms in the Jacobiator are zero, being the bracket with a constant function.
 * Any bivector field on a 2-dimensional manifold is automatically a Poisson structure; indeed, $$ [\pi,\pi] $$ is a 3-vector field, which is always zero in dimension 2.
 * Given any Poisson bivector field $$ \pi $$ on a 3-dimensional manifold $$ M $$, the bivector field $$ f \pi $$, for any $$ f \in \mathcal{C}^\infty(M) $$, is automatically Poisson.
 * The Cartesian product $$ (M_{0} \times M_{1},\pi_{0} \times \pi_{1}) $$ of two Poisson manifolds $$ (M_{0},\pi_{0}) $$ and $$ (M_{1},\pi_{1}) $$ is again a Poisson manifold.
 * Let $$ \mathcal{F} $$ be a (regular) foliation of dimension $$ 2 r $$ on $$ M $$ and $$ \omega \in {\Omega^{2}}(\mathcal{F}) $$ a closed foliation two-form for which the power $$ \omega^{r} $$ is nowhere-vanishing. This uniquely determines a regular Poisson structure on $$ M $$ by requiring the symplectic leaves of $$ \pi $$ to be the leaves $$ S $$ of $$ \mathcal{F} $$ equipped with the induced symplectic form $$ \omega|_S $$.
 * Let $$ G $$ be a Lie group acting on a Poisson manifold $$ (M,\pi) $$ by Poisson diffeomorphisms. If the action is free and proper, the quotient manifold $$ M/G $$ inherits a Poisson structure $$ \pi_{M/G} $$ from $$ \pi $$ (namely, it is the only one such that the submersion $$ (M,\pi) \to (M/G,\pi_{M/G}) $$ is a Poisson map).

Poisson cohomology
The Poisson cohomology groups $$ H^k(M,\pi) $$ of a Poisson manifold are the cohomology groups of the cochain complex$$ \ldots \xrightarrow{d_\pi} \mathfrak{X}^\bullet(M) \xrightarrow{d_\pi} \mathfrak{X}^{\bullet+1}(M) \xrightarrow{d_\pi} \ldots \color{white}{\sum^i} $$

where the operator $$ d_\pi = [\pi,-] $$ is the Schouten-Nijenhuis bracket with $$ \pi $$. Notice that such a sequence can be defined for every bivector on $$ M $$; the condition $$ d_\pi \circ d_\pi = 0 $$ is equivalent to $$ [\pi,\pi]=0 $$, i.e. $$ M $$ being Poisson.

Using the morphism $$ \pi^{\sharp}: T^{*} M \to T M $$, one obtains a morphism from the de Rham complex $$ (\Omega^\bullet(M),d_{dR}) $$ to the Poisson complex $$ (\mathfrak{X}^\bullet(M), d_\pi) $$, inducing a group homomorphism $$ H_{dR}^\bullet(M) \to H^\bullet(M,\pi) $$. In the nondegenerate case, this becomes an isomorphism, so that the Poisson cohomology of a symplectic manifold fully recovers its de Rham cohomology.

Poisson cohomology is difficult to compute in general, but the low degree groups contain important geometric information on the Poisson structure:

 * $$ H^0(M,\pi) $$ is the space of the Casimir functions, i.e. smooth functions Poisson-commuting with all others (or, equivalently, smooth functions constant on the symplectic leaves);
 * $$ H^1(M,\pi) $$ is the space of Poisson vector fields modulo Hamiltonian vector fields;
 * $$ H^2(M,\pi) $$ is the space of the infinitesimal deformations of the Poisson structure modulo trivial deformations;
 * $$ H^3(M,\pi) $$ is the space of the obstructions to extend infinitesimal deformations to actual deformations.

Modular class
The modular class of a Poisson manifold is a class in the first Poisson cohomology group, which is the obstruction to the existence of a volume form invariant under the Hamiltonian flows. It was introduced by Koszul and Weinstein.

Recall that the divergence of a vector field $$X \in \mathfrak{X}(M)$$ with respect to a given volume form $$\lambda$$ is the function $${\rm div}_\lambda (X) \in \mathcal{C}^\infty(M)$$ defined by $$ {\rm div}_\lambda (X) = \frac{\mathcal{L}_{X} \lambda}{\lambda}$$. The modular vector field of a Poisson manifold, with respect to a volume form $$\lambda$$, is the vector field $$X_\lambda$$ defined by the divergence of the Hamiltonian vector fields: $$X_\lambda: f \mapsto {\rm div}_\lambda (X_f)$$.

The modular vector field is a Poisson 1-cocycle, i.e. it satisfies $$\mathcal{L}_{X_\lambda} \pi = 0$$. Moreover, given two volume forms $$\lambda_1$$ and $$\lambda_2$$, the difference $$X_{\lambda_1} - X_{\lambda_2}$$ is a Hamiltonian vector field. Accordingly, the Poisson cohomology class $$[X_\lambda]_\pi \in H^1 (M,\pi) $$ does not depend on the original choice of the volume form $$\lambda$$, and it is called the modular class of the Poisson manifold.

A Poisson manifold is called unimodular if its modular class vanishes. Notice that this happens if and only if there exists a volume form $$\lambda$$ such that the modular vector field $$X_\lambda$$ vanishes, i.e. $$ {\rm div}_\lambda (X_f) = 0$$ for every $$f$$; in other words, $$\lambda$$ is invariant under the flow of any Hamiltonian vector field. For instance:

 * Symplectic structures are always unimodular, since the Liouville form is invariant under all Hamiltonian vector fields;
 * For linear Poisson structures the modular class is the infinitesimal modular character of $$\mathfrak{g}$$, since the modular vector field associated to the standard Lebesgue measure on $$\mathfrak{g}^*$$ is the constant vector field on $$\mathfrak{g}^*$$. Then $$\mathfrak{g}^*$$ is unimodular as Poisson manifold if and only if it is unimodular as Lie algebra;
 * For regular Poisson structures the modular class is related to the Reeb class of the underlying symplectic foliation (an element of the first leafwise cohomology group, which obstructs the existence of a volume normal form invariant by vector fields tangent to the foliation).

Poisson homology
Poisson cohomology was introduced in 1977 by Lichnerowicz himself; a decade later, Brylinski introduced a homology theory for Poisson manifolds, using the operator $$\partial_\pi = [d, \iota_\pi]$$.

Several results have been proved relating Poisson homology and cohomology. For instance, for orientable unimodular Poisson manifolds, Poisson homology turns out to be isomorphic to Poisson cohomology: this was proved independently by Xu and Evans-Lu-Weinstein.

Poisson maps
A smooth map $$ \varphi: M \to N $$ between Poisson manifolds is called a  if it respects the Poisson structures, i.e. one of the following equivalent conditions holds (compare with the equivalent definitions of Poisson structures above):

 * the Poisson brackets $$ \{ \cdot,\cdot \}_{M} $$ and $$ \{ \cdot,\cdot \}_{N} $$ satisfy $$ {\{ f,g \}_{N}}(\varphi(x)) = {\{ f \circ \varphi,g \circ \varphi \}_{M}}(x) $$ for every $$ x \in M $$ and smooth functions $$ f,g \in {C^{\infty}}(N) $$
 * the bivector fields $$ \pi_{M} $$ and $$ \pi_{N} $$ are $$ \varphi $$-related, i.e. $$ \pi_N = \varphi_* \pi_M $$
 * the Hamiltonian vector fields associated to every smooth function $$ H \in \mathcal{C}^\infty(N) $$ are $$ \varphi $$-related, i.e. $$X_H = \varphi_* X_{H \circ \phi}$$
 * the differential $$ d\varphi: (TM,{\rm Graph}(\pi_M)) \to (TN,{\rm Graph}(\pi_N)) $$ is a Dirac morphism.

An anti-Poisson map satisfies analogous conditions with a minus sign on one side.

Poisson manifolds are the objects of a category $$ \mathfrak{Poiss} $$, with Poisson maps as morphisms. If a Poisson map $$\varphi: M\to N$$ is also a diffeomorphism, then we call $$\varphi$$ a Poisson-diffeomorphism.

Examples

 * Given the product Poisson manifold $$ (M_{0} \times M_{1},\pi_{0} \times \pi_{1}) $$, the canonical projections $$ \mathrm{pr}_{i}: M_{0} \times M_{1} \to M_{i} $$, for $$ i \in \{ 0,1 \} $$, are Poisson maps.
 * The inclusion mapping of a symplectic leaf, or of an open subspace, is a Poisson map.
 * Given two Lie algebras $$ \mathfrak{g} $$ and $$ \mathfrak{h} $$, the dual of any Lie algebra homomorphism $$ \mathfrak{g} \to \mathfrak{h} $$ induces a Poisson map $$ \mathfrak{h}^* \to \mathfrak{g}^* $$ between their linear Poisson structures.
 * Given two Lie algebroids $$ A \to M $$ and $$ B \to M $$, the dual of any Lie algebroid morphism $$ A \to B $$ over the identity induces a Poisson map $$ B^* \to A^* $$ between their fibrewise linear Poisson structure.

One should notice that the notion of a Poisson map is fundamentally different from that of a symplectic map. For instance, with their standard symplectic structures, there exist no Poisson maps $$ \mathbb{R}^{2} \to \mathbb{R}^{4} $$, whereas symplectic maps abound.

Symplectic realisations
A symplectic realisation on a Poisson manifold M consists of a symplectic manifold $$ (P,\omega) $$ together with a Poisson map $$ \phi: (P,\omega) \to (M,\pi) $$ which is a surjective submersion. Roughly speaking, the role of a symplectic realisation is to "desingularise" a complicated (degenerate) Poisson manifold by passing to a bigger, but easier (non-degenerate), one.

Notice that some authors define symplectic realisations without this last condition (so that, for instance, the inclusion of a symplectic leaf in a symplectic manifold is an example) and call full a symplectic realisation where $$ \phi $$ is a surjective submersion. Examples of (full) symplectic realisations include the following:

A symplectic realisation $$ \phi $$ is called complete if, for any complete Hamiltonian vector field $$X_H$$, the vector field $$X_{H \circ \phi}$$ is complete as well. While symplectic realisations always exist for every Poisson manifold (and several different proofs are available), complete ones do not, and their existence plays a fundamental role in the integrability problem for Poisson manifolds (see below).
 * For the trivial Poisson structure $$ (M,0 ) $$, one takes as $$ P $$ the cotangent bundle $$ T^*M $$, with its canonical symplectic structure, and as $$ \phi $$the projection $$ T^*M \to M $$.
 * For a non-degenerate Poisson structure $$ (M,\omega) $$ one takes as $$ P $$ the manifold $$ M $$ itself and as $$ \phi $$ the identity $$ M \to M $$.
 * For the Lie-Poisson structure on $$ \mathfrak{g}^* $$, one takes as $$ P $$ the cotangent bundle $$ T^*G $$ of a Lie group $$ G $$ integrating $$ \mathfrak{g} $$ and as $$ \phi $$ the dual map $$ \phi: T^*G \to \mathfrak{g}^* $$ of the differential at the identity of the (left or right) translation $$ G \to G $$.

Integration of Poisson manifolds
Any Poisson manifold $$ (M,\pi) $$ induces a structure of Lie algebroid on its cotangent bundle $$ T^*M \to M $$, also called the cotangent algebroid. The anchor map is given by $$ \pi^{\sharp}: T^{*} M \to T M $$ while the Lie bracket on $$ \Gamma(T^*M) = \Omega^1(M) $$ is defined as$$ [\alpha, \beta] := \mathcal{L}_{\pi^\sharp(\alpha)} (\beta) - \iota_{\pi^\sharp(\beta)} d\alpha = \mathcal{L}_{\pi^\sharp(\alpha)} (\beta) - \mathcal{L}_{\pi^\sharp(\beta)} (\alpha) - d\pi (\alpha, \beta). $$Several notions defined for Poisson manifolds can be interpreted via its Lie algebroid $$ T^*M $$:

 * the symplectic foliation is the usual (singular) foliation induced by the anchor of the Lie algebroid;
 * the symplectic leaves are the orbits of the Lie algebroid;
 * a Poisson structure on $$ M $$ is regular precisely when the associated Lie algebroid $$ T^*M $$ is;
 * the Poisson cohomology groups coincide with the Lie algebroid cohomology groups of $$ T^*M $$ with coefficients in the trivial representation;
 * the modular class of a Poisson manifold coincides with the modular class of the associated Lie algebroid $$ T^*M $$.

It is of crucial importance to notice that the Lie algebroid $$ T^*M $$ is not always integrable to a Lie groupoid.

Symplectic groupoids
A  is a Lie groupoid $$ \mathcal{G} \rightrightarrows M $$ together with a symplectic form $$ \omega \in \Omega^2(\mathcal{G}) $$ which is also multiplicative, i.e. it satisfies the following algebraic compatibility with the groupoid multiplication: $$ m^*\omega = {\rm pr}_1^* \omega + {\rm pr}_2^* \omega $$. Equivalently, the graph of $$ m $$ is asked to be a Lagrangian submanifold of $$ (\mathcal{G} \times \mathcal{G} \times \mathcal{G}, \omega \oplus \omega \oplus - \omega) $$. Among the several consequences, the dimension of $$ \mathcal{G} $$ is automatically twice the dimension of $$ M $$. The notion of symplectic groupoid was introduced at the end of the 80's independently by several authors.

A fundamental theorem states that the base space of any symplectic groupoid admits a unique Poisson structure $$ \pi $$ such that the source map $$ s: (\mathcal{G}, \omega) \to (M,\pi) $$ and the target map $$ t: (\mathcal{G}, \omega) \to (M,\pi) $$ are, respectively, a Poisson map and an anti-Poisson map. Moreover, the Lie algebroid $$ {\rm Lie}(\mathcal{G}) $$ is isomorphic to the cotangent algebroid $$ T^*M $$ associated to the Poisson manifold $$ (M,\pi) $$. Conversely, if the cotangent bundle $$ T^*M $$ of a Poisson manifold is integrable to some Lie groupoid $$ \mathcal{G} \rightrightarrows M $$, then $$ \mathcal{G} $$ is automatically a symplectic groupoid.

Accordingly, the integrability problem for a Poisson manifold consists in finding a (symplectic) Lie groupoid which integrates its cotangent algebroid; when this happens, the Poisson structure is called integrable.

While any Poisson manifold admits a local integration (i.e. a symplectic groupoid where the multiplication is defined only locally), there are general topological obstructions to its integrability, coming from the integrability theory for Lie algebroids. Using such obstructions, one can show that a Poisson manifold is integrable if and only if it admits a complete symplectic realisation.

The candidate $$ \Pi(M,\pi) $$ for the symplectic groupoid integrating a given Poisson manifold $$ (M,\pi) $$ is called Poisson homotopy groupoid and is simply the Weinstein groupoid of the cotangent algebroid $$ T^*M \to M $$, consisting of the quotient of the Banach space of a special class of paths in $$ T^*M $$ by a suitable equivalent relation. Equivalently, $$ \Pi(M,\pi) $$ can be described as an infinite-dimensional symplectic quotient.

Examples of integrations

 * The trivial Poisson structure $$ (M,0) $$ is always integrable, the symplectic groupoid being the bundle of abelian (additive) groups $$ T^*M \rightrightarrows M $$ with the canonical symplectic form.
 * A non-degenerate Poisson structure on $$ M $$ is always integrable, the symplectic groupoid being the pair groupoid $$ M \times M \rightrightarrows M $$ together with the symplectic form $$ s^* \omega - t^* \omega $$ (for $$ \pi^\sharp = (\omega^{\flat})^{-1} $$).
 * A Lie-Poisson structure on $$ \mathfrak{g}^* $$ is always integrable, the symplectic groupoid being the (coadjoint) action groupoid $$ G \times \mathfrak{g}^* \rightrightarrows \mathfrak{g}^* $$, for $$ G $$ the simply connected integration of $$ \mathfrak{g} $$, together with the canonical symplectic form of $$ T^*G \cong G \times \mathfrak{g}^* $$.
 * A Lie-Poisson structure on $$ A^* $$ is integrable if and only if the Lie algebroid $$ A \to M $$ is integrable to a Lie groupoid $$ \mathcal{G} \rightrightarrows M $$, the symplectic groupoid being the cotangent groupoid $$ T^*\mathcal{G} \rightrightarrows A^* $$ with the canonical symplectic form.

Submanifolds
A Poisson submanifold of $$ (M, \pi) $$ is an immersed submanifold $$ N \subseteq M $$ such that the immersion map $$ (N,\pi_{\mid N}) \hookrightarrow (M,\pi) $$ is a Poisson map. Equivalently, one asks that every Hamiltonian vector field $$ X_f $$, for $$ f \in \mathcal{C}^\infty(M) $$, is tangent to $$ N $$.

This definition is very natural and satisfies several good properties, e.g. the transverse intersection of two Poisson submanifolds is again a Poisson submanifold. However, it has also a few problems:

 * Poisson submanifolds are rare: for instance, the only Poisson submanifolds of a symplectic manifold are the open sets;
 * the definition does not behave functorially: if $$ \Phi: (M,\pi_M) \to (N,\pi_N) $$ is a Poisson map transverse to a Poisson submanifold $$ Q $$ of $$ N $$, the submanifold $$ \Phi^{-1} (Q) $$ of $$ M $$ is not necessarily Poisson.

In order to overcome these problems, one often uses the notion of a Poisson transversal (originally called cosymplectic submanifold). This can be defined as a submanifold $$ X \subseteq M $$ which is transverse to every symplectic leaf $$ S $$ and such that the intersection $$ X \cap S $$ is a symplectic submanifold of $$ (S,\omega_S) $$. It follows that any Poisson transversal $$ X \subseteq (M,\pi) $$ inherits a canonical Poisson structure $$ \pi_X $$ from $$ \pi $$. In the case of a nondegenerate Poisson manifold $$ (M, \pi) $$ (whose only symplectic leaf is $$ M $$ itself), Poisson transversals are the same thing as symplectic submanifolds.

More general classes of submanifolds play an important role in Poisson geometry, including Lie–Dirac submanifolds, Poisson–Dirac submanifolds, coisotropic submanifolds and pre-Poisson submanifolds.

Books and surveys

 * Previous version available on.
 * See also the review by Ping Xu in the Bulletin of the AMS.
 * Previous version available on.
 * See also the review by Ping Xu in the Bulletin of the AMS.
 * See also the review by Ping Xu in the Bulletin of the AMS.
 * See also the review by Ping Xu in the Bulletin of the AMS.
 * See also the review by Ping Xu in the Bulletin of the AMS.

title: Poisson manifold


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Talk:The Missouri Bootheel

On 1 August 2004, this article was nominated for deletion. It was subsequently turned into a redirect. The decision was to keep the now-harmless redirect. See Votes for deletion/The Missouri Bootheel for a record of the discussion. Rossami 05:00, 7 Aug 2004 (UTC)

title: Talk:The Missouri Bootheel


text: Category:South Asia

title: Category:South Asia


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Talk:Strigolniki

Where would one go to learn more about this regrettable incident of the drowning of the Strigolniki? Do we know the names or offices of the church officials who instigated the mob? Wesley 05:03, 7 Aug 2004 (UTC)

Russian Orthodox?
I assume that they were a sect of the Russian Orthodox Church. Shouldn't the article say so with a link? --Samuel Wantman 05:54, 8 Aug 2004 (UTC)

No, they weren't. KNewman 22:33, Aug 9, 2004 (UTC)

Actually, the Strigol'niki are considered the first Russian heretics. This means that they did not accept certain teachings or practices of the official Orthodox Church. A link would be helpful.

Several etymologies have been suggested for the word strigol'niki. One of the most persuasive is that of A. I. Alekseev, who cites Rule 21 of the Sixth Universal [Church] Council to the effect that those who are defrocked as priests and repent have the right to cut their hair as do the clergy. If they do not repent they must grow their hair as do all other laymen. Mr Alekseev's discussion of the strigol'niki can be found in his article On the Study of the Strigol'nikovs' Heresy [in Russian]. It appeared in Papers of the Participants of the Second International Conference "The Complex Approach to the study of Ancient Rus'" [in Russian]. His paper can be found at http://www.drevnyaya.ru/vyp/stat/s4_18_2.pdf --Brambeus (talk) 01:21, 10 January 2011 (UTC)

title: Talk:Strigolniki


text:
I Only Have Eyes for You (Buffy the Vampire Slayer)

"I Only Have Eyes for You" is episode 19 of season two of  the television series Buffy the Vampire Slayer. It was written by Marti Noxon, directed by James Whitmore Jr., and first broadcast on April 28, 1998, on The WB. In the episode, Sunnydale High is haunted by the ghosts of a teacher and a student who fell in love years ago, recreating their tragedy by possessing the bodies of students and staff preparing for the upcoming Sadie Hawkins dance.

Written as a ghost story dealing with the themes of second chance and regret, the episode received positive reviews from critics.

Plot
Buffy stops a male student from shooting a female student. Afterwards, they have no recollection of why they were arguing and the gun disappears. Principal Snyder blames Buffy for the incident. While waiting in his office, a yearbook from 1955 falls off the shelf. In class later that day, Buffy starts daydreaming about a relationship that a student had with his teacher. As she comes back to the present, she finds that her teacher has unknowingly written "Don't walk away from me, bitch!" on the black board, a line the male student had shouted. Later, Xander is grabbed by a monster arm from inside his locker. Buffy helps him break away, but the arm then disappears.

Giles is intrigued by the possible presence of a poltergeist. Later that night, Giles sees the janitor shoot a teacher, though only moments earlier they were cordial. Giles is convinced that Jenny Calendar is haunting the school.

Willow finds information on her laptop about a killing in 1955, where student James Stanley killed his teacher Grace Newman after she tried to break off their affair. In the cafeteria, chaos erupts as the food turns into snakes. Outside, Snyder talks to the police chief about the incident and it is revealed that they know about the Hellmouth under the school and the possibility of Buffy being the Slayer. Drusilla gets a vision about Buffy meeting with death.

Willow devises a plan to contain the spirits, and they head off to the school where they prepare, though Giles has already arrived and is trying to summon Jenny's spirit. Buffy hears music coming from the music room and sees Grace and James dancing there. James' face suddenly changes to a gory mess, startling Buffy. On the stairwell, Willow begins to sink into the floor and Giles saves her. Willow finally convinces him that the spirit is not Jenny. A swarm of wasps arrives and surrounds the school.

Everyone recuperates at Buffy's while she continues to show her anger towards James. Buffy goes to the kitchen where she finds a sign for the 1955 Sadie Hawkins dance in her pocket as James' voice whispers, "I need you." She heads to the school where the wasps part for her to enter. Willow finds the ad and everyone rushes after Buffy, but they cannot enter the school.

Angelus appears in the halls as Buffy, now possessed by James, talks to him as if he were Grace. They repeat the ghosts' argument until, at the climax, Buffy pulls out a gun and shoots Angelus. He falls off of the balcony as though dead. Buffy, still possessed by James, goes to the music room where "James" plans to kill himself. The possessed Angelus arrives just in time to stop Buffy from pulling the trigger. They exchange apologies and kiss. The spirits leave their bodies and are finally finding peace. Buffy and Angelus break away from the kiss, and Angelus, disgusted at what he has been doing, pushes Buffy away and flees. Angelus and Drusilla leave Spike after Angelus taunts him and Spike breaks free from his wheelchair.

Production details
Scriptwriter Marti Noxon said that prior to working on Buffy, she had written several times about ghosts, which for her are figurative expressions of the need for "repentance and second chances", drawing on her own family background. "I realize that I was constantly telling the story of my family and fears," she says. Noxon was influenced in her storytelling by the movies Poltergeist and Truly, Madly, Deeply, which featured a widow who was unable to move on after the loss of her husband.

Cultural references
In the episode "Out of Mind, Out of Sight," their English class discussed William Shakespeare's The Merchant of Venice. Here, Xander alludes to the play by saying, "The quality of mercy is not Buffy."

When Willow suggests an exorcism of the ghosts, Cordelia replies, "Are you crazy? I saw that movie. Even the priest died!" She refers to The Exorcist.

The website Women at Warp compares the "past love affair gone bad plays out again and again" theme with the Star Trek: The Next Generation episode "Eye of the Beholder." "In both episodes, the spirit left behind by a man whose love affair turned tragic inhabits the minds of key female characters. When Troi investigates a crew member's suicide, she becomes entranced by the telepathic memory of Lt. Pierce, who killed his lover and then killed himself. Seeing through his eyes, she seems to live out his memories, killing Lt. Worf then attempting to kill herself. In the Buffyverse, the ghost of a young man who—that's right—killed his lover then himself, causes Buffy to attempt to act out the same events with her lover, Angel. Both are ultimately saved by the lovers they thought they murdered."

Reception
Vox rated it #45 in a list of all 144 episodes, writing, "The smart, moving twist, as penned by Marti Noxon, is that when Buffy and Angel (excuse me, Angelus) end up possessed, the gender roles are reversed, with Buffy consumed by the guilt and self-loathing James feels for killing the person he loved. It's an unexpected, effective window into Buffy's psyche in the aftermath of Angel's transformation, and while James's spirit eventually makes peace with his actions, the episode shows Buffy still has a long way to go toward forgiving herself for her part in Angel losing his soul."

The A.V. Club described the episode as "a really beautiful episode that raises the subtext of the season to the surface in a surprising and emotionally powerful way" and "a story about obsession and repetition". BBC praised the direction and the script, and said that "it's impossible not to be charmed by the whole picture. The ending, where Buffy and Angel re-enact James and Grace's final moments, is such a compelling twist that it suddenly seems that the rest of the episode was written to explain the finale, rather than the other way around." The review opined that the final scene of the episode, in which Spike regained his health, was a "very welcome development" but also an "ill-judged addition" because it broke the "spell" of the screenplay, and because it teased the season finale, it should have been used in the next episode.


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Wikipedia:Articles for deletion/Yelf

Yelf was proposed for deletion. This page is an archive of the discussion about the proposed deletion. This page is no longer live. Further comments should be made on the article's talk page rather than here so that this page is preserved as an historic record. The result of the debate was delete

Stubby history of a non-notable family name. RickK 05:08, Aug 7, 2004 (UTC) This page is now preserved as an archive of the debate and, like other '/delete' pages is no longer 'live'. Subsequent comments on the issue, the deletion or on the decision-making process should be placed on the relevant 'live' pages. Please do not edit this page.
 * Delete: belongs in m:GlobalFamilyTree if anything (if and when that project begins). My opinion is the same for all these not-particularly-notable families and family names that are cropping up. TPK 11:50, 7 Aug 2004 (UTC)
 * Delete: We're not a genealogy project.  It needs to be fleshed out a bit before really being contributory on the Family Tree project, I'd guess.  Saying that it's a Danelaw family that went forth and was fruitful is nebulous to the point of meaninglessness. Geogre 12:18, 7 Aug 2004 (UTC)
 * Delete. Either move it to another area or get rid of it altogether, but it doesn't seem particularly noteworthy or interesting to anyone other than someone with that particular last name. Not Wikipedia material. Skyler 03:05, Aug 9, 2004 (UTC)
 * Delete --Vik-Thor 03:53, 12 Aug 2004 (UTC)
 * Delete. Not encyclopedic. -- Ian Maxwell 05:05, 2004 Aug 12 (UTC)

title: Wikipedia:Articles for deletion/Yelf


text:
Talk:Welsh Assembly Election 2007

On 1 Aug 2004, this article was nominated for deletion as a "speculative future event." The discussion did not reach a clear consensus to delete this article. See Votes for deletion/Welsh Assembly Election 2007 for a record of the discussion. Rossami 05:12, 7 Aug 2004 (UTC)

title: Talk:Welsh Assembly Election 2007


text:
Derrick Thomas

Derrick Vincent Thomas (January 1, 1967 – February 8, 2000), nicknamed "D. T.", was an American football linebacker for the Kansas City Chiefs of the National Football League (NFL). Considered one of the greatest pass rushers of all time, he played 11 seasons with the Chiefs until his death in 2000. Thomas played college football for the Alabama Crimson Tide, where he won the Butkus Award, and was selected fourth overall by Kansas City in the 1989 NFL draft. During his career, he received nine Pro Bowl and two first-team All-Pro selections, and set the single-game sacks record.

After the Chiefs' 1999 season, Thomas was rendered paraplegic by a car crash and died two weeks later from a pulmonary embolism. He was posthumously inducted to the Pro Football Hall of Fame in 2009 and the College Football Hall of Fame in 2014.

Early life
Born in Miami, Florida, Thomas was raised by his mother Edith Morgan. His father, Air Force Captain and B-52 pilot Robert James Thomas, died during a mission in the Vietnam War when Derrick was five years old. Thomas played his high school football at South Miami Senior High School.

College career
Alongside Cornelius Bennett and later Keith McCants at Alabama, Thomas spearheaded one of the best defensive lines in college football and smashed many Crimson Tide defensive records, including sacks in a single season. He was awarded the Butkus Award in 1988. He was also selected as a consensus All-American at the conclusion of the 1988 season, a season which culminated in the Crimson Tide's thrilling 29-28 victory over Army in the 1988 Sun Bowl. In 2000, Thomas was named a Sun Bowl Legend. He was awarded the Sington Soaring Spirit Award by the Lakeshore Foundation. This annual award is named for University of Alabama football legend Fred Sington. Thomas was posthumously inducted to the College Football Hall of Fame in 2014.

Professional career
Thomas was selected fourth overall in the first round of the 1989 NFL draft by the Kansas City Chiefs. He was the first selection made by new head coach Marty Schottenheimer.

Thomas would record his first career sack in the Chiefs week 2 game against the Los Angeles Raiders. That game was also his first multi-sack game as he finished with 2.5 sacks. He would record another 7.5 sacks that season finishing with 10 his rookie year. He was named AP Defensive Rookie of the Year. He was also named to the Pro Bowl.

In his second year, Thomas recorded at least a half of a sack in each of the Chiefs first 5 games. In the Chiefs week 10 game against the Seattle Seahawks, he broke Fred Dean's 7-year old record of sacks in a single game with 7 sacks. The quarterback he sacked in that game, Dave Krieg, would become his teammate 2 seasons later. However, on the game's final play, Thomas had a clear shot for an eighth sack, but missed it and the Seahawks would throw a game winning touchdown after the missed sack. He would later call that play the one play in his career he wished he could have a second chance at. He would finish the season with what would prove to be a career high 20 sacks, setting a franchise record that stood until it was broken by Justin Houston in 2014. He finished 2nd in defensive player of the year voting and was named 1st team All-Pro.

The following season, he would record his first career touchdown on a 23-yard fumble return for a touchdown in the Chiefs week 11 game against the Los Angeles Rams.

Thomas would record double digits sacks for each of the first four seasons in his career. In total, 7 of his 11 seasons in the NFL he recorded double digit sack totals.

In the Chiefs 1999 season, Thomas recorded a career low for sacks with 7. He would also record his first career interception in the Chiefs week 8 win over the San Diego Chargers. He would record the final sack of his career in the Chiefs week 15 against the Pittsburgh Steelers. In what would be the final game of his career, as he would die 37 days later, the Chiefs played their rival the Oakland Raiders. With a victory in the game, the Chiefs would qualify for the playoffs. The Chiefs lost in overtime 41–38. He didn't manage to record a sack on his former teammate Rich Gannon, but he did record 6 total tackles.

Legacy
Thomas was named 1st team All-Pro 2 times and was named to the Pro Bowl 9 times. He is 17th all-time in sacks and at the time of his death in 2000, he was 9th all-time with 126.5. He remains the Chiefs' all-time leader in sacks, safeties, forced fumbles, and fumble recoveries. During his career, he recorded 1 interception and recovered 19 fumbles, returning them for 161 yards and 4 touchdowns. Thomas said in interviews his favorite quarterback to sack was Denver Broncos quarterback John Elway, whom he sacked 26 times. The sack total over Elway is most against any quarterback in Thomas' career and the most any individual player sacked Elway. He was posthumously inducted into the Chiefs Hall of Fame in 2000, with the Chiefs breaking the tradition of waiting four years after the end of the player's career. In 2009, he was posthumously inducted into the Pro Football Hall of Fame. Later that same year, the Chiefs retired the number 58 in honor of him. Following his death until it was officially retired, the Chiefs did not reissue the number. The Chiefs also named their player of the year award in Thomas' honor, an award he won twice himself prior to the award being named in his honor. He is the only NFL player to die during their career that was later inducted to the Hall of Fame.

Death
On January 23, 2000, Thomas' SUV went off Interstate 435 in Clay County as he and two passengers were driving to Kansas City International Airport during a snowstorm for a flight to St. Louis to watch the NFC Championship Game between the St. Louis Rams and the Tampa Bay Buccaneers. Police reports indicated that Thomas, who was driving, was speeding at approximately 70 mph even though snow and ice were rapidly accumulating on the roadway. Thomas and one of the passengers were not wearing seat belts and both were thrown from the car; the passenger, Michael Tellis, was killed instantly. The second passenger, who was wearing his safety belt, walked away from the scene uninjured. Thomas was left paralyzed from the chest down. By early February, Thomas was being treated at Miami's Jackson Memorial Hospital. The morning of February 8, 2000, while being transferred from his hospital bed to a wheelchair on his way to therapy, Thomas told his mother he was not feeling well. His eyes then rolled back, recalled Frank Eismont, an orthopedic surgeon at Jackson Memorial Hospital. Eismont said Thomas went into cardiorespiratory arrest and died as a result of a pulmonary embolism, a massive blood clot that developed in his legs and traveled to his lungs. Months later, Thomas' family sued General Motors for $73 million in damages stemming from the accident. In 2004, a jury ruled that the family was not entitled to any money.

Charity work
In 1990, Thomas founded the Derrick Thomas Third and Long Foundation. The foundation's mission is to "sack illiteracy" and change the lives of 9- to 13-year-old urban children facing challenging and life-threatening situations in the Kansas City area.

The Derrick Thomas Academy, a charter school in Kansas City, Missouri, opened in September 2001. It served nearly 1,000 children from kindergarten through eighth grade until it closed in 2013.

title: Derrick Thomas


text:
Talk:Hot Topic

Hot Topic Foundation
Hello everyone, I will be editing the article this week to include some information about the Hot Topic Foundation. Just wanted to let you all know! Kylee Truoccolo (talk) 23:25, 23 September 2019 (UTC)

 * Hello. Thanks for starting a discussion on the talk page. I'm sorry I didn't notice this sooner. Because Wikipedia is not a platform for promotion, we need to make sure any such information is explained appropriately. The best way to do that is by citing reliable sources. These sources will almost certainly also be independent sources. Press releases and Hot Topic's own website are only of limited use for this. Corporate donations are not likely to be significant without a good reason. This reason would have to be supported by a reliable source. That context could, potentially, be included in the article. Without such a source, it is too routine to be worth mentioning. Grayfell (talk) 04:07, 28 September 2019 (UTC)

question
what is there mission statement? 142.202.125.228 (talk) 15:07, 4 January 2024 (UTC)

title: Talk:Hot Topic


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Symplectic space

A symplectic space may refer to:
 * Symplectic manifold
 * Symplectic vector space

title: Symplectic space


text: Talk:Symplectic space

title: Talk:Symplectic space


text:
User talk:Juromaru

Hello and welcome to Wikipedia! Hope you like it here, and stick around.

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Juromaru picture
You appear to have uploaded a picture to the Juromaru article, presumably in AOL ART format.

Did you make it yourself? If so, could you save it in PNG format instead, or at least JPEG? Many people have browsers that lack support for ART format (including me), and it is not really a very good quality format anyway.

If you did not make it, please let me know where you found it, so I can convert it to a PNG file.

In either case, please add copyright information to the image page. For details on how to do that, see the image use policy. The image copyright tags are useful for that as well.

-- Josh 06:20, Aug 7, 2004 (UTC) (Talk)

AfD nomination of Kaijinbō
Kaijinbō, an article you created, has been nominated for deletion. We appreciate your contributions. However, an editor does not feel that Kaijinbō satisfies Wikipedia's criteria for inclusion and has explained why in the nomination space (see also "What Wikipedia is not" and the Wikipedia deletion policy). Your opinions on the matter are welcome; please participate in the discussion by adding your comments at Articles for deletion/Kaijinbō and please be sure to sign your comments with four tildes ( ~ ). You are free to edit the content of Kaijinbō during the discussion but should not remove the articles for deletion template from the top of the article; such removal will not end the deletion discussion. Thank you. Pilotbob (talk) 03:28, 22 November 2007 (UTC)

Gatenmaru
A proposed deletion template has been added to the article Gatenmaru, suggesting that it be deleted according to the proposed deletion process. All contributions are appreciated, but this article may not satisfy Wikipedia's criteria for inclusion, and the deletion notice should explain why (see also "What Wikipedia is not" and Wikipedia's deletion policy). You may prevent the proposed deletion by removing the  notice, but please explain why you disagree with the proposed deletion in your edit summary or on its talk page.

Please consider improving the article to address the issues raised because even though removing the deletion notice will prevent deletion through the proposed deletion process, the article may still be deleted if it matches any of the speedy deletion criteria or it can be sent to Articles for Deletion, where it may be deleted if consensus to delete is reached. Do you want to opt out of receiving this notice? Collectonian (talk) 02:18, 21 April 2008 (UTC)

RfD nomination of Sota Higurashi
I have nominated for discussion. Your opinions on the matter are welcome; please participate in the discussion by adding your comments at the discussion page. Thank you. Elen of the Roads (talk) 23:03, 28 September 2009 (UTC)

title: User talk:Juromaru


text:
User talk:Jaques O. Carvalho

Hello and welcome to Wikipedia! Hope you like it here, and stick around.

Here are some tips to help you get started:

 * To sign your posts (on talk pages, for example) use the '~' symbol. To insert just your name, type ~~~ (3 tildes), or, to insert your name and timestamp, use ~~~~ (4 tildes).
 * Try the Tutorial, and feel free to experiment in the test area.
 * If you need help, post a question at the Help Desk
 * Follow the Simplified Ruleset
 * Eventually, you might want to read the Manual of Style and Policies and Guidelines.
 * Remember Neutral point of view
 * Explore, be bold in editing pages, and, most importantly, have fun!

Good luck!

ArbCom elections are now open!
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 Hello! Voting in the 2022 Arbitration Committee elections is now open until 23:59 (UTC) on. All eligible users are allowed to vote. Users with alternate accounts may only vote once.

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The Arbitration Committee is the panel of editors responsible for conducting the Wikipedia arbitration process. It has the authority to impose binding solutions to disputes between editors, primarily for serious conduct disputes the community has been unable to resolve. This includes the authority to impose site bans, topic bans, editing restrictions, and other measures needed to maintain our editing environment. The arbitration policy describes the Committee's roles and responsibilities in greater detail.

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title: User talk:Jaques O. Carvalho


text:
Al Duerr

Alfred Herman Duerr (born January 29, 1951) is a former Canadian politician who served as the 34th mayor of Calgary, Alberta from 1989 to 2001, and a city alderman from 1983 to 1989.

Early life and education
Duerr was born January 29, 1951, in Humboldt, Saskatchewan, the fourth of six children. He attended Humboldt Collegiate.

Duerr graduated from the University of Saskatchewan with a degree in regional and urban development in 1973. He holds a Master of Business Administration degree from the University of Calgary.

Political career
Duerr was first elected as an alderman for Ward 9 on Calgary City Council in 1983 and was re-elected in 1986. Incumbent Mayor Ralph Klein resigned in the months before the 1989 municipal election to pursue provincial office and Duerr contested and won the mayor's seat in the 1989 election. Duerr would go on to be re-elected as mayor three more times in 1992, 1995, and 1998 before deciding not to contest the 2001 election, which was won by former Alderman Dave Bronconnier.

Duerr's time as mayor coincided with significant spending cuts by the federal government and austerity policies by Ralph Klein's provincial government. Provincial austerity would result in a significant reduction in grant funding for municipalities which in turn resulted in lower capital spending and public construction projects in Calgary.

In 1991, Duerr would proclaim gay pride week in Calgary; however, he would deny future proclamations due to public pressure.

During his four terms, Duerr was considered one of the most popular mayors in the city's history. His conciliatory, managerial style and likeable personality made him well respected. Duerr was given the nickname of "Mr. Beige" by local media columnists, and in the 1998 election would be challenged by local political columnist Rick Bell, although Bell's candidacy was not a serious attempt to contest the mayor's office.

In 1997 while serving as mayor, Duerr played the character of Mayor Wilkins in an episode of the television show Viper.

Later life
He is the Chief Executive Officer of Emergo Projects International, and serves on the board of directors for two other companies in Calgary. He is a member of the Community and Partners Advisory Committee of the Libin Cardiovascular Institute of Alberta.

In 2014, Duerr was appointed Chair of the independent Calgary Ward Boundary Commission to examine the boundaries of Calgary City Council wards in time for the 2017 Calgary municipal election. The Commission recommended against increasing the number of councillors for the City and instead recommended creating satellite offices for larger constituencies and increasing the budget of the councillor's office.

Personal life
Duerr is married to Kit Chan and has two children.

title: Al Duerr


text: User talk:69.111.72.245

title: User talk:69.111.72.245


text:
Dana Stubblefield

Dana William Stubblefield (born November 14, 1970) is an American former professional football player and convicted sex offender. He was a defensive tackle in the National Football League (NFL). After graduating from Taylor High School in North Bend, Ohio, Stubblefield attended the University of Kansas. He is serving a sentence of 15 years to life in prison, after being convicted of rape.

Professional career
Stubblefield was drafted in the first round (26th overall) of the 1993 NFL Draft by the San Francisco 49ers who went to a 4–3 defense at the start of the season. In his rookie year, Stubblefield led the 49ers with 10.5 sacks (making himself the NFL Defensive Rookie of the Year), and recorded 8.5 the following year. He appeared in the Pro Bowl in 1994 and 1995. The 1996 season was less successful for Stubblefield, but his 1997 season found him recording 15 sacks and being named 1997's NFL Defensive Player of the Year by the Associated Press.

After the 1997 season, he became an unrestricted free agent and signed with the Washington Redskins, where his numbers greatly diminished despite the fact that he played opposite Dan Wilkinson, who often drew double-teams. He returned to San Francisco in 2001 and 2002, and played with the Oakland Raiders as a free agent in 2003. In 2004, he was signed by the New England Patriots, but he was injured and was released before the start of the season.

BALCO incident
Stubblefield's name and those of several of his Oakland Raiders team members were found on the list of clients of the Bay Area Laboratory Co-operative that had given performance-enhancing drugs to Marion Jones and others. Although initially he lied to federal investigators about using both EPO and THG, after he was charged on January 18, 2008, Stubblefield cooperated with both federal and NFL investigators and turned over the names of players, agents, and trainers that he suspected of using drugs.

Because of his cooperation with the investigation, he received a fairly light sentence after pleading guilty to lying to federal investigators on February 6, 2009. He served two years' probation.

Personal life
He was formerly the varsity defensive line coach at Valley Christian High School in San Jose, California.

On October 12, 2010, former NFL agent Josh Luchs mentioned in an article for Sports Illustrated that he offered Stubblefield $10,000 cash while he was still playing football at the University of Kansas, but Stubblefield refused to accept it.

On December 9, 2010, San Francisco U.S. District Court judge Susan Illston sentenced Stubblefield to 90 days in jail for stealing his ex-girlfriend's mail by way of fraudulent submission of a change-of-address form.

On May 2, 2016, he was charged with sexually assaulting a disabled woman, who at the time of the incident, was 31 years old. The incident occurred April 9, 2015. Stubblefield originally contacted her through a babysitter website. On July 27, 2020, Stubblefield was convicted of rape and on October 22, 2020, was sentenced to 15 years to life in prison.

title: Dana Stubblefield


text:
Nambu mechanics

In mathematics, Nambu mechanics is a generalization of Hamiltonian mechanics involving multiple Hamiltonians. Recall that Hamiltonian mechanics is based upon the flows generated by a smooth Hamiltonian over a symplectic manifold. The flows are symplectomorphisms and hence obey Liouville's theorem. This was soon generalized to flows generated by a Hamiltonian over a Poisson manifold. In 1973, Yoichiro Nambu suggested a generalization involving Nambu–Poisson manifolds with more than one Hamiltonian.

Nambu bracket
Specifically, consider a differential manifold $M$, for some integer $N ≥ 2$; one has a smooth $N$-linear map from $N$ copies of   $C^{∞} (M)$ to itself, such that it is completely antisymmetric: the Nambu bracket,

 * $$\{h_1,\ldots,h_{N-1}, \cdot \} : C^\infty(M) \times \cdots C^\infty(M) \rightarrow C^\infty(M),$$

which acts as a derivation

 * $$\{h_1,\ldots,h_{N-1},fg\} = \{h_1,\ldots,h_{N-1},f\}g + f\{h_1,\ldots,h_{N-1},g\},$$

whence the Filippov Identities (FI) (evocative of  the  Jacobi identities, but unlike them, not antisymmetrized in all arguments, for  $N ≥ 2$ ):

$$ \{ f_1,\cdots, ~f_{N-1},~ \{ g_1,\cdots,~ g_N\}\} = \{ \{ f_1, \cdots, ~ f_{N-1},~g_1\},~g_2,\cdots,~g_N\}+\{g_1, \{f_1,\cdots,f_{N-1}, ~g_2\},\cdots,g_N\}+\dots $$ $$+\{g_1,\cdots, g_{N-1},\{f_1,\cdots,f_{N-1},~g_N\}\},   $$

so that ${ f_{1}, ..., f_{N−1}, • }$  acts as a generalized derivation over the $N$-fold product ${ . ,..., . }$.

Hamiltonians and flow
There are N − 1 Hamiltonians, $H_{1}, ..., H_{N−1}$, generating an incompressible flow,

 * $$\frac{d}{dt}f = \{f, H_1, \ldots, H_{N-1}\},$$

The generalized phase-space velocity is divergenceless, enabling Liouville's theorem. The case $N = 2$ reduces to  a Poisson manifold, and conventional Hamiltonian mechanics.

For larger even $N$, the $N−1$ Hamiltonians identify with the maximal number of independent invariants of motion (cf. Conserved quantity) characterizing a superintegrable system that evolves in $N$-dimensional phase space. Such systems are also describable by conventional Hamiltonian dynamics; but their description in the framework of Nambu mechanics is substantially more elegant and intuitive, as all invariants enjoy the same geometrical status as the Hamiltonian: the trajectory in phase space is the intersection of the $N − 1$ hypersurfaces specified by these invariants. Thus, the flow is perpendicular to all $N − 1$ gradients of these Hamiltonians, whence parallel to the generalized cross product specified by the respective Nambu bracket.

Nambu mechanics can be extended to fluid dynamics, where the resulting Nambu brackets are non-canonical and the Hamiltonians are identified with the Casimir of the system, such as enstrophy or helicity. Quantizing Nambu dynamics leads to intriguing structures that coincide with conventional quantization ones when superintegrable systems are involved—as they must.

title: Nambu mechanics


text:
Wikipedia:Articles for deletion/Arghoslent (0th nomination)

Arghoslent was proposed for deletion. This page is an archive of the discussion about the proposed deletion. This page is no longer live. Further comments should be made on the article's talk page rather than here so that this page is preserved as an historic record. The result of the debate was delete

Obscure heavy metal band, appears to be neo-nazi. Delete with extreme prejudice. --Shibboleth 05:36, 7 Aug 2004 (UTC)

This page is now preserved as an archive of the debate and, like other '/delete' pages is no longer 'live'. Subsequent comments on the issue, the deletion or on the decision-making process should be placed on the relevant 'live' pages. Please do not edit this page.
 * No entry at all music guide, nor ultimate band list delete Dunc_Harris|☺ 10:58, 7 Aug 2004 (UTC)
 * Delete: Oh, it's a two-for-one: death metal and third reich 'n roll. Not notable, link spam. Geogre 12:16, 7 Aug 2004 (UTC)
 * Delete this nazi spam.  — Chameleon Main/Talk/Images 13:46, 7 Aug 2004 (UTC)
 * Delete. Noteworthy and/or encyclopedic content? Skyler 03:06, Aug 9, 2004 (UTC)
 * Argh indeed. Delete.  Postdlf 03:20, 12 Aug 2004 (UTC)


text:
Parallel communication

In data transmission, parallel communication is a method of conveying multiple binary digits (bits) simultaneously using multiple conductors. This contrasts with serial communication, which conveys only a single bit at a time; this distinction is one way of characterizing a communications link.

The basic difference between a parallel and a serial communication channel is the number of electrical conductors used at the physical layer to convey bits. Parallel communication implies more than one such conductor. For example, an 8-bit parallel channel will convey eight bits (or a byte) simultaneously, whereas a serial channel would convey those same bits sequentially, one at a time. If both channels operated at the same clock speed, the parallel channel would be eight times faster. A parallel channel may have additional conductors for other signals, such as a clock signal to pace the flow of data, a signal to control the direction of data flow, and handshaking signals.

Parallel communication is and always has been widely used within integrated circuits, in peripheral buses, and in memory devices such as RAM. Computer system buses, on the other hand, have evolved over time: parallel communication was commonly used in earlier system buses, whereas serial communications are prevalent in modern computers.

Examples of parallel communication systems

 * Internal buses: memory bus, system bus, and front-side bus
 * IBM System/360 Direct Control Feature (1964).  Standard System/360 had an eight-bit wide port.  The process-control variant Model 44 had a 32-bit width.
 * Legacy computer peripheral buses: ISA, ATA, SCSI, PCI, and the once-ubiquitous IEEE-1284 / Centronics "printer port"
 * Laboratory Instrumentation bus IEEE-488
 * (see more examples at computer bus)

Comparison with serial links
Before the development of high-speed serial technologies, the choice of parallel links over serial links was driven by these factors:

 * Speed: Superficially, the speed of a parallel data link is equal to the number of bits sent at one time times the bit rate of each individual path; doubling the number of bits sent at once doubles the data rate.  In practice, clock skew reduces the speed of every link to the slowest of all of the links.
 * Cable length: Crosstalk creates interference between the parallel lines, and the effect worsens with the length of the communication link. This places an upper limit on the length of a parallel data connection that is usually shorter than a serial connection.
 * Complexity: Parallel data links are easily implemented in hardware, making them a logical choice. Creating a parallel port in a computer system is relatively simple, requiring only a latch to copy data onto a data bus.  In contrast, most serial communication must first be converted back into parallel form by a universal asynchronous receiver/transmitter (UART) before they may be directly connected to a data bus.

The decreasing cost and better performance of integrated circuits has led to serial links being used in favor of parallel links; for example, IEEE 1284 printer ports vs. USB, Parallel ATA vs. Serial ATA, and FireWire or Thunderbolt are now the most common connectors for transferring data from audiovisual (AV) devices such as digital cameras or professional-grade scanners that used to require purchasing a SCSI HBA years ago.

One huge advantage of having fewer wires/pins in a serial cable is the significant reduction in the size, the complexity of the connectors, and the associated costs. Designers of devices such as smartphones benefit from the development of connectors/ports that are small, durable, and still provide adequate performance.

On the other hand, there has been a resurgence of parallel data links in RF communication. Rather than transmitting one bit at a time (as in Morse code and BPSK), well-known techniques such as PSM, PAM, and Multiple-input multiple-output communication send a few bits in parallel. (Each such group of bits is called a "symbol"). Such techniques can be extended to send an entire byte at once (256-QAM).

title: Parallel communication