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In [[mathematics]], a '''Poisson algebra''' is an [[associative algebra]] together with a [[Lie algebra|Lie bracket]] that also satisfies [[product rule|Leibniz' law]]; that is, the bracket is also a [[derivation (abstract algebra)|derivation]]. Poisson algebras appear naturally in [[Hamiltonian mechanics]], and are also central in the study of  [[quantum group]]s. [[Manifold]]s with a Poisson algebra structure are known as [[Poisson manifold]]s, of which the [[symplectic manifold]]s and the [[Poisson-Lie group]]s are a special case.  The algebra is named in honour of [[Siméon Denis Poisson]].
It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you loved this article and you would like to be given more info regarding [http://www.youtube.com/watch?v=90z1mmiwNS8 Washington DC Dentist] nicely visit the web site.
 
==Definition==
A Poisson algebra is a [[vector space]] over a [[field (mathematics)|field]] ''K'' equipped with two [[bilinear map|bilinear]] products, ⋅ and {, }, having the following properties:
 
* The product ⋅ forms an [[associative algebra|associative ''K''-algebra]].
 
* The product {, }, called the [[Poisson bracket]], forms a [[Lie algebra]], and so it is anti-symmetric, and obeys the [[Jacobi identity]].
 
* The Poisson bracket acts as a [[Derivation (abstract algebra)|derivation]] of the associative product ⋅, so that for any three elements ''x'', ''y'' and ''z'' in the algebra, one has {''x'', ''y'' ⋅ ''z''} = {''x'', ''y''} ⋅ ''z'' + ''y'' ⋅ {''x'', ''z''}.
 
The last property often allows a variety of different formulations of the algebra to be given, as noted in the examples below.
 
== Examples ==
Poisson algebras occur in various settings.
 
===Symplectic manifolds===
The space of real-valued [[smooth function]]s over a [[symplectic manifold]] forms a Poisson algebra. On a symplectic manifold, every real-valued  function ''H'' on the manifold induces a vector field ''X<sub>H</sub>'', the [[Hamiltonian vector field]]. Then, given any two smooth functions ''F'' and ''G'' over the symplectic manifold, the Poisson bracket may be defined as:
 
:<math>\{F,G\}=dG(X_F) = X_F(G)\,</math>.
 
This definition is consistent in part because the Poisson bracket acts as a derivation. Equivalently, one may define the bracket {,} as
 
:<math>X_{\{F,G\}}=[X_F,X_G]\,</math>
 
where [,] is the [[Lie derivative]]. When the symplectic manifold is '''R'''<sup>2''n''</sup> with the standard symplectic structure, then the Poisson bracket takes on the well-known form
 
:<math>\{F,G\}=\sum_{i=1}^n \frac{\partial F}{\partial q_i}\frac{\partial G}{\partial p_i}-\frac{\partial F}{\partial p_i}\frac{\partial G}{\partial q_i}.</math>
 
Similar considerations apply for [[Poisson manifold]]s, which generalize symplectic manifolds by allowing the symplectic bivector to be vanishing on some (or trivially, all) of the manifold.
 
===Associative algebras===
If ''A'' is an [[associative algebra]], then the commutator [''x'',''y'']≡''xy''&minus;''yx'' turns it into a Poisson algebra.
 
===Vertex operator algebras===
For a [[vertex operator algebra]] ''(V,Y, ω, 1)'', the space ''V/C<sub>2</sub>(V)'' is a Poisson algebra with ''{a, b}'' = ''a<sub>0</sub>b'' and ''a'' ⋅ ''b'' = ''a<sub>−1</sub>b''.  For certain vertex operator algebras, these Poisson algebras are finite dimensional.
 
==See also==
*[[Poisson superalgebra]]
*[[Antibracket algebra]]
*[[Moyal bracket]]
 
==References==
*{{springer|id=p/p110170|title=Poisson algebra|author=Y. Kosmann-Schwarzbach}}
*{{cite book|first = K. H.|last = Bhaskara|first2 = K.|last2 = Viswanath|title = Poisson algebras and Poisson manifolds|location = |publisher = Longman|year = 1988|isbn = 0-582-01989-3}}
 
[[Category:Algebras]]
[[Category:Symplectic geometry]]

Revision as of 21:42, 20 February 2014

It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.

Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.

At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.

Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.

Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.

Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.

Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.

Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.

In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.

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