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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 have any thoughts pertaining to exactly where and how to use [http://www.youtube.com/watch?v=90z1mmiwNS8 Best Dentists in DC], you can speak to us at the page.
In [[mathematics]], particularly in [[linear algebra]], a '''flag''' is an increasing sequence of [[Linear subspace|subspaces]] of a finite-dimensional [[vector space]] ''V''. Here "increasing" means each is a proper subspace of the next (see [[filtration (abstract algebra)|filtration]]):
:<math>\{0\} = V_0 \sub V_1 \sub V_2 \sub \cdots \sub V_k = V.</math>
If we write the dim ''V''<sub>''i''</sub> = ''d''<sub>''i''</sub> then we have
:<math>0 = d_0 < d_1 < d_2 < \cdots < d_k = n,</math>
where ''n'' is the [[dimension (linear algebra)|dimension]] of ''V'' (assumed to be finite-dimensional). Hence, we must have ''k'' ≤ ''n''. A flag is called a '''complete flag''' if ''d''<sub>''i''</sub> = ''i'', otherwise it is called a '''partial flag'''.
 
A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed (in many different ways) by inserting suitable subspaces.
 
The '''signature''' of the flag is the sequence (''d''<sub>1</sub>, … ''d''<sub>''k''</sub>).
 
Under certain conditions the resulting sequence resembles a [[flag]] with a point connected to a line connected to a surface.
 
==Bases==
An ordered [[basis (linear algebra)|basis]] for ''V'' is said to be adapted to a flag if the first ''d''<sub>''i''</sub> basis vectors form a basis for ''V''<sub>''i''</sub> for each 0 ≤ ''i'' ≤ ''k''. Standard arguments from linear algebra can show that any flag has an adapted basis.
 
Any ordered basis gives rise to a complete flag by letting the ''V''<sub>''i''</sub> be the span of the first ''i'' basis vectors. For example, the '''{{Visible anchor|standard flag}}''' in '''R'''<sup>''n''</sup> is induced from the [[standard basis]] (''e''<sub>1</sub>, ..., ''e''<sub>''n''</sub>) where ''e''<sub>''i''</sub> denotes the vector with a 1 in the ''i''th slot and 0's elsewhere. Concretely, the standard flag is the subspaces:
:<math>0 < \left\langle e_1\right\rangle < \left\langle e_1,e_2\right\rangle < \cdots < \left\langle e_1,\ldots,e_n \right\rangle = K^n.</math>
 
An adapted basis is almost never unique (trivial counterexamples); see below.
 
A complete flag on an [[inner product space]] has an essentially unique [[orthonormal basis]]: it is unique up to multiplying each vector by a unit (scalar of unit length, like 1, -1, ''i''). This is easiest to prove inductively, by noting that <math>v_i \in V_{i-1}^\perp < V_i</math>, which defines it uniquely up to unit.
 
More abstractly, it is unique up to an action of the [[maximal torus]]: the flag corresponds to the [[Borel group]], and the inner product corresponds to the [[maximal compact subgroup]].<ref>Harris, Joe (1991). ''Representation Theory: A First Course'', p. 95. Springer. ISBN 0387974954.</ref>
 
==Stabilizer==
The stabilizer subgroup of the standard flag is the group of invertible [[upper triangular]] matrices.
 
More generally, the stabilizer of a flag (the [[linear operators]] on ''V'' such that <math>T(V_i) < V_i</math> for all ''i'') is, in matrix terms, the [[algebra]] of block [[upper triangular]] matrices (with respect to an adapted basis), where the block sizes <math>d_i-d_{i-1}</math>. The stabilizer subgroup of a complete flag is the set of invertible [[upper triangular]] matrices with respect to any basis adapted to the flag. The subgroup of [[lower triangular]] matrices  with respect to such a basis depends on that basis, and can therefore ''not'' be characterized in terms of the flag only.
 
The stabilizer subgroup of any complete flag is a [[Borel subgroup]] (of the [[general linear group]]), and the stabilizer of any partial flags is a [[parabolic subgroup]].
 
The stabilizer subgroup of a flag acts [[simply transitive]]ly on adapted bases for the flag, and thus these are not unique unless the stabilizer is trivial. That is a very exceptional circumstance: it happens only for a vector space of dimension 0, or for a vector space over <math>\mathbf{F}_2</math> of dimension 1 (precisely the cases where only one basis exists, independently of any flag).
 
==Subspace nest==
In an infinite-dimensional space ''V'', as used in [[functional analysis]], the flag idea generalises to a '''subspace nest''', namely a collection of subspaces of ''V'' that is a [[total order]] for inclusion and which further is closed under  arbitrary intersections and closed linear spans. See [[nest algebra]].
 
==Set-theoretic analogs==
{{See|Field with one element}}
From the point of view of the [[field with one element]], a set can be seen as a vector space over the field with one element: this formalizes various analogies between [[Coxeter group]]s and [[algebraic group]]s.
 
Under this correspondence, an ordering on a set corresponds to a maximal flag: an ordering is equivalent to a maximal filtration of a set. For instance, the filtration (flag) <math>\{0\} \subset \{0,1\} \subset \{0,1,2\}</math> corresponds to the ordering <math>\{0,1,2\}</math>.
 
==See also==
* [[Filtration (mathematics)]]
* [[Flag manifold]]
* [[Grassmannian]]
 
== References ==
{{Reflist}}
 
* {{cite book
  | last = Shafarevich
  | first = I. R.
  | authorlink = Igor Shafarevich
  | coauthors = A. O. Remizov
  | title = Linear Algebra and Geometry
  | publisher = [[Springer Science+Business Media|Springer]]
  | year = 2012
  | url = http://www.springer.com/mathematics/algebra/book/978-3-642-30993-9
  | isbn = 978-3-642-30993-9}}
 
{{DEFAULTSORT:Flag (Linear Algebra)}}
[[Category:Linear algebra]]

Latest revision as of 22:51, 27 September 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.

If you have any thoughts pertaining to exactly where and how to use Best Dentists in DC, you can speak to us at the page.

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