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[[File:Inclusion ordering.svg|thumb|The [[power set]] of a 2-element set ordered by [[subset|inclusion]] ]]
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 enjoyed this post and you would certainly such as to obtain more info regarding [http://www.youtube.com/watch?v=90z1mmiwNS8 Dentists in DC] kindly visit the site.
In [[order theory]], a '''Hasse diagram''' ({{IPAc-en|ˈ|h|æ|s|ə}}; [[German language|German]]: {{IPA|/ˈhasə/}}) is a type of [[mathematical diagram]] used to represent a finite [[partially ordered set]], in the form of a [[Graph drawing|drawing]] of its [[transitive reduction]]. Concretely, for a partially ordered set ''(S, ≤)'' one represents each element of ''S'' as a [[vertex (graph theory)|vertex]] in the plane and draws a [[line segment]] or curve that goes ''upward'' from ''x'' to ''y'' whenever ''y'' [[Covering relation|covers]] ''x'' (that is, whenever ''x'' &lt; ''y'' and there is no ''z'' such that ''x'' &lt; ''z'' &lt; ''y'').
These curves may cross each other but must not touch any vertices other than their endpoints. Such a diagram, with labeled vertices, uniquely determines its partial order.
 
Hasse diagrams are named after [[Helmut Hasse]] (1898&ndash;1979); according to {{harvtxt|Birkhoff|1948}}, they are so-called because of the effective use Hasse made of them. However, Hasse was not the first to use these diagrams; they appear, e.g., in {{harvtxt|Vogt|1895}}. Although Hasse diagrams were originally devised as a technique for making drawings of partially ordered sets by hand, they have more recently been created automatically using [[graph drawing]] techniques.<ref>E.g., see {{harvtxt|Di Battista|Tamassia|1988}} and {{harvtxt|Freese|2004}}.</ref>
 
The phrase "Hasse diagram" may also refer to the transitive reduction as an abstract [[directed acyclic graph]], independently of any drawing of that graph, but this usage is eschewed here.
 
== A "good" Hasse diagram ==
 
Although Hasse diagrams are simple as well as intuitive tools for dealing with finite [[Partially ordered set|posets]], it turns out to be rather difficult to draw "good" diagrams. The reason is that there will in general be many possible ways to draw a Hasse diagram for a given poset. The simple technique of just starting with the [[minimal element]]s of an order and then drawing greater elements incrementally often produces quite poor results: symmetries and internal structure of the order are easily lost.
 
The following example demonstrates the issue. Consider the [[power set]] of a 4-element set ordered by inclusion <math>\subseteq</math>. Below are four different Hasse diagrams for this partial order. Each subset has a node labelled with a binary encoding that shows whether a certain element is in the subset (1) or not (0):
 
{| style="margin: 0 auto;"
| [[File:Hypercubeorder binary.svg|230px|]] ||&nbsp;&nbsp;&nbsp;|| [[File:Hypercubecubes binary.svg|260px|]] ||&nbsp;&nbsp;&nbsp;|| [[File:Hypercubestar binary.svg|240px|]] ||&nbsp;&nbsp;&nbsp;|| [[File:Hypercubematrix binary.svg|180px|]]
|}
 
The first diagram makes clear that the power set is a [[graded poset]]. The second diagram has the same graded structure, but by making some edges longer than others, it emphasizes that the [[tesseract|4-dimensional cube]] is a union of two 3-dimensional cubes. The third diagram shows some of the internal symmetry of the structure. In the fourth diagram the vertices are arranged like the fields of a 4×4 [[Matrix (mathematics)|matrix]].
 
==Upward planarity==
{{main|Upward planar drawing}}
[[File:Dih4 subgroups.svg|thumb|This Hasse diagram of the [[lattice of subgroups]] of the [[dihedral group]] [[Dihedral group of order 8|Dih<sub>4</sub>]] has no crossing edges.]]
If a partial order can be drawn as a Hasse diagram in which no two edges cross, its covering graph is said to be ''upward planar''. A number of results on upward planarity and on crossing-free Hasse diagram construction are known:
*If the partial order to be drawn is a [[lattice (order)|lattice]], then it can be drawn without crossings if and only if it has [[order dimension]] at most two.<ref>{{harvtxt|Garg|Tamassia|1995a}}, Theorem 9, p. 118; {{harvtxt|Baker|Fishburn|Roberts|1971}}, theorem 4.1, page 18.</ref> In this case, a non-crossing drawing may be found by deriving Cartesian coordinates for the elements from their positions in the two linear orders realizing the order dimension, and then rotating the drawing counterclockwise by a 45-degree angle.
*If the partial order has at most one [[minimal element]], or it has at most one [[maximal element]], then it may be tested in [[linear time]] whether it has a non-crossing Hasse diagram.<ref>{{harvtxt|Garg|Tamassia|1995a}}, Theorem 15, p. 125; {{harvtxt|Bertolazzi|Di Battista|Mannino|Tamassia|1993}}.</ref>
*It is [[NP-complete]] to determine whether a partial order with multiple sources and sinks can be drawn as a crossing-free Hasse diagram.<ref>{{harvtxt|Garg|Tamassia|1995a}}, Corollary 1, p. 132; {{harvtxt|Garg|Tamassia|1995b}}.</ref> However, finding a crossing-free Hasse diagram is [[fixed-parameter tractable]] when parametrized by the number of [[articulation point]]s and [[triconnected component]]s of the transitive reduction of the partial order.<ref>{{harvtxt|Chan|2004}}.</ref>
*If the ''y''-coordinates of the elements of a partial order are specified, then a crossing-free Hasse diagram respecting those coordinate assignments can be found in linear time, if such a diagram exists.<ref>{{harvtxt|Jünger|Leipert|1999}}.</ref> In particular, if the input poset is a [[graded poset]], it is possible to determine in linear time whether there is a crossing-free Hasse diagram in which the height of each vertex is proportional to its rank.
 
==Notes==
{{reflist|2}}
 
==References==
*{{citation|first1=K. A.|last1=Baker|first2=P.|last2=Fishburn|author2-link=Peter C. Fishburn|first3=F. S.|last3=Roberts|author3-link=Fred S. Roberts|title=Partial orders of dimension 2|journal=Networks|volume=2|pages=11–28|issue=1|doi=10.1002/net.3230020103|year=1971}}.
*{{citation|last1=Bertolazzi|first1=R|last2=Di Battista|first2=G.|last3=Mannino|first3=C.|last4=Tamassia|first4=R.|author4-link=Roberto Tamassia|year=1993|contribution=Optimal upward planarity testing of single-source digraphs|title=[[European Symposium on Algorithms|Proc. 1st European Symposium on Algorithms (ESA '93)]]|volume=726|series=Lecture Notes in Computer Science|publisher=Springer-Verlag|pages=37–48|doi=10.1007/3-540-57273-2_42}}.
*{{citation|first=Garrett|last=Birkhoff|authorlink=Garrett Birkhoff|title=Lattice Theory|edition=Revised|publisher=[[American Mathematical Society]]|year=1948}}.
*{{citation|first=Hubert|last=Chan|contribution=A parameterized algorithm for upward planarity testing|title=[[European Symposium on Algorithms|Proc. 12th European Symposium on Algorithms (ESA '04)]]|year=2004|volume=3221|series=Lecture Notes in Computer Science|publisher=Springer-Verlag|pages=157–168|url=http://www.springerlink.com/content/pbxglecx113c6axl/}}.
*{{citation|first1=G.|last1=Di Battista|first2=R.|last2=Tamassia|author2-link=Roberto Tamassia|title=Algorithms for plane representation of acyclic digraphs|journal=Theoretical Computer Science|volume=61|year=1988|pages=175–178|doi=10.1016/0304-3975(88)90123-5|issue=2–3}}.
*{{citation|first=Ralph|last=Freese|contribution=Automated lattice drawing|title=Concept Lattices|publisher=Springer-Verlag|series=Lecture Notes in Computer Science|volume=2961|pages=589–590|year=2004}}. An extended preprint is available online: [http://www.math.hawaii.edu/~ralph/Preprints/latdrawing.pdf].
*{{citation|first1=Ashim|last1=Garg|first2=Roberto|last2=Tamassia|author2-link=Roberto Tamassia|title=Upward planarity testing|journal=[[Order (journal)|Order]]|volume=12|pages=109–133|year=1995a|doi=10.1007/BF01108622|issue=2}}.
*{{citation|first1=Ashim|last1=Garg|first2=Roberto|last2=Tamassia|author2-link=Roberto Tamassia|year=1995b|contribution=On the computational complexity of upward and rectilinear planarity testing|title=[[International Symposium on Graph Drawing|Graph Drawing (Proc. GD '94)]]|volume=894|series=LectureNotes in Computer Science|publisher=Springer-Verlag|pages=286–297|doi=10.1007/3-540-58950-3_384}}.
*{{citation|first1=Michael|last1=Jünger|first2=Sebastian|last2=Leipert|contribution=Level planar embedding in linear time|title=[[International Symposium on Graph Drawing|Graph Drawing (Proc. GD '99)]]|year=1999|volume=1731|pages=72–81|doi=10.1007/3-540-46648-7_7|series=Lecture Notes in Computer Science|isbn=978-3-540-66904-3}}.
*{{citation|first=Henri Gustav|last=Vogt|publisher=Nony|year=1895|page=91|title=Leçons sur la résolution algébrique des équations}}.
 
==External links==
* {{Commons-inline|list=
:[[commons:Hasse diagram|Hasse diagram]] (Gallery) &nbsp;&nbsp;&nbsp;&nbsp; [[commons:Category:Hasse diagrams|Hasse diagrams]] (Category)}}
 
* {{mathworld|urlname=HasseDiagram|title=Hasse Diagram}}
 
[[Category:Order theory]]
[[Category:Diagrams]]
[[Category:Graph drawing]]

Latest revision as of 02:48, 21 December 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 enjoyed this post and you would certainly such as to obtain more info regarding Dentists in DC kindly visit the site.