Koopmans' theorem: Difference between revisions

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In [[graph theory]], a '''cut''' is a [[Partition_of_a_set|partition]] of the vertices of a graph into two disjoint subsets that are joined by at least one edge. The '''cut-set''' of the cut is the set of edges whose end points are in different subsets of the partition. Edges are said to be ''crossing'' the cut if they are in its cut-set.
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><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 type of concerns relating to where and ways to utilize [http://www.youtube.com/watch?v=90z1mmiwNS8 dentist DC], you can contact us at our web site.
(Many graph theorists use the term '''cut''' to refer to the edge set instead of the vertex partition.)
 
In an unweighted undirected graph, the ''size'' or ''weight'' of a cut is the number of edges crossing the cut. In a [[Graph_(mathematics)#Weighted_graph| weighted graph]], the same term is defined by the sum of the weights of the edges crossing the cut.
 
In a [[flow network]], an '''s-t cut''' is a cut that requires the [[Glossary_of_graph_theory#Direction|''source'']] and the [[Glossary_of_graph_theory#Direction|''sink'']] to be in different subsets, and its ''cut-set'' only consists of edges going from the source's side to the sink's side. The ''capacity'' of an s-t cut is defined as the sum of [[Capacity of a set|capacity]] of each edge in the ''cut-set''.
 
==Definition==
: A '''cut''' <math>C=(S,T)</math> is a partition of <math>V</math> of a graph <math>G=(V,E)</math>.  
 
: An '''s-t cut''' <math>C=(S,T)</math> of a network <math>N=(V,E)</math> is a cut of <math>N</math> such that <math>s\in S</math> and <math>t \in T</math>, where <math>s</math> and <math>t</math> are the [[Glossary_of_graph_theory#Direction|''source'']] and the [[Glossary_of_graph_theory#Direction|''sink'']] of <math>N</math> respectively.  
 
: The '''cut-set''' of a cut <math>C=(S,T)</math> is the set <math>\{(u,v)\in E | u\in S, v \in T\}</math>.
 
:The '''size''' of a cut <math>C=(S,T)</math> is the number of edges in the cut-set. If the edges are weighted, the '''value''' (or '''weight''') of the cut is the sum of the weights.
 
==Minimum cut==
[[File:Min-cut.svg|frame|right|A minimum cut.]]
{{main|Minimum cut}}
 
A cut is ''minimum'' if the size of the cut is not larger than the size of any other cut. The illustration on the right shows a minimum cut: the size of this cut is 2, and there is no cut of size 1 because the graph is [[Bridge (graph theory)|bridgeless]].
 
The [[max-flow min-cut theorem]] proves that the maximum [[flow network|network flow]] and the sum of the cut-edge weights of any minimum cut that separates the source and the sink are equal. There are [[polynomial time|polynomial-time]] methods to solve the min-cut problem, notably the [[Edmonds-Karp algorithm]].
 
{{clear}}
 
==Maximum cut==
[[File:Max-cut.svg|frame|right|A maximum cut.]]
{{main|Maximum cut}}
 
A cut is ''maximum'' if the size of the cut is not smaller than the size of any other cut. The illustration on the right shows a maximum cut: the size of the cut is equal to 5, and there is no cut of size |''E''| because the graph is not [[Bipartite graph|bipartite]] (there is an [[Cycle graph#Terminology|odd cycle]]).
 
In general, finding a maximum cut is computationally hard. The max-cut problem is one of [[Karp's 21 NP-complete problems]]. The max cut problem is also [[Constant-factor approximation algorithm|APX-hard]], meaning that there is no polynomial-time approximation scheme for it unless P&nbsp;=&nbsp;NP.
 
Note that min-cut and max-cut are ''not'' [[Linear programming#Duality|dual]] problems in the [[linear programming]] sense, even though one gets from one problem to other by changing min to max in the [[objective function]]. The max-flow problem is the dual of the min-cut problem
 
== Sparsest cut ==
The '''Sparsest cut problem''' is to bipartition the vertices so as to minimize the ratio of the number of edges across the cut divided by the number of vertices in the smaller half of the partition. This objective function favors solutions that are both sparse (few edges crossing the cut) and balanced (close to a bisection). The problem is known to be NP-Hard, and the best known algorithm is an <math>O(\sqrt{\log n})</math> approximation due to {{Harvtxt|Arora|Rao|Vazirani|2009}}.
 
== See also ==
* [[Connectivity (graph theory)]]
* [[Graph cuts in computer vision]]
* [[Vertex separator]]
 
== References ==
*{{Citation | last1=Arora | first1=Sanjeev | author1-link=Sanjeev Arora | last2=Rao | first2=Satish | last3=Vazirani | first3=Umesh | author3-link=Umesh Vazirani | title=Expander flows, geometric embeddings and graph partitioning | publisher=ACM | year=2009 | journal=J. ACM | issn=0004-5411 | doi=10.1145/1502793.1502794 | volume=56 | issue=2 | pages=1–37}}
*{{cite book | author = [[Thomas H. Cormen]], [[Charles E. Leiserson]], [[Ronald L. Rivest]], and [[Clifford Stein]] | title=[[Introduction to Algorithms]], Second Edition | year = 2001 | publisher = MIT Press and McGraw-Hill | isbn = 0-262-03293-7 | pages=563,655,1043}}
*{{cite book|author = [[Michael R. Garey]] and [[David S. Johnson]] | year = 1979 | title = [[Computers and Intractability: A Guide to the Theory of NP-Completeness]] | publisher = W.H. Freeman | isbn = 0-7167-1045-5}} A2.2: ND16, pg.210.
*M. X. Goemans,  and D. P. Williamson, [http://portal.acm.org/citation.cfm?id=227684 ''Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming''], Journal of the ACM, 42, 6 (Nov. 1995), 1115-1145
*[[Richard Karp|R. M. Karp]], ''Reducibility among combinatorial problems'', in R. E. Miller and J. W. Thacher (eds.), ''Complexity of Computer Computation'', Plenum Press, New York, 85-103 (1972)
*[[Subhash Khot|S. Khot]], G. Kindler, E. Mossel, and R. O’Donnell, [http://www.cs.cmu.edu/~odonnell/papers/maxcut.pdf ''Optimal inapproximability results for MAX-CUT and other two-variable CSPs?''], In Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, pp. 146–154, 2004.
*{{cite book | author = [[Vijay_Vazirani | Vijay V. Vazirani]] | title = Approximation Algorithms | year = 2004 | publisher = Springer | isbn = 3-540-65367-8 | pages = 97–98}}
*{{cite web|last=Meira|first=Luis A. A.|title=Semidefinite Programming Based Algorithms for the Sparsest Cut Problem|url=http://www.ic.unicamp.br/~fkm/publication/Rairo11.pdf|accessdate=6 September 2011}}
 
[[Category:Graph connectivity]]
[[Category:Combinatorial optimization]]

Revision as of 02:24, 8 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.

If you have any type of concerns relating to where and ways to utilize dentist DC, you can contact us at our web site.

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