Wien bridge oscillator: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>Trappist the monk
m →References: Remove redundant |year= parameter from CS1 citations; using AWB
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
In [[mathematics]], specifically in [[algebraic geometry]], the '''Grothendieck–Riemann–Roch theorem''' is a far-reaching result on [[coherent cohomology]]. It is a generalisation of the [[Hirzebruch–Riemann–Roch theorem]], about [[complex manifold]]s, which is itself a generalisation of the classical [[Riemann–Roch theorem]] for [[line bundle]]s on [[compact Riemann surface]]s.  
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 adored this write-up and you would such as to receive even more details relating to [http://www.youtube.com/watch?v=90z1mmiwNS8 Washington DC Dentist] kindly visit our own page.
 
Riemann–Roch type theorems relate [[Euler characteristic]]s of the [[cohomology]] of a [[vector bundle]] with their [[topological degree]]s, or more generally their characteristic classes in (co)homology or algebraic analogues thereof. The classical Riemann–Roch theorem does this for curves and line bundles, whereas the Hirzebruch–Riemann–Roch theorem generalises this to vector bundles over manifolds. The Grothendieck–Riemann–Roch theorem sets both theorems in a relative situation of a [[morphism]] between two manifolds (or more general [[scheme (mathematics)|schemes]]) and changes the theorem from a statement about a single bundle, to one applying to [[chain complex]]es of [[sheaf (mathematics)|sheaves]].  
 
The theorem has been very influential, not least for the development of the [[Atiyah–Singer index theorem]]. Conversely, [[complex analysis|complex analytic]] analogues of the Grothendieck–Riemann–Roch theorem can be proved using the index theorem for families. [[Alexander Grothendieck]] gave a first proof in a 1957 manuscript, later published.<ref>A. Grothendieck. Classes de faisceaux et théorème de Riemann-Roch (1957). Published in SGA 6, Springer-Verlag (1971), 20-71.</ref> [[Armand Borel]] and [[Jean-Pierre Serre]] wrote up and published Grothendieck's proof in 1958.<ref>A. Borel and J.-P. Serre. Bull. Soc. Math. France 86 (1958), 97-136.</ref> Later, Grothendieck and his collaborators simplified and generalized the proof.<ref>SGA 6, Springer-Verlag (1971).</ref>
 
==Formulation==
Let ''X'' be a [[smooth scheme|smooth]] [[quasi-projective scheme]] over a [[Field (mathematics)|field]]. Under these assumptions, the [[Grothendieck group]]
 
:<math>K_0(X)\,</math>
 
of [[bounded complex]]es of [[coherent sheaf|coherent sheaves]] is canonically isomorphic to the Grothendieck group of bounded complexes of finite-rank vector bundles. Using this isomorphism, consider the [[Chern character]] (a rational combination of [[Chern classes]]) as a [[functor]]ial transformation
 
:<math>\mbox{ch} \colon K_0(X) \to A(X, {\Bbb Q}),</math>  
 
where
 
:<math>A_d(X,{\Bbb Q})\,</math>
 
is the [[Chow ring|Chow group]] of cycles on ''X'' of dimension ''d'' modulo [[Chow ring#Rational equivalence|rational equivalence]], [[tensor product|tensor]]ed with the [[rational number]]s. In case ''X'' is defined over the [[complex number]]s, the latter group maps to the topological [[cohomology group]]
 
:<math>H^{2 \mathrm{dim}(X) - 2d}(X, {\Bbb Q}).</math>
 
Now consider a [[proper morphism]]
 
:<math>f \colon X \to Y\,</math>
 
between smooth quasi-projective schemes and a bounded complex of sheaves <math>{\mathcal F^\bull}.</math>
 
The '''Grothendieck–Riemann–Roch theorem''' relates the pushforward map
 
:<math>f_{\mbox{!}} = \sum (-1)^i R^i f_* \colon K_0(X) \to K_0(Y)</math>
 
and the pushforward
 
:<math>f_* \colon A(X) \to A(Y),\,</math>
 
by the formula
 
:<math> \mbox{ch}(f_{\mbox{!}}{\mathcal F}^\bull)\mbox{td}(Y) = f_* (\mbox{ch}({\mathcal F}^\bull) \mbox{td}(X) ). </math>
Here td(''X'') is the [[Todd genus]] of (the [[tangent bundle]] of) ''X''. Thus the theorem gives a precise measure for the lack of commutativity of taking the push forwards in the above senses and the Chern character and shows that the needed correction factors depend on ''X'' and ''Y'' only. In fact, since the Todd genus is functorial and multiplicative in [[exact sequence]]s, we can rewrite the Grothendieck–Riemann–Roch formula as
 
:<math> \mbox{ch}(f_{\mbox{!}}{\mathcal F}^\bull) = f_* (\mbox{ch}({\mathcal F}^\bull) \mbox{td}(T_f) ),</math>
 
where ''T''<sub>''f''</sub> is the relative tangent sheaf of ''f'', defined as the element ''TX'' − ''f''<sup>*</sup>''TY'' in ''K''<sub>0</sub>(''X''). For example, when ''f'' is a [[smooth morphism]], ''T''<sub>''f''</sub> is simply a vector bundle, known as the tangent bundle along the fibers of ''f''.
 
==Generalising and specialising==
 
Generalisations of the theorem can be made to the non-smooth case by considering an appropriate generalisation of the combination ch(&mdash;)td(''X'') and to the non-proper case by considering [[cohomology with compact support]].  
 
The [[arithmetic Riemann–Roch theorem]] extends the Grothendieck–Riemann–Roch theorem to [[arithmetic scheme]]s.
 
The [[Hirzebruch–Riemann–Roch theorem]] is (essentially) the special case where ''Y'' is a point and the field is the field of complex numbers.
 
== History ==
 
[[Alexander Grothendieck]]'s version of the Riemann–Roch theorem was originally conveyed in a letter to [[Jean-Pierre Serre]] around 1956–7. It was made public at the initial [[Bonn Arbeitstagung]], in 1957. Serre and [[Armand Borel]] subsequently organized a seminar at Princeton to understand it. The final published paper was in effect the Borel–Serre exposition.  
 
The significance of Grothendieck's approach rests on several points.  First, Grothendieck changed the statement itself: the theorem was, at the time, understood to be a theorem about a [[algebraic variety|variety]], whereas Grothendieck saw it as a theorem about a morphism between varieties. By finding the right generalization, the proof became simpler while the conclusion became more general. In short, Grothendieck applied a strong [[category theory|categorical]] approach to a hard piece of [[Mathematical analysis|analysis]]. Moreover, Grothendieck introduced [[Algebraic K-theory|K-groups]], as discussed above, which paved the way for [[algebraic K-theory]].
 
==Notes==
{{reflist}}
 
==References==
* {{Citation | last1=Borel | first1=Armand | author1-link=Armand Borel | last2=Serre | first2=Jean-Pierre | author2-link=Jean-Pierre Serre | title=Le théorème de Riemann–Roch | mr=0116022 | year=1958 | journal=Bulletin de la Société Mathématique de France | volume=86 | pages=97–136 | issn=0037-9484 | language=French }}
* {{Citation | last1=Fulton | first1=William | author1-link=William Fulton (mathematician) | title=Intersection theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=3-540-62046-X | mr=1644323 | year=1998 | zbl=0885.14002 | edition=2nd | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | volume=2 }}
*{{cite book
| last = Berthelot
| first = Pierre
| authorlink = Pierre Berthelot (mathematician)
| coauthors = [[Alexandre Grothendieck]], [[Luc Illusie]], eds.
| title = Séminaire de Géométrie Algébrique du Bois Marie - 1966-67 - Théorie des intersections et théorème de Riemann-Roch - (SGA 6) (Lecture notes in mathematics '''225''')
| year = 1971
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| location = Berlin; New York
| language = French
| pages = xii+700
| nopp = true
|doi=10.1007/BFb0066283
|isbn= 978-3-540-05647-8
}}
 
==External links==
* The [http://mathoverflow.net/questions/63095/how-does-one-understand-grr-grothendieck-riemann-roch thread] "how does one understand GRR? (Grothendieck Riemann Roch)" on [[MathOverflow]].
 
{{DEFAULTSORT:Grothendieck-Hirzebruch-Riemann-Roch theorem}}
[[Category:Topological methods of algebraic geometry]]
[[Category:Theorems in algebraic geometry]]

Latest revision as of 18:20, 26 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 adored this write-up and you would such as to receive even more details relating to Washington DC Dentist kindly visit our own page.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Wien_bridge_oscillator&oldid=238762"