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| In [[mathematics]], a '''characteristic class''' is a way of associating to each [[principal bundle]] on a [[topological space]] ''X'' a [[cohomology]] class of ''X''. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses [[fiber bundle|section]]s or not. In other words, characteristic classes are global [[topological invariant|invariant]]s which measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in [[algebraic topology]], [[differential geometry]] and [[algebraic geometry]].
| | 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 issues concerning where and how to use [http://www.youtube.com/watch?v=90z1mmiwNS8 Dentists in DC], you can contact us at the web page. |
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| The notion of characteristic class arose in 1935 in the work of [[Eduard Stiefel|Stiefel]] and [[Hassler Whitney|Whitney]] about vector fields on manifolds.
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| ==Definition==
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| Let ''G'' be a [[topological group]], and for a topological space ''X'', write ''b''<sub>''G''</sub>(''X'') for the set of [[isomorphism class]]es of [[principal bundle|principal ''G''-bundles]]. This ''b''<sub>''G''</sub> is a contravariant [[functor]] from '''Top''' (the [[category (mathematics)|category]] of topological spaces and [[continuous function]]s) to '''Set''' (the category of [[set (mathematics)|set]]s and [[function (mathematics)|function]]s), sending a map ''f'' to the [[pullback bundle|pullback]] operation ''f''*.
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| A '''characteristic class''' ''c'' of principal ''G''-bundles is then a [[natural transformation]] from ''b''<sub>''G''</sub> to a cohomology functor ''H''*, regarded also as a functor to '''Set'''.
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| In other words, a characteristic class associates to any principal ''G''-bundle ''P'' → ''X'' an element ''c(P)'' in ''H''*(''X'') such that, if ''f'' : ''Y'' → ''X'' is a continuous map, then ''c''(''f''*''P'') = ''f''*''c''(''P''). On the left is the class of the pullback of ''P'' to ''Y''; on the right is the image of the class of ''P'' under the induced map in cohomology.
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| ==Characteristic numbers==
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| {{For|characteristic numbers in fluid dynamics|characteristic number (fluid dynamics)}}
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| Characteristic classes are elements of cohomology groups;<ref>Informally, characteristic classes "live" in cohomology.</ref> one can obtain integers from characteristic classes, called '''characteristic numbers'''. Respectively:
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| [[Stiefel-Whitney_class#Stiefel–Whitney_numbers|Stiefel-Whitney numbers]],
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| [[Chern_class#Chern_numbers|Chern numbers]], [[Pontryagin_class#Pontryagin_numbers|Pontryagin numbers]], and the
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| [[Euler_class#Relations_to_other_invariants|Euler characteristic]].
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| Given an oriented manifold ''M'' of dimension ''n'' with [[fundamental class]] <math>[M] \in H_n(M)</math>, and a ''G''-bundle with characteristic classes <math>c_1,\dots,c_k</math>, one can pair a product of characteristic classes of total degree ''n'' with the fundamental class. The number of distinct characteristic numbers is the number of [[monomial]]s of degree ''n'' in the characteristic classes, or equivalently the partitions of ''n'' into <math>\mbox{deg}\,c_i</math>.
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| Formally, given <math>i_1,\dots,i_l</math> such that <math>\sum \mbox{deg}\,c_{i_j} = n</math>, the corresponding characteristic number is:
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| :<math>c_{i_1}\smile c_{i_2}\smile \dots \smile c_{i_m}([M])</math>
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| where <math>\smile</math> denotes the [[cup product]] of cohomology classes.
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| These are notated various as either the product of characteristic classes, such as <math>c_1^2</math> or by some alternative notation, such as <math>P_{1,1}</math> for the [[Pontryagin_class#Pontryagin_numbers|Pontryagin number]] corresponding to <math>p_1^2</math>, or <math>\chi</math> for the Euler characteristic.
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| From the point of view of [[de Rham cohomology]], one can take [[differential form]]s representing the characteristic classes,<ref>By [[Chern-Weil theory]], these are polynomials in the curvature; by [[Hodge theory]], one can take harmonic form.</ref> take a wedge product so that one obtains a top dimensional form, then integrates over the manifold; this is analogous to taking the product in cohomology and pairing with the fundamental class.
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| This also works for non-orientable manifolds, which have a <math>\mathbf{Z}/2</math>-orientation, in which case one obtains <math>\mathbf{Z}/2</math>-valued characteristic numbers, such as the Stiefel-Whitney numbers.
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| Characteristic numbers solve the oriented and unoriented [[Cobordism#Cobordism_classes|bordism question]]s: two manifolds are (respectively oriented or unoriented) cobordant if and only if their characteristic numbers are equal.
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| ==Motivation==
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| Characteristic classes are in an essential way phenomena of [[cohomology theory]] — they are [[contravariant]] constructions, in the way that a [[Section (category theory)|section]] is a kind of function ''on'' a space, and to lead to a contradiction from the existence of a section we do need that variance. In fact cohomology theory grew up after [[Homology (mathematics)|homology]] and [[homotopy theory]], which are both [[Covariance|covariant]] theories based on mapping ''into'' a space; and characteristic class theory in its infancy in the 1930s (as part of [[obstruction theory]]) was one major reason why a 'dual' theory to homology was sought. The characteristic class approach to [[curvature]] invariants was a particular reason to make a theory, to prove a general [[Gauss-Bonnet theorem]].
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| When the theory was put on an organised basis around 1950 (with the definitions reduced to homotopy theory) it became clear that the most fundamental characteristic classes known at that time (the [[Stiefel-Whitney class]], the [[Chern class]], and the [[Pontryagin class]]es) were reflections of the classical linear groups and their [[maximal torus]] structure. What is more, the Chern class itself was not so new, having been reflected in the [[Schubert calculus]] on [[Grassmannian]]s, and the work of the [[Italian school of algebraic geometry]]. On the other hand there was now a framework which produced families of classes, whenever there was a [[vector bundle]] involved.
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| The prime mechanism then appeared to be this: Given a space ''X'' carrying a vector bundle, that implied in the [[CW complex|homotopy category]] a mapping from ''X'' to a [[classifying space]] ''BG'', for the relevant linear group ''G''. For the homotopy theory the relevant information is carried by compact subgroups such as the [[orthogonal group]]s and [[unitary group]]s of ''G''. Once the cohomology ''H''*(''BG'') was calculated, once and for all, the contravariance property of cohomology meant that characteristic classes for the bundle would be defined in ''H''*(''X'') in the same dimensions. For example the [[Chern class]] is really one class with graded components in each even dimension.
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| This is still the classic explanation, though in a given geometric theory it is profitable to take extra structure into account. When cohomology became 'extraordinary' with the arrival of [[K-theory]] and [[cobordism theory]] from 1955 onwards, it was really only necessary to change the letter ''H'' everywhere to say what the characteristic classes were.
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| Characteristic classes were later found for [[foliation]]s of [[manifold]]s; they have (in a modified sense, for foliations with some allowed singularities) a classifying space theory in [[homotopy]] theory.
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| In later work after the ''rapprochement'' of mathematics and [[physics]], new characteristic classes were found by [[Simon Donaldson]] and [[Dieter Kotschick]] in the [[instanton]] theory. The work and point of view of [[Shiing-Shen Chern|Chern]] have also proved important: see [[Chern-Simons|Chern-Simons theory]].
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| ==Stability==
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| In the language of [[stable homotopy theory]], the [[Chern class]], [[Stiefel-Whitney class]], and [[Pontryagin class]] are ''stable'', while the [[Euler class]] is ''unstable''.
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| Concretely, a stable class is one that does not change when one adds a trivial bundle: <math>c(V \oplus 1) = c(V)</math>. More abstractly, it means that the cohomology class in the [[classifying space]] for <math>BG(n)</math> pulls back from the cohomology class in <math>BG(n+1)</math> under the inclusion <math>BG(n) \to BG(n+1)</math> (which corresponds to the inclusion <math>\mathbf{R}^n \to \mathbf{R}^{n+1}</math> and similar). Equivalently, all finite characteristic classes pull back from a stable class in <math>BG</math>.
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| This is not the case for the Euler class, as detailed there, not least because the Euler class of a ''k''-dimensional bundle lives in <math>H^k(X)</math> (hence pulls back from <math>H^k(BO(k))</math>, so it can’t pull back from a class in <math>H^{k+1}</math>, as the dimensions differ.
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| From the perspective of the [[splitting principle]], this corresponds to the stability of [[symmetric polynomials]], and the instability of [[alternating polynomials]], specifically the [[Vandermonde polynomial]], which represents the Euler class.
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| ==See also==
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| *[[Chern class]]
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| *[[Segre class]]
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| *[[Stiefel-Whitney class]]
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| *[[Pontryagin class]]
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| *[[Euler class]]
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| *[[Euler characteristic]]
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| == Notes ==
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| <references/> | |
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| == References ==
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| * [[Allen Hatcher]], [http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html Vector Bundles & K-Theory]
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| * [[John Milnor|Milnor, John W.]]; [[Jim Stasheff|Stasheff, James D.]] ''Characteristic classes''. Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. vii+331 pp. ISBN 0-691-08122-0.
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| * Shiing-Shen Chern, ''Complex Manifolds Without Potential Theory'' (Springer-Verlag Press, 1995) ISBN 0-387-90422-0, ISBN 3-540-90422-0.
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| *:The appendix of this book: "Geometry of Characteristic Classes" is a very neat and profound introduction to the development of the ideas of characteristic classes.
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| {{DEFAULTSORT:Characteristic Class}}
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| [[Category:Algebraic topology]]
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| [[Category:Characteristic classes]]
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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.
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 issues concerning where and how to use Dentists in DC, you can contact us at the web page.