Constructive proof - Revision history

en>Erel Segal: /* See also */

2014-11-26T13:56:59Z

See also

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	~~In [[axiomatic set theory]]~~, the ~~'''gimel function'''~~ is the ~~following function mapping [[cardinal number]]s~~ to ~~cardinal numbers:~~		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 liked this write-up and you would certainly such as to receive even more info relating to [http://www.youtube.com/watch?v=90z1mmiwNS8 Best Dentists in DC] kindly go to our page.

	:<~~math~~>~~\gimel\colon\kappa\mapsto\kappa^{\mathrm{cf}(\kappa)}~~<~~/math~~>

	~~where cf denotes~~ the ~~[[cofinality]] function;~~ the ~~gimel function is used for studying~~ the ~~[[continuum function]] and~~ the ~~[[cardinal number#Cardinal exponentiation\|cardinal exponentiation]] function~~.

	~~==Values~~ of the ~~Gimel function==~~

	~~The gimel function has the property~~ <~~math>\gimel(\kappa)~~>~~\kappa~~<~~/math~~> ~~for all infinite cardinals κ~~ by ~~[[König's theorem~~ (~~set theory~~)~~\|König's theorem]].~~

	~~For regular cardinals~~
	~~<math>\kappa</math>,~~
	~~<math>\gimel~~(~~\kappa~~)~~= 2^\kappa</math>, and [[Easton's theorem]] says we don't know much about~~ the ~~values of this function~~. ~~For singular~~
	<~~math~~>~~\kappa~~<~~/math~~>~~, upper bounds for <math>\gimel(\kappa)</math> can~~ be ~~found from [[Saharon Shelah\|Shelah]]'s [[PCF theory]]~~.

	~~==Reducing~~ the ~~exponentiation function~~ to the ~~gimel function==~~

	~~All cardinal exponentiation is determined~~ (~~recursively~~) ~~by the gimel function as follows~~.
	*If ~~κ is~~ an ~~infinite successor cardinal then~~ <~~math~~>~~2^\kappa = \gimel(\kappa)~~<~~/math~~>
	*If κ is a ~~limit~~ and the ~~continuum function is eventually constant below κ then <math>2^\kappa=2^{<\kappa}\times\gimel(\kappa)</math>~~
	*If ~~κ~~ is a ~~limit and~~ the ~~continuum function is~~ not ~~eventually constant below κ then~~ <~~math~~>~~2^\kappa=\gimel(2^{<\kappa})~~<~~/math~~>
	The ~~remaining rules hold whenever κ~~ and ~~λ are both infinite:~~
	*If ~~&alefsym;~~<~~sub~~>0<~~/sub~~>~~≤κ≤λ then κ<sup>λ</sup> = 2<sup>λ</sup>~~
	*If ~~μ~~<~~sup~~>~~λ~~<~~/sup~~>~~≥κ for some μ<κ then κ<sup>λ</sup> = μ<sup>λ</sup>~~
	*If κ> λ and ~~μ~~<~~sup~~>~~λ~~<~~/sup~~>~~<κ for all μ<κ and cf~~(~~κ~~)~~≤λ then κ~~<~~sup~~>~~λ~~<~~/sup~~> ~~= κ<sup>cf(κ)</sup>~~
	*If ~~κ> λ~~ and ~~μ<sup>λ<~~/~~sup><κ for all μ<κ and cf(κ)>λ then κ<sup>λ<~~/~~sup>~~ = ~~κ~~

	~~==References==~~
	*[[Thomas Jech]~~], ''Set Theory'', 3rd millennium ed~~.~~, 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2.~~

	~~[[Category:Cardinal numbers]]~~

en>Jochen Burghardt: /* Brouwerian counterexamples */

2013-10-12T10:52:10Z

Brouwerian counterexamples

← Older revision		Revision as of 12:52, 12 October 2013
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	~~All of~~ the ~~trophies from all of~~ the ~~members in your kin get added up and as well as divided by 2 find out your clans overall awards. Playing many different kinds of~~ [~~http~~:/~~/mondediplo.com/spip.php?page=recherche&recherche=games+sells games sells~~] ~~your gaming time more fun. and your league also determines any battle win bonus. 5 star rating and that~~ is ~~known to be unbelievably addictive as players are known to devote several hours experiencing~~ the ~~game. If you enjoyed this short article~~ and ~~you would such as to receive additional info concerning~~ [~~http://circuspartypanama~~.~~com hack clash~~ of ~~clans no survey] kindly see~~ the ~~webpage. She centers on beauty salon business investment and client fascination.~~<br><br>~~Video games are fun to explore your kids. Assist you learn much another recommendation of your kid~~'s ~~interests. Sharing interests with children like this can on top of that create great conversations. It also gives you an opportunity to monitor progress of their skills~~.<br><br>~~To take pleasure from unlimited points~~, ~~resources~~, ~~loose change or gems, you must download~~ the ~~clash~~ of ~~clans compromise tool by clicking on his or her button~~. ~~Depending regarding the operating system that are generally using~~, ~~you will be required to run the downloaded file~~ for ~~as administrator. Give you log in ID and judge the device. Correct after this, you are want enter the number pointing to gems or coins that you desire to get.~~<br><br>In the ~~first instance clash of clans compromise tool no survey devises believe in among ones people. Society may be definitely powered~~ by ~~fellow pressure, one of the particular most powerful forces about~~ the ~~planet. As long~~ as ~~peer power utilizes its power to gain good, clash of clans hack tool no overview will have its house in community~~.<br><br>~~Regardless of the reason, computer game power tips are widespread~~ and ~~sent out fairly rapidly over~~ the ~~internet. The gaming community~~ is ~~intending to find means cease cheaters from overrunning whichever game~~; having lots off cheaters playing a game can really produce honest players to remaining playing, or play only with friends they trust. This poses a huge problem particularly for request games for example EverQuest, wherein a loss behind players ultimately result in the loss of income.<br><br>~~Tribe wars can alone find yourself started by market responsable or co-leaders. As soon started,~~ the ~~bold would chase to have any good adversary association of agnate durability. Backbone~~ is not ~~that bent because of~~ some ~~of the cardinal of trophies, but alternatively by anniversary members advancing ability~~ (~~troops, army affected capacity, spells clash with regards to clans Cheats and heroes~~) ~~in addition to arresting backbone~~ (~~security buildings, walls, accessories~~ and ~~heroes~~).<br><br>~~Loose time waiting for game of the yr versions of major leaders~~. ~~These often come out a year or maybe more after the original title~~, ~~but form lots of the downloadable and extra content that has been released~~ in ~~stages right after the first title. Of these games offer a much more bang for the dollars~~.		In [[axiomatic set theory]], the '''gimel function''' is the following function mapping [[cardinal number]]s to cardinal numbers:

			:<math>\gimel\colon\kappa\mapsto\kappa^{\mathrm{cf}(\kappa)}</math>

			where cf denotes the [[cofinality]] function; the gimel function is used for studying the [[continuum function]] and the [[cardinal number#Cardinal exponentiation\|cardinal exponentiation]] function.

			==Values of the Gimel function==

			The gimel function has the property <math>\gimel(\kappa)>\kappa</math> for all infinite cardinals κ by [[König's theorem (set theory)\|König's theorem]].

			For regular cardinals
			<math>\kappa</math>,
			<math>\gimel(\kappa)= 2^\kappa</math>, and [[Easton's theorem]] says we don't know much about the values of this function. For singular
			<math>\kappa</math>, upper bounds for <math>\gimel(\kappa)</math> can be found from [[Saharon Shelah\|Shelah]]'s [[PCF theory]].

			==Reducing the exponentiation function to the gimel function==

			All cardinal exponentiation is determined (recursively) by the gimel function as follows.
			*If κ is an infinite successor cardinal then <math>2^\kappa = \gimel(\kappa)</math>
			*If κ is a limit and the continuum function is eventually constant below κ then <math>2^\kappa=2^{<\kappa}\times\gimel(\kappa)</math>
			*If κ is a limit and the continuum function is not eventually constant below κ then <math>2^\kappa=\gimel(2^{<\kappa})</math>
			The remaining rules hold whenever κ and λ are both infinite:
			*If &alefsym;<sub>0</sub>≤κ≤λ then κ<sup>λ</sup> = 2<sup>λ</sup>
			*If μ<sup>λ</sup>≥κ for some μ<κ then κ<sup>λ</sup> = μ<sup>λ</sup>
			*If κ> λ and μ<sup>λ</sup><κ for all μ<κ and cf(κ)≤λ then κ<sup>λ</sup> = κ<sup>cf(κ)</sup>
			*If κ> λ and μ<sup>λ</sup><κ for all μ<κ and cf(κ)>λ then κ<sup>λ</sup> = κ

			==References==
			*[[Thomas Jech]], ''Set Theory'', 3rd millennium ed., 2003, Springer Monographs in Mathematics, Springer, ISBN 3-540-44085-2.

			[[Category:Cardinal numbers]]

en>FrescoBot: Bot: misplaced invisible LTR marks

2012-07-22T23:15:15Z

Bot: misplaced invisible LTR marks

New page

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