Cycles and fixed points - Revision history

en>David Eppstein: rm redundant see-also

2014-11-28T22:00:48Z

rm redundant see-also

← Older revision		Revision as of 00:00, 29 November 2014
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	~~In [[mathematics]], '''Somos' quadratic recurrence constant''', named after [[Michael Somos]],~~ is ~~the number~~		My name is Eusebia and I am studying American Politics and Environmental Management at Les Abymes / France.<br><br>My webpage :: [http://Freaky-Monsterradio.de/infusions/guest_book/guest_book.php fifa 15 Coin generator]

	~~:<math>\sigma = \sqrt {1 \sqrt {2 \sqrt{3 \cdots}}} =~~
	~~1^{1~~/~~2}\;2^{1/4}\; 3^{1/8} \cdots~~.~~\,</math>~~

	~~This can be easily re-written into the far more quickly converging product representation~~

	:<~~math~~>~~\sigma = \sigma^2/\sigma =~~
	~~\left(\frac{2}{1} \right)^{1/2}~~
	~~\left(\frac{3}{2} \right)^{1/4}~~
	~~\left(\frac{4}{3} \right)^{1/8}~~
	~~\left(\frac{5}{4} \right)^{1/16}~~
	~~\cdots.~~<~~/math~~>

	~~The constant σ arises when studying the asymptotic behaviour of the sequence~~

	:~~<math>g_0=1\, ; \, g_n = ng_{n-1}^2, \qquad n > 1, \, </math>~~

	~~with first few terms 1, 1, 2, 12, 576, 1658880 ... {{OEIS\|id=A052129}}. This sequence can be shown to have asymptotic behaviour as follows~~:~~<ref>{{MathWorld\|title=Somos's Quadratic Recurrence Constant\|urlname=SomossQuadraticRecurrenceConstant}}</ref>~~

	~~:<math>g_n \sim \frac {\sigma^{2^n}}{n + 2 + O(\frac{1}{n})}. </math>~~

	~~Guillera and Sondow give a representation in terms of the [[derivative]] of the [~~[~~Lerch transcendent]]~~:

	~~:<math>\ln \sigma = \frac{-1}{2}~~
	~~\frac {\partial \Phi} {\partial s}~~
	~~\left( \frac{1}{2}, 0, 1 \right)<~~/~~math>~~

	~~where ln is the [[natural logarithm]] and <math>\Phi<~~/~~math>(''z'', ''s'', ''q'') is the Lerch transcendent.~~

	~~Using [[series acceleration]] it is the sum of the n~~-~~th differences of ln(k) at k=1 as given by:~~

	~~:<math>\ln \sigma = \sum_{n=1}^\infty \sum_{k=0}^n (-1)^{n-k} {n \choose k} \ln (k+1)~~. </~~math>~~

	~~Finally,~~

	~~:<math> \sigma = 1.661687949633594121296\dots\;<~~/~~math> {{OEIS\|id=A112302}}.~~

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	~~{{refbegin}}~~
	* Steven R. Finch, ''Mathematical Constants'' (2003), [[Cambridge University Press]], p.~~ 446. ISBN 0-521-81805-2.~~
	* Jesus Guillera and [[Jonathan Sondow]], "Double integrals and infinite products for some classical constants via analytic continuations of [[Lerch's transcendent]]", ''Ramanujan Journal'' 16 (2008), 247–270 (Provides an integral and a series representation). {{arxiv\|math/0506319}}
	~~{{refend}}~~

	~~[[Category:Mathematical constants]~~]

en>Glphvgacs at 20:45, 22 September 2013

2013-09-22T20:45:31Z

← Older revision		Revision as of 22:45, 22 September 2013
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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.<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 article and also you would like to obtain more info relating to~~ [~~http~~:~~//www.youtube.com/watch?v=90z1mmiwNS8 dentist DC~~] ~~nicely visit our website.~~		In [[mathematics]], '''Somos' quadratic recurrence constant''', named after [[Michael Somos]], is the number

			:<math>\sigma = \sqrt {1 \sqrt {2 \sqrt{3 \cdots}}} =
			1^{1/2}\;2^{1/4}\; 3^{1/8} \cdots.\,</math>

			This can be easily re-written into the far more quickly converging product representation

			:<math>\sigma = \sigma^2/\sigma =
			\left(\frac{2}{1} \right)^{1/2}
			\left(\frac{3}{2} \right)^{1/4}
			\left(\frac{4}{3} \right)^{1/8}
			\left(\frac{5}{4} \right)^{1/16}
			\cdots.</math>

			The constant σ arises when studying the asymptotic behaviour of the sequence

			:<math>g_0=1\, ; \, g_n = ng_{n-1}^2, \qquad n > 1, \, </math>

			with first few terms 1, 1, 2, 12, 576, 1658880 ... {{OEIS\|id=A052129}}. This sequence can be shown to have asymptotic behaviour as follows:<ref>{{MathWorld\|title=Somos's Quadratic Recurrence Constant\|urlname=SomossQuadraticRecurrenceConstant}}</ref>

			:<math>g_n \sim \frac {\sigma^{2^n}}{n + 2 + O(\frac{1}{n})}. </math>

			Guillera and Sondow give a representation in terms of the [[derivative]] of the [[Lerch transcendent]]:

			:<math>\ln \sigma = \frac{-1}{2}
			\frac {\partial \Phi} {\partial s}
			\left( \frac{1}{2}, 0, 1 \right)</math>

			where ln is the [[natural logarithm]] and <math>\Phi</math>(''z'', ''s'', ''q'') is the Lerch transcendent.

			Using [[series acceleration]] it is the sum of the n-th differences of ln(k) at k=1 as given by:

			:<math>\ln \sigma = \sum_{n=1}^\infty \sum_{k=0}^n (-1)^{n-k} {n \choose k} \ln (k+1). </math>

			Finally,

			:<math> \sigma = 1.661687949633594121296\dots\;</math> {{OEIS\|id=A112302}}.

			==Notes==
			{{reflist}}

			==References==
			{{refbegin}}
			* Steven R. Finch, ''Mathematical Constants'' (2003), [[Cambridge University Press]], p. 446. ISBN 0-521-81805-2.
			* Jesus Guillera and [[Jonathan Sondow]], "Double integrals and infinite products for some classical constants via analytic continuations of [[Lerch's transcendent]]", ''Ramanujan Journal'' 16 (2008), 247–270 (Provides an integral and a series representation). {{arxiv\|math/0506319}}
			{{refend}}

			[[Category:Mathematical constants]]

en>Cydebot: Robot - Speedily moving category Fixed points to :Category:Fixed points (mathematics) per CFDS.

2012-02-14T23:20:09Z

Robot - Speedily moving category Fixed points to Category:Fixed points (mathematics) per CFDS.

New page

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 liked this article and also you would like to obtain more info relating to [http://www.youtube.com/watch?v=90z1mmiwNS8 dentist DC] nicely visit our website.