Rotational partition function - Revision history

en>GangofOne: E_j instead of E_i, to be consistent with rest

2014-11-12T09:34:36Z

E_j instead of E_i, to be consistent with rest

← Older revision		Revision as of 11:34, 12 November 2014
Line 1:		Line 1:
	The '''Duffin–Schaeffer conjecture''' is an important conjecture in [[metric number theory]] proposed by [[Richard Duffin\|R. J. Duffin]] and [[Albert Charles Schaeffer\|A. C. Schaeffer]] in 1941.<ref>R. J. Duffin and A. C. Schaeffer, ''Khintchine's problem in metric [[Diophantine approximation]]'', [[Duke Mathematical Journal]], '''8''' (1941), 243–255</ref> It states that if <math>f : \mathbb{N} \rightarrow \mathbb{R}^+</math> is a real-valued function taking on positive values, then for [[almost all]] <math> \alpha </math> (with respect to [[Lebesgue measure]]), the inequality

	~~: <math> \left \| \alpha - \frac{p}{q} \right \| < \frac{f(q)}{q} </math>~~

	~~has infinitely many solutions in [[co~~-~~prime]] integers~~ <~~math~~>~~p,q~~<~~/math~~> ~~with <math>q > 0</math> if~~ and ~~only if the sum~~		Eyelash Growth - The Way To Having Beautiful Eyelashes<br><br>Are you stressed when you can't grow longer lashes, since they're thin and brittle and seem to drop out as soon as they grow? We all know there are solutions like false eyelash extenders, but they are merely a short-run answer, which enable it to are expensive as they still disappear after a a few a month. The real response is to learn how you can grow longer lashes which are healthy and natural.<br><br>The most appalling reaction of trichotillomania is alopecia. This is a condition whereby excessive hair thinning on account of pulling hair out leads to balding.  Moreover, several patients eat their very own hair after pulling it. This condition is called as "trichophagia". Trichophagia can further bring about bezoar formation, followed by anemia, abdominal pain, nausea, vomiting, bowel obstruction, perforation, pancreatitis and obstructive jaundice. Hence you ought to clearly take hint when you see a person removing hair around the scalp or pulling out eyelashes. Besides these, there are many other symptoms like pulling out others hair, increased levels of stress and anxiety, avoidance of social situations, gastrointestinal complaints and so on so forth.<br><br>With an eyelash grower, it's easy and quick. You just apply a grower to your eyelashes at night and within three to five weeks there's a remarkable difference.A� The product that I consider is natural and that means you usually are not using harmful chemicals on or near up your eyes anymore and you may use this system for those who have eyelash extensions.<br><br>A host of companies have launched eyelash growth serums. Be cautious, however, because so many companies usually do not list ingredients at all. Even if a company simply lists their ingredients without making up which key ingredients will assist you to regrow and thicken lashes, it may plainly certainly be a nice conditioning product without true claims for growth and thickening.<br><br>Some of you do not like cream foundation because you might think there is no way on your skin to breathe. In fact, the high tech cream foundation achieves the transformation from cream to liquid and to powder. You can feel the smoothness and thinness when applying and after applying the cream foundation. The tip is usually to control the amount of cream foundation the application of.<br><br>If you loved this informative article and you would want to receive much more information with regards to heated Rollers for short hair please visit our site.

	~~: <math> \sum_{q=1}^\infty f(q) \frac{\varphi(q)}{q} = \infty~~. <~~/math~~>

	~~Here~~ <~~math> \varphi(q)</math~~> is ~~the [[Euler totient function]]~~.

	~~The full conjecture remains unsolved. However,~~ a ~~higher dimensional analogue~~ of ~~this conjecture has been resolved~~.~~<ref name=Mon204/><ref>{{cite journal \| first1=A~~.D. ~~\| last1=Pollington \| first2=R~~.C. \| last2=Vaughan \| author2-link=Bob Vaughan \| title=The ''k'' dimensional Duffin–Schaeffer conjecture \| journal=Mathematika \| volume=37 \| number=2 \| year=1990 \| pages=190–200 \| zbl=0715.11036 \| issn=0025-5793 }}</ref><ref name=Har0269>Harman (2002) p.69</ref>

	~~==Progress==~~
	~~The implication from the existence~~ of ~~the rational approximations to the divergence~~ of ~~the series follows from the [[Borel–Cantelli lemma]]~~.<~~ref name=Har0268~~>~~Harman (2002) p.68~~<~~/ref~~> ~~The converse implication is the crux of the conjecture~~.~~<ref name=Mon204/>~~
	~~There have been many partial results of the Duffin–Schaeffer conjecture established~~ to ~~date~~. ~~[[Paul Erdős]] established in 1970~~ that ~~the conjecture holds if there exists a constant <math>c > 0 </math> such~~ that for ~~every integer~~ <~~math~~> n <~~/math~~> we have ~~either <math>f(n) = c/n</math> or <math>f(n) = 0</math>~~.~~<ref name=Mon204>{{ cite book \| last=Montgomery \| first=Hugh L~~. ~~\| author-link=Hugh Montgomery (mathematician) \| title=Ten lectures on the interface between analytic number theory~~ and ~~harmonic analysis \| series=Regional Conference Series in Mathematics \| volume=84 \| location=Providence~~, ~~RI \| publisher=[[American Mathematical Society]] \| year=1994 \| isbn=0-8218-0737-4 \| zbl=0814~~.~~11001 \| page=204 }}~~<~~/ref~~><~~ref name=Har27~~>~~Harman (1998) p.27</ref> This was strengthened by Jeffrey Vaaler in 1978~~ to ~~the case <math>f(n) = O(n^{-1})</math>~~.~~<ref>http://www.math.osu.edu/files/duffin-schaeffer%20conjecture.pdf</ref><ref name=Har28>Harman (1998) p.28</ref> More recently~~, ~~this was strengthened to~~ the ~~conjecture being true whenever there exists some <math>\epsilon > 0 </math> such that~~ the ~~series <math>\sum_{n=1}^\infty \left(\frac{f(n)}{n}\right)^{1 + \epsilon} \varphi(n) = \infty </math>~~. ~~This was done by Haynes, Pollington,~~ and ~~Velani.<ref>A. Haynes, A. Pollington,~~ and S. ~~Velani, ''~~The ~~Duffin-Schaeffer Conjecture with extra divergence'', arXiv, (2009), http://arxiv.org/abs/0811.1234</ref>~~

	~~In 2006, Beresnevich and Velani proved that a [[Hausdorff dimension]] analogue of the Duffin–Schaeffer conjecture~~ is ~~equivalent~~ to the ~~original Duffin–Schaeffer conjecture, which is a priori weaker. This result is published in~~ the ~~''[[Annals~~ of ~~Mathematics]]''~~.<~~ref~~>~~{{cite journal \| last1=Beresnevich \| first1=Victor \| last2=Velani \| first2=Sanju \| title=A mass transference principle~~ and ~~the Duffin-Schaeffer conjecture~~ for ~~Hausdorff measures \| journal=[[Annals of Mathematics]] (2) \| volume=164 \| number=3 \| year=2006 \| pages=971–992 \| url=http://arxiv.org/abs/math/0412141 \| zbl=1148.11033 \| issn=0003-486X }}</ref>~~

	~~==Notes==~~
	~~{{Reflist}}~~

	~~==References==~~
	* {{cite book \| last=Harman \| first=Glyn \| author-link= \| title=Metric number theory \| series=London Mathematical Society Monographs. ~~New Series \| volume=18 \| location=Oxford \| publisher=[[Clarendon Press]] \| year=1998 \| isbn=0-19-850083-1 \| zbl=1081.11057 }}~~
	* {{cite book \| last=Harman \| first=Glyn \| authorlink=Glyn Harman \| chapter=One hundred years of normal numbers \| editor1-last=Bennett \| editor1-first=M. A. \| editor2-last=Berndt \| editor2-first=B.C. \| editor2-link=Bruce C. Berndt \| editor3-last=Boston \| editor3-first=N. \| editor3-link=Nigel Boston \| editor4-last=Diamond \| editor4-first=H.G. \| editor5-last=Hildebrand \| editor5-first=A.J. \| editor6-last=Philipp \| editor6-first=W. \| title=Surveys in number theory: Papers from the millennial conference on number theory \| location=Natick, MA \| publisher=A K Peters \| pages=57–74 \| year=2002 \| isbn=1-56881-162-4 \| zbl=1062.11052 }}

	~~{{DEFAULTSORT:Duffin-Schaeffer conjecture}}~~
	~~[[Category:Conjectures]]~~
	~~[[Category:Diophantine approximation]]~~

en>ChrisGualtieri: General Fixes using AWB

2013-10-22T01:13:14Z

General Fixes using AWB

← Older revision		Revision as of 03:13, 22 October 2013
Line 1:		Line 1:
	~~What~~'s ~~contained~~ in ~~this post can alter your daily life. If you~~'~~re contemplating investing~~, ~~you want what~~ is ~~presented~~ on ~~this page~~, to ~~be successful. If income are whatever you want and you have the drive and willpower to maintain you experiencing both the bad and good instances~~, ~~keep reading to determine~~ the ~~trick to achievement.~~<br><br>~~Keep~~ in ~~mind that personal shares do not really represent the whole market place. A good supply might soar while the all round market tanks~~, ~~while an unsatisfactory stock might jump in importance when~~ the ~~other industry is successful~~. ~~That~~ is ~~why it's a good idea to diversify~~ the ~~kinds of inventory you have~~, ~~picking stocks and shares from~~ a ~~variety~~ of ~~firms in many different market sectors~~.<br><br>~~When selecting stocks and shares, locate a strategy you enjoy and stay with it~~. ~~As an illustration, you might want to ignore the market~~'~~s behavior typically and concentration only with a company~~'~~s profits prospective. Once you choose a private pair of regulations, you may seek out well known buyers or financial specialists who discuss your approach, and you could gain knowledge from them~~.<br><br>~~Your making an investment strategy ought to include a long list~~ of ~~reasons behind committing. Identifying why you want~~ to ~~invest, and what you might do with~~ the ~~money you get may help you come up with all~~ of ~~your expenditure strategy. It will also help you stay encouraged to contribute to your investments~~.<br><br>~~Adjust your margin~~ of ~~security depending on~~ the ~~status, profitability, and size of a specific firm~~. ~~Whilst businesses like Yahoo and google or Johnson & Johnson are hardy and~~ have ~~a tendency~~ to ~~put close to,~~ there ~~are actually certain firms~~ that ~~could do perfectly~~ for ~~a while before crashing. Take this into account when picking shares.~~<br><br>~~Discover in order to make use of a brokerage to buy stocks and shares,~~ or ~~if you want to get right from a Immediate Expense Strategy or Dividend Reinvestment prepare~~. ~~If you do not believe, you really can afford a brokerage service, there are several discounted brokerages readily available~~. ~~Just keep~~ in ~~mind some businesses usually do not provide a Straight Expense Program~~.<br><br>~~It is important that you never ever consider committing as being a pastime~~. ~~It is really an extremely competing business, and if you keep that~~ in ~~mind it will be possible~~ to ~~experience a a lot more valuable perspective~~. ~~You need to seriously fully grasp your profits and failures together with the firms you happen to be making an investment in~~.<br><br>~~Expect to wait around it~~. ~~If you are investing in stocks~~, ~~expect~~ to ~~leave them by yourself for no less than five-years. Ensure~~ that ~~you can easily deal with without those funds, since it is~~ the ~~only way you will notice an effective earnings~~. ~~If the market place begins to do improperly~~, ~~try and keep levelheaded~~, and ~~realize that just like the marketplace falls, it will come back, nevertheless it needs time to work~~.<br>~~<br>You need to in no way make investments your entire dollars into 1 enterprise~~. ~~It makes no difference how much you cherish a certain industry~~. ~~To be able to develop an outstanding expense profile~~, ~~you will need to broaden~~. ~~Diversity is definitely the established approach to tremendously [~~http://~~browse~~.~~Deviantart.com~~/~~?q=increasing increasing] the chances of you profiting out of your inventory transactions~~.<br>~~<br>Use ranking methods cautiously in the keep industry. These score systems could be untrustworthy during this period~~, ~~so you could find yourself shedding~~ a ~~lot~~ of ~~money if you rely solely upon them. Rather than utilizing them like a information~~, ~~make use of them~~ a ~~method of second information and component~~ the ~~ranking into your choices using a grain~~ of ~~salt~~.<br>~~<br>As elementary as looking at this report~~ and ~~using~~ the ~~tips a part~~ of ~~it, start converting revenue in your assets and undertaking what it takes to achieve success in shelling out~~. ~~Whatever industry you decide on, or what ever degree of purchase you are feeling comfortable with, it doesn't acquire significantly hard work, when you have the knowledge needed~~. ~~These guidelines can aid you to appreciate your income supply!~~<br>~~<br>If you cherished this article therefore you would like to be given more info with regards to~~ [~~http://www~~.~~crh~~.~~noaa~~.~~gov/nwsexit~~.~~php?url~~=~~http://www~~.~~silverlinetrade~~.in~~/ commodity broker in chennai] kindly visit~~ the ~~web page~~.		The '''Duffin–Schaeffer conjecture''' is an important conjecture in [[metric number theory]] proposed by [[Richard Duffin\|R. J. Duffin]] and [[Albert Charles Schaeffer\|A. C. Schaeffer]] in 1941.<ref>R. J. Duffin and A. C. Schaeffer, ''Khintchine's problem in metric [[Diophantine approximation]]'', [[Duke Mathematical Journal]], '''8''' (1941), 243–255</ref> It states that if <math>f : \mathbb{N} \rightarrow \mathbb{R}^+</math> is a real-valued function taking on positive values, then for [[almost all]] <math> \alpha </math> (with respect to [[Lebesgue measure]]), the inequality

			: <math> \left \| \alpha - \frac{p}{q} \right \| < \frac{f(q)}{q} </math>

			has infinitely many solutions in [[co-prime]] integers <math>p,q</math> with <math>q > 0</math> if and only if the sum

			: <math> \sum_{q=1}^\infty f(q) \frac{\varphi(q)}{q} = \infty. </math>

			Here <math> \varphi(q)</math> is the [[Euler totient function]].

			The full conjecture remains unsolved. However, a higher dimensional analogue of this conjecture has been resolved.<ref name=Mon204/><ref>{{cite journal \| first1=A.D. \| last1=Pollington \| first2=R.C. \| last2=Vaughan \| author2-link=Bob Vaughan \| title=The ''k'' dimensional Duffin–Schaeffer conjecture \| journal=Mathematika \| volume=37 \| number=2 \| year=1990 \| pages=190–200 \| zbl=0715.11036 \| issn=0025-5793 }}</ref><ref name=Har0269>Harman (2002) p.69</ref>

			==Progress==
			The implication from the existence of the rational approximations to the divergence of the series follows from the [[Borel–Cantelli lemma]].<ref name=Har0268>Harman (2002) p.68</ref> The converse implication is the crux of the conjecture.<ref name=Mon204/>
			There have been many partial results of the Duffin–Schaeffer conjecture established to date. [[Paul Erdős]] established in 1970 that the conjecture holds if there exists a constant <math>c > 0 </math> such that for every integer <math> n </math> we have either <math>f(n) = c/n</math> or <math>f(n) = 0</math>.<ref name=Mon204>{{ cite book \| last=Montgomery \| first=Hugh L. \| author-link=Hugh Montgomery (mathematician) \| title=Ten lectures on the interface between analytic number theory and harmonic analysis \| series=Regional Conference Series in Mathematics \| volume=84 \| location=Providence, RI \| publisher=[[American Mathematical Society]] \| year=1994 \| isbn=0-8218-0737-4 \| zbl=0814.11001 \| page=204 }}</ref><ref name=Har27>Harman (1998) p.27</ref> This was strengthened by Jeffrey Vaaler in 1978 to the case <math>f(n) = O(n^{-1})</math>.<ref>http://www.math.osu.edu/files/duffin-schaeffer%20conjecture.pdf</ref><ref name=Har28>Harman (1998) p.28</ref> More recently, this was strengthened to the conjecture being true whenever there exists some <math>\epsilon > 0 </math> such that the series <math>\sum_{n=1}^\infty \left(\frac{f(n)}{n}\right)^{1 + \epsilon} \varphi(n) = \infty </math>. This was done by Haynes, Pollington, and Velani.<ref>A. Haynes, A. Pollington, and S. Velani, ''The Duffin-Schaeffer Conjecture with extra divergence'', arXiv, (2009), http://arxiv.org/abs/0811.1234</ref>

			In 2006, Beresnevich and Velani proved that a [[Hausdorff dimension]] analogue of the Duffin–Schaeffer conjecture is equivalent to the original Duffin–Schaeffer conjecture, which is a priori weaker. This result is published in the ''[[Annals of Mathematics]]''.<ref>{{cite journal \| last1=Beresnevich \| first1=Victor \| last2=Velani \| first2=Sanju \| title=A mass transference principle and the Duffin-Schaeffer conjecture for Hausdorff measures \| journal=[[Annals of Mathematics]] (2) \| volume=164 \| number=3 \| year=2006 \| pages=971–992 \| url=http://arxiv.org/abs/math/0412141 \| zbl=1148.11033 \| issn=0003-486X }}</ref>

			==Notes==
			{{Reflist}}

			==References==
			* {{cite book \| last=Harman \| first=Glyn \| author-link= \| title=Metric number theory \| series=London Mathematical Society Monographs. New Series \| volume=18 \| location=Oxford \| publisher=[[Clarendon Press]] \| year=1998 \| isbn=0-19-850083-1 \| zbl=1081.11057 }}
			* {{cite book \| last=Harman \| first=Glyn \| authorlink=Glyn Harman \| chapter=One hundred years of normal numbers \| editor1-last=Bennett \| editor1-first=M. A. \| editor2-last=Berndt \| editor2-first=B.C. \| editor2-link=Bruce C. Berndt \| editor3-last=Boston \| editor3-first=N. \| editor3-link=Nigel Boston \| editor4-last=Diamond \| editor4-first=H.G. \| editor5-last=Hildebrand \| editor5-first=A.J. \| editor6-last=Philipp \| editor6-first=W. \| title=Surveys in number theory: Papers from the millennial conference on number theory \| location=Natick, MA \| publisher=A K Peters \| pages=57–74 \| year=2002 \| isbn=1-56881-162-4 \| zbl=1062.11052 }}

			{{DEFAULTSORT:Duffin-Schaeffer conjecture}}
			[[Category:Conjectures]]
			[[Category:Diophantine approximation]]

en>Cydebot: Robot - Moving category Fundamental physics concepts to :Category:Concepts in physics per CFD at Wikipedia:Categories for discussion/Log/2012 July 12.

2012-07-19T00:41:54Z

Robot - Moving category Fundamental physics concepts to Category:Concepts in physics per CFD at Wikipedia:Categories for discussion/Log/2012 July 12.

New page

What's contained in this post can alter your daily life. If you're contemplating investing, you want what is presented on this page, to be successful. If income are whatever you want and you have the drive and willpower to maintain you experiencing both the bad and good instances, keep reading to determine the trick to achievement. Keep in mind that personal shares do not really represent the whole market place. A good supply might soar while the all round market tanks, while an unsatisfactory stock might jump in importance when the other industry is successful. That is why it's a good idea to diversify the kinds of inventory you have, picking stocks and shares from a variety of firms in many different market sectors. When selecting stocks and shares, locate a strategy you enjoy and stay with it. As an illustration, you might want to ignore the market's behavior typically and concentration only with a company's profits prospective. Once you choose a private pair of regulations, you may seek out well known buyers or financial specialists who discuss your approach, and you could gain knowledge from them. Your making an investment strategy ought to include a long list of reasons behind committing. Identifying why you want to invest, and what you might do with the money you get may help you come up with all of your expenditure strategy. It will also help you stay encouraged to contribute to your investments. Adjust your margin of security depending on the status, profitability, and size of a specific firm. Whilst businesses like Yahoo and google or Johnson & Johnson are hardy and have a tendency to put close to, there are actually certain firms that could do perfectly for a while before crashing. Take this into account when picking shares. Discover in order to make use of a brokerage to buy stocks and shares, or if you want to get right from a Immediate Expense Strategy or Dividend Reinvestment prepare. If you do not believe, you really can afford a brokerage service, there are several discounted brokerages readily available. Just keep in mind some businesses usually do not provide a Straight Expense Program. It is important that you never ever consider committing as being a pastime. It is really an extremely competing business, and if you keep that in mind it will be possible to experience a a lot more valuable perspective. You need to seriously fully grasp your profits and failures together with the firms you happen to be making an investment in. Expect to wait around it. If you are investing in stocks, expect to leave them by yourself for no less than five-years. Ensure that you can easily deal with without those funds, since it is the only way you will notice an effective earnings. If the market place begins to do improperly, try and keep levelheaded, and realize that just like the marketplace falls, it will come back, nevertheless it needs time to work. You need to in no way make investments your entire dollars into 1 enterprise. It makes no difference how much you cherish a certain industry. To be able to develop an outstanding expense profile, you will need to broaden. Diversity is definitely the established approach to tremendously [http://browse.Deviantart.com/?q=increasing increasing] the chances of you profiting out of your inventory transactions. Use ranking methods cautiously in the keep industry. These score systems could be untrustworthy during this period, so you could find yourself shedding a lot of money if you rely solely upon them. Rather than utilizing them like a information, make use of them a method of second information and component the ranking into your choices using a grain of salt. As elementary as looking at this report and using the tips a part of it, start converting revenue in your assets and undertaking what it takes to achieve success in shelling out. Whatever industry you decide on, or what ever degree of purchase you are feeling comfortable with, it doesn't acquire significantly hard work, when you have the knowledge needed. These guidelines can aid you to appreciate your income supply! If you cherished this article therefore you would like to be given more info with regards to [http://www.crh.noaa.gov/nwsexit.php?url=http://www.silverlinetrade.in/ commodity broker in chennai] kindly visit the web page.