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2014-11-20T19:36:38Z

Bot: Fixing broken section link

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	~~Quick fat reduction diets have been around since~~ the ~~world's initially dieter looked at their abdomen plus thought, "I need~~ to ~~lose~~ a ~~limited pounds - NOW." Even though experience has shown which quick "fad" diets usually cause temporary fat reduction~~, ~~dieters are still searching for the Holy Grail: A diet that lets them lose fat quick and keep it off~~.<br><br>~~Then regarding calorie intake~~, the ~~bulky bodybuilder must reduce it. What you need to do initially is determine what your basal metabolic rate is (BMR) is. This really is~~ the ~~calorie amount~~ the ~~body burns at rest~~. ~~There is~~ a ~~fast and easy-to-use free~~ [http://~~safedietplans~~.~~com~~/~~bmr-calculator bmr calculator~~] ~~we can employ on my webpage. Simply click on the link below this article~~ to ~~check it out. Once~~ you ~~discover what a BMR is we want to take in at least 500 however, no more than 1000 calories lower than this amount~~. ~~Keeping your calorie expenditure~~ in ~~this range might help we burn~~ the ~~fat without muscle together with it~~. ~~Plus~~, you~~'ll keep~~ the ~~metabolic rate high for maximum fat reduction. This is how to~~ get ~~ripped rapidly with calorie consumption should you happe to be bulky~~.<br><br>~~Naturally~~, ~~there are more factors to the numbers game then simply catching a formula; including~~, ~~genetics~~, ~~family history~~, ~~pre-existing conditions~~, ~~etc. If~~ you~~'d like~~ to get ~~an exact reading of the basal metabolic rate then please consult a dietician~~.<br><br>~~A wise part~~ of the ~~carbs for each day ought~~ to ~~be consumed with breakfast~~. ~~The time we spend snoozing~~ is ~~time~~ a ~~liver has~~ to ~~make stamina utilizing ATP plus~~ the ~~associated porphyria toxins with it~~. ~~You could eat little food every 2~~-~~3 hours rather~~ of ~~3 main food~~. ~~Just like~~ with ~~breakfast, a carb-heavy late dinner could enable a liver out overnight~~.<br><br>~~Body Surface Area:~~ Your ~~height plus fat lead~~ a ~~lot inside determining bmr~~. ~~The greater is a body surface area~~, ~~the high is a BMR. Thin~~, ~~tall people have a high BMR~~.<br><br>~~The BMI Calculator or BMI is a worthwhile tool~~, ~~however, it's based on fat plus refuses to measure fat vs muscle. There is a amount~~ of ~~people whose BMI~~ is ~~actually okay~~, ~~but have nevertheless have too much fat due~~ to a ~~reduction of muscle when you reach this age group. So, one suggested idea~~ is to ~~consider your waist~~ to ~~cool ratio. Measuring a waistline first, as a guide ladies ought~~ to ~~be 32inches (80cm) or less plus guys must be 37inches (94cm) or less. Anything over means you are at a high risk~~ of ~~diabetes plus heart disease~~. ~~Then measure a hips plus split the waist measuring by the cool measuring. Guys~~ you ~~need to be no high than 0.9~~, ~~ladies~~ no ~~higher than 0~~.8.<br><br>~~To sum everything up the simple formula~~ for ~~losing fat is having~~ the ~~calories burned better than your calories consumed~~. ~~Calculate your estimated RMR then decide on daily calorie intake~~.		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71.42.216.93 at 22:13, 15 February 2014

2014-02-15T22:13:13Z

← Older revision		Revision as of 00:13, 16 February 2014
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	~~In [[number theory]], a branch of [[mathematics]],~~ the '~~''Stark–Heegner theorem''' states precisely which [[quadratic field\|quadratic imaginary number fields]] admit [[unique factorization domain\|unique factorisation]] in~~ their ~~[[ring of integers]]~~. ~~It solves a special case of Gauss's [[Class number problem~~ for ~~imaginary quadratic fields\|class number problem]] of determining~~ the ~~number of imaginary quadratic fields~~ that ~~have a given fixed [[ideal class group\|class number]].~~		Quick fat reduction diets have been around since the world's initially dieter looked at their abdomen plus thought, "I need to lose a limited pounds - NOW." Even though experience has shown which quick "fad" diets usually cause temporary fat reduction, dieters are still searching for the Holy Grail: A diet that lets them lose fat quick and keep it off.<br><br>Then regarding calorie intake, the bulky bodybuilder must reduce it. What you need to do initially is determine what your basal metabolic rate is (BMR) is. This really is the calorie amount the body burns at rest. There is a fast and easy-to-use free [http://safedietplans.com/bmr-calculator bmr calculator] we can employ on my webpage. Simply click on the link below this article to check it out. Once you discover what a BMR is we want to take in at least 500 however, no more than 1000 calories lower than this amount. Keeping your calorie expenditure in this range might help we burn the fat without muscle together with it. Plus, you'll keep the metabolic rate high for maximum fat reduction. This is how to get ripped rapidly with calorie consumption should you happe to be bulky.<br><br>Naturally, there are more factors to the numbers game then simply catching a formula; including, genetics, family history, pre-existing conditions, etc. If you'd like to get an exact reading of the basal metabolic rate then please consult a dietician.<br><br>A wise part of the carbs for each day ought to be consumed with breakfast. The time we spend snoozing is time a liver has to make stamina utilizing ATP plus the associated porphyria toxins with it. You could eat little food every 2-3 hours rather of 3 main food. Just like with breakfast, a carb-heavy late dinner could enable a liver out overnight.<br><br>Body Surface Area: Your height plus fat lead a lot inside determining bmr. The greater is a body surface area, the high is a BMR. Thin, tall people have a high BMR.<br><br>The BMI Calculator or BMI is a worthwhile tool, however, it's based on fat plus refuses to measure fat vs muscle. There is a amount of people whose BMI is actually okay, but have nevertheless have too much fat due to a reduction of muscle when you reach this age group. So, one suggested idea is to consider your waist to cool ratio. Measuring a waistline first, as a guide ladies ought to be 32inches (80cm) or less plus guys must be 37inches (94cm) or less. Anything over means you are at a high risk of diabetes plus heart disease. Then measure a hips plus split the waist measuring by the cool measuring. Guys you need to be no high than 0.9, ladies no higher than 0.8.<br><br>To sum everything up the simple formula for losing fat is having the calories burned better than your calories consumed. Calculate your estimated RMR then decide on daily calorie intake.

	~~Let '''Q''' denote the set of [[rational number]]s,~~ and ~~let ''d'' be a square-free [[integer]] (i.e~~., ~~a product of distinct [[prime number\|primes]]) other than 1~~. ~~Then [[algebraic number field\|'''Q'''~~(~~√''d''~~)]] is ~~a [[field extension\|finite extension]] of '''Q''', called a quadratic extension~~. ~~The [[ideal class group\|class number]] of '''Q'''(√''d'')~~ is the ~~number of [[equivalence class]]es of [[ring ideal\|ideal]]s of~~ the ~~ring of integers of '''Q'''(√''d''), where two ideals ''I'' and ''J'' are equivalent [[if and only if]] there exist [[principal ideal]]s (''a'') and (''b'') such that (''a'')''I'' = (''b'')''J''~~. ~~Thus, the ring of integers of '''Q'''(√''d'')~~ is a ~~[[principal ideal domain]] (~~and ~~hence a~~ [~~[unique factorization domain~~]~~]) if and only if~~ the ~~class number of '''Q'''(√''d'') is equal~~ to 1. ~~The Stark–Heegner theorem can then be stated as follows:~~

	~~:If ''d'' < 0, then the class number of '''Q'''(√''d'')~~ is ~~equal~~ to ~~1 if and only if~~

	~~:: <math>d \~~in ~~\{\~~, -1, ~~-2, -3, -7, -11, -19, -43, -67, -163\,\}.</math>~~
	~~These are known as~~ the ~~[[Heegner number]]s~~.

	This ~~list~~ is ~~also written, replacing −1~~ with ~~−4 and −2 with −8 (which does not change the field), as:<ref>{{harvtxt\|Elkies\|1999}}, p. 93~~.<~~/ref~~>

	:<~~math~~>~~D = -3~~, -4, -7, -8, -11, ~~-19, -43, -67, -163,\,</math>~~
	~~where~~ '~~'D'' is interpreted as the [[Discriminant of~~ an ~~algebraic number field\|discriminant]] (either~~ of the ~~[[number field]] or of an [[elliptic curve]] with [[complex multiplication]]).~~

	~~==History==~~
	This result was first conjectured by [[Carl Friedrich Gauss\|Gauss]]. It was essentially proven by [[Kurt Heegner]] in 1952, but Heegner's proof had some minor gaps and the theorem was not accepted until [[Harold Stark]] gave a ~~complete proof in 1967, which Stark showed was actually equivalent to Heegner's. Heegner "died before anyone really understood what he had done"~~.<~~ref~~>~~{{harvtxt\|Goldfeld\|1985}}.~~<~~/ref~~> ~~Stark formally filled in~~ the ~~gap in Heegner's proof in 1969~~.~~<ref>{{harvtxt\|Stark\|1969}}.</ref> [[Alan Baker (mathematician)\|Alan Baker]] gave~~ a ~~completely different proof at about~~ the ~~same time (or more precisely reduced the result to a finite amount~~ of ~~computation)~~. ~~Stark's 1969 paper also cited the widely known 1895 text by [[Heinrich Martin Weber]] and noted that had Weber "only made the observation that the reducibility of [~~a ~~certain equation] would lead to~~ a ~~[[Diophantine equation]], the class-number one problem would have been solved 60 years ago"~~.

	~~In 1985, Monsur Kenku~~<~~ref~~>~~{{harvtxt\|Kenku\|1985}}.~~<~~/ref~~> ~~gave~~ a ~~novel proof using~~ the ~~[[Klein quartic]]~~. ~~[[Noam Elkies]] gives an exposition of this result~~.<~~ref~~>~~{{harvtxt\|Elkies\|1999}}, section 4.3.~~<~~/ref~~>

	~~==Real case==~~
	~~On the other hand~~, it ~~is unknown whether there are infinitely many ''d'' > 0 for which '''Q'''(√''d'') has class number 1. Computational results indicate that there are many such fields.~~

	~~==Notes==~~
	~~{{reflist\|2}}~~

	~~==References==~~

	*{{citation\|authorlink=Noam Elkies\|last=Elkies\|first=Noam D.\|year=1999\|contribution-url=http://library.msri.org/books/Book35/files/elkies.pdf\|contribution=The Klein Quartic in Number Theory\|pages=51–101\|editor-last=Levy\|editor-first=Silvio\|url=http://library.msri.org/books/Book35\|title=The Eightfold Way: The Beauty of Klein's ~~Quartic Curve\|publisher=Cambridge University Press\|series=MSRI Publications\|volume=35}}~~.
	*{{citation\|last=Goldfeld\|first=Dorian\|authorlink=Dorian M. Goldfeld\|title=The Gauss Class Number Problem For Imaginary Quadratic Fields\|year=1985\|mr=788386\|journal=[[Bulletin of the American Mathematical Society]]\|volume=13\|pages=23–37\|doi=10.1090/S0273-0979-1985-15352-2}}.
	*{{citation \| last=Heegner \| first=Kurt \| authorlink=Kurt Heegner \| doi=10.1007/BF01174749\|mr=0053135 \| title=Diophantische Analysis und Modulfunktionen \| journal=[[Mathematische Zeitschrift]] \| volume=56 \| year=1952 \| pages=227–253}}
	*{{citation\|last=Kenku\|first=M. Q.\|year=1985\|title=A note on the integral points of a ~~modular curve~~ of ~~level 7\|journal=Mathematika\|volume=32\|pages=45–48\|doi=10~~.~~1112/S0025579300010846}}~~.
	*{{citation\|editor-last=Levy\|editor-first~~=Silvio\|year=1999\|url=http://library~~.~~msri.org/books/Book35/contents.html\|title=The Eightfold Way: The Beauty~~ of ~~Klein's Quartic Curve\|publisher=Cambridge University Press\|series=MSRI Publications\|volume=35}}~~.
	*{{citation\|last=Stark\|first=H. M.~~\|authorlink=Harold Stark\|year=1969\|url=http://deepblue~~.~~lib~~.~~umich.edu/bitstream/2027.42/33039/1/0000425.pdf\|title=On~~ the ~~gap in~~ the ~~theorem of Heegner\|journal=[[Journal of Number Theory]]\|volume=1\|pages=16–27\|doi=10~~.~~1016/0022-314X(69)90023-7}}~~.

	~~{{DEFAULTSORT:Stark-Heegner theorem}}~~
	~~[[Category:Theorems in algebraic number theory]]~~

en>Mark viking: Added wl

2013-10-01T04:43:20Z

Added wl

← Older revision		Revision as of 06:43, 1 October 2013
Line 1:		Line 1:
	~~Hi!~~ I ~~am Dalton. Acting has been~~ a ~~thing~~ that I ~~appreciate totally addicted~~ to. ~~Vermont~~ is where ~~my home could be described~~ as. ~~Managing people has been my 24-hour period job for~~ a ~~while but I plan inside changing it~~. I've been ~~training on my website to produce some time now~~. ~~Study~~ it ~~out here~~: http://~~prometeu~~.~~net<br><br>Here is my website~~ ... [http://~~prometeu~~.~~net clash~~ of ~~clans hack no download~~]		In [[number theory]], a branch of [[mathematics]], the '''Stark–Heegner theorem''' states precisely which [[quadratic field\|quadratic imaginary number fields]] admit [[unique factorization domain\|unique factorisation]] in their [[ring of integers]]. It solves a special case of Gauss's [[Class number problem for imaginary quadratic fields\|class number problem]] of determining the number of imaginary quadratic fields that have a given fixed [[ideal class group\|class number]].

			Let '''Q''' denote the set of [[rational number]]s, and let ''d'' be a square-free [[integer]] (i.e., a product of distinct [[prime number\|primes]]) other than 1. Then [[algebraic number field\|'''Q'''(√''d'')]] is a [[field extension\|finite extension]] of '''Q''', called a quadratic extension. The [[ideal class group\|class number]] of '''Q'''(√''d'') is the number of [[equivalence class]]es of [[ring ideal\|ideal]]s of the ring of integers of '''Q'''(√''d''), where two ideals ''I'' and ''J'' are equivalent [[if and only if]] there exist [[principal ideal]]s (''a'') and (''b'') such that (''a'')''I'' = (''b'')''J''. Thus, the ring of integers of '''Q'''(√''d'') is a [[principal ideal domain]] (and hence a [[unique factorization domain]]) if and only if the class number of '''Q'''(√''d'') is equal to 1. The Stark–Heegner theorem can then be stated as follows:

			:If ''d'' < 0, then the class number of '''Q'''(√''d'') is equal to 1 if and only if

			:: <math>d \in \{\, -1, -2, -3, -7, -11, -19, -43, -67, -163\,\}.</math>
			These are known as the [[Heegner number]]s.

			This list is also written, replacing −1 with −4 and −2 with −8 (which does not change the field), as:<ref>{{harvtxt\|Elkies\|1999}}, p. 93.</ref>

			:<math>D = -3, -4, -7, -8, -11, -19, -43, -67, -163,\,</math>
			where ''D'' is interpreted as the [[Discriminant of an algebraic number field\|discriminant]] (either of the [[number field]] or of an [[elliptic curve]] with [[complex multiplication]]).

			==History==
			This result was first conjectured by [[Carl Friedrich Gauss\|Gauss]]. It was essentially proven by [[Kurt Heegner]] in 1952, but Heegner's proof had some minor gaps and the theorem was not accepted until [[Harold Stark]] gave a complete proof in 1967, which Stark showed was actually equivalent to Heegner's. Heegner "died before anyone really understood what he had done".<ref>{{harvtxt\|Goldfeld\|1985}}.</ref> Stark formally filled in the gap in Heegner's proof in 1969.<ref>{{harvtxt\|Stark\|1969}}.</ref> [[Alan Baker (mathematician)\|Alan Baker]] gave a completely different proof at about the same time (or more precisely reduced the result to a finite amount of computation). Stark's 1969 paper also cited the widely known 1895 text by [[Heinrich Martin Weber]] and noted that had Weber "only made the observation that the reducibility of [a certain equation] would lead to a [[Diophantine equation]], the class-number one problem would have been solved 60 years ago".

			In 1985, Monsur Kenku<ref>{{harvtxt\|Kenku\|1985}}.</ref> gave a novel proof using the [[Klein quartic]]. [[Noam Elkies]] gives an exposition of this result.<ref>{{harvtxt\|Elkies\|1999}}, section 4.3.</ref>

			==Real case==
			On the other hand, it is unknown whether there are infinitely many ''d'' > 0 for which '''Q'''(√''d'') has class number 1. Computational results indicate that there are many such fields.

			==Notes==
			{{reflist\|2}}

			==References==

			*{{citation\|authorlink=Noam Elkies\|last=Elkies\|first=Noam D.\|year=1999\|contribution-url=http://library.msri.org/books/Book35/files/elkies.pdf\|contribution=The Klein Quartic in Number Theory\|pages=51–101\|editor-last=Levy\|editor-first=Silvio\|url=http://library.msri.org/books/Book35\|title=The Eightfold Way: The Beauty of Klein's Quartic Curve\|publisher=Cambridge University Press\|series=MSRI Publications\|volume=35}}.
			*{{citation\|last=Goldfeld\|first=Dorian\|authorlink=Dorian M. Goldfeld\|title=The Gauss Class Number Problem For Imaginary Quadratic Fields\|year=1985\|mr=788386\|journal=[[Bulletin of the American Mathematical Society]]\|volume=13\|pages=23–37\|doi=10.1090/S0273-0979-1985-15352-2}}.
			*{{citation \| last=Heegner \| first=Kurt \| authorlink=Kurt Heegner \| doi=10.1007/BF01174749\|mr=0053135 \| title=Diophantische Analysis und Modulfunktionen \| journal=[[Mathematische Zeitschrift]] \| volume=56 \| year=1952 \| pages=227–253}}
			*{{citation\|last=Kenku\|first=M. Q.\|year=1985\|title=A note on the integral points of a modular curve of level 7\|journal=Mathematika\|volume=32\|pages=45–48\|doi=10.1112/S0025579300010846}}.
			*{{citation\|editor-last=Levy\|editor-first=Silvio\|year=1999\|url=http://library.msri.org/books/Book35/contents.html\|title=The Eightfold Way: The Beauty of Klein's Quartic Curve\|publisher=Cambridge University Press\|series=MSRI Publications\|volume=35}}.
			*{{citation\|last=Stark\|first=H. M.\|authorlink=Harold Stark\|year=1969\|url=http://deepblue.lib.umich.edu/bitstream/2027.42/33039/1/0000425.pdf\|title=On the gap in the theorem of Heegner\|journal=[[Journal of Number Theory]]\|volume=1\|pages=16–27\|doi=10.1016/0022-314X(69)90023-7}}.

			{{DEFAULTSORT:Stark-Heegner theorem}}
			[[Category:Theorems in algebraic number theory]]

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