Kirchhoff integral theorem - Revision history

en>Fgnievinski at 01:14, 13 October 2014

2014-10-13T01:14:07Z

← Older revision		Revision as of 03:14, 13 October 2014
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	~~In mathematics,~~ the ~~'''Noether inequality''', named after [[Max Noether]], is a property of [[compact space\|compact]] minimal [[complex surface]]s~~ that ~~restricts~~ the ~~topological type~~ of the ~~underlying topological [[4~~-~~manifold]]~~. ~~It holds more generally for minimal projective surfaces of general type over~~ an ~~algebraically closed field~~.		CMS provides the best platform to create websites that fulfill all the specifications of SEO. You may discover this probably the most time-consuming part of building a Word - Press MLM website. The Word - Press Dashboard : an administrative management tool that supports FTP content upload 2. Word - Press also provides protection against spamming, as security is a measure issue. It's as simple as hiring a Wordpress plugin developer or learning how to create what is needed. <br><br>Luckily, for Word - Press users, WP Touch plugin transforms your site into an IPhone style theme. If you wish to sell your services or products via internet using your website, you have to put together on the website the facility for trouble-free payment transfer between customers and the company. You are able to set them within your theme options and so they aid the search engine to get a suitable title and description for the pages that get indexed by Google. Furthermore, with the launch of Windows 7 Phone is the smart phone market nascent App. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself. <br><br>The least difficult and very best way to do this is by acquiring a Word - Press site. By using Word - Press MLM websites or blogs, you and your companies presence on the internet can be created swiftly and simply. If Gandhi was empowered with a blogging system, every event in his life would have been minutely documented so that it could be recounted to the future generations. Every single Theme might be unique, providing several alternatives for webpage owners to reap the benefits of in an effort to instantaneously adjust their web page appear. If you have any questions on starting a Word - Press food blog or any blog for that matter, please post them below and I will try to answer them. <br><br>Word - Press has plenty of SEO benefits over Joomla and Drupal. In case you need to hire PHP developers or hire Offshore Code - Igniter development services or you are looking for Word - Press development experts then Mindfire Solutions would be the right choice for a Software Development partner. This allows for keeping the content editing toolbar in place at all times no matter how far down the page is scrolled. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. Digital digital cameras now function gray-scale configurations which allow expert photographers to catch images only in black and white. <br><br>Yet, overall, less than 1% of websites presently have mobile versions of their websites. If you cherished this article therefore you would like to receive more info regarding [http://GET7.pw/backup_plugin_79171 backup plugin] kindly visit our web page. As a website owner, you can easily manage CMS-based website in a pretty easy and convenient style. This allows updates to be sent anyone who wants them via an RSS reader or directly to their email. It is a fact that Smartphone using online customers do not waste much of their time in struggling with drop down menus. Likewise, professional publishers with a multi author and editor setup often find that Word - Press lack basic user and role management capabilities.

	~~==Formulation of the inequality==~~
	~~Let~~ '~~'X'' be~~ a ~~smooth [[Minimal model (birational geometry)\|minimal]] [[Algebraic variety#Projective varieties\|projective]] [[surface of general type]] defined over an [[algebraically closed field]] (~~or ~~a smooth minimal compact complex surface of general type) with canonical divisor ''K'' = −''c''~~<~~sub~~>1<~~/sub~~>~~(''X'')~~, ~~and let ''p''<sub>g</sub> = ''h''<sup>0</sup>(''K'') be the dimension of the space of holomorphic two forms~~, ~~then~~
	~~:<math> p_g \le \frac{1}{2} c_1(X)^2 + 2~~. ~~</math>~~

	~~For complex surfaces~~, ~~an alternative formulation expresses this inequality in terms of topological invariants of~~ the ~~underlying real oriented four manifold~~. ~~Since~~ a ~~surface of general type is a [[Kaehler manifold\|Kähler]] surface,~~ the ~~dimension of the maximal positive subspace in intersection form on the second cohomology is given~~ by ~~''b''<sub>+</sub> = 1 + 2''p''<sub>g</sub>~~. ~~Moreover by the [[Hirzebruch signature theorem]] ''c''<sub>1</sub><sup>2</sup> (''X'') = 2''e'' + 3''σ''~~, ~~where ''e'' = ''c''<sub>2</sub>(''X'') is~~ the ~~topological [[Euler characteristic]] and ''σ'' = ''b''<sub>+</sub> − ''b''<sub>−</sub>~~ is the ~~signature of the [[intersection form]]~~. ~~Therefore~~ the ~~Noether inequality~~ can ~~also be expressed~~ as
	~~:<math> b_+ \le 2 e + 3 \sigma + 5 \~~, <~~/math~~>

	~~or equivalently using ''e'' = 2 – 2 ''b''~~<~~sub>1</sub> + ''b''<sub>+</sub> + ''b''<sub~~>-~~</sub>~~
	~~:<math> b_~~- ~~+ 4 b_1 \le 4b_+ + 9. \~~, ~~</math>~~

	~~Combining~~ the ~~Noether inequality~~ with ~~the [[Noether formula]] 12χ=''c''<sub>1</sub><sup>2</sup>+''c''<sub>2</sub> gives~~
	~~:<math> 5 c_1(X)^2 - c_2(X) + 36 \ge 12q </math>~~
	~~where ''q'' is the [[irregularity of~~ a ~~surface]]~~, ~~which leads~~ to
	~~a slightly weaker inequality, which is also often called~~ the ~~Noether inequality:~~
	~~:<math> 5 c_1(X)^2 - c_2(X) + 36 \ge 0 \quad (c_1^2(X)\text{ even}) </math>~~
	~~:<math> 5 c_1(X)^2 - c_2(X) + 30 \ge 0 \quad (c_1^2(X)\text{ odd})~~. ~~</math>~~

	~~Surfaces where equality holds (i.e. on~~ the ~~Noether line) are called [[Surfaces~~ of ~~general type#Horikawa surfaces\|Horikawa surfaces]]~~.

	~~==Proof sketch==~~
	~~It follows from the minimal general type condition~~ that ~~''K''<sup>2</sup> > 0. We may thus assume that ''p''<sub>g</sub> > 1~~, ~~since the inequality is otherwise automatic~~. ~~In particular, we may assume there is an effective divisor ''D'' representing ''K''. We then have an exact sequence~~
	~~:<math> 0 \to H^0(\mathcal{O}_X) \to H^0(K) \to H^0( K\|_D) \to H^1(\mathcal{O}_X) \to~~ <~~/math~~>

	so <~~math~~> ~~p_g~~ - ~~1 \le h^0(K\|_D). </math>~~

	~~Assume that ''D'' is smooth. By the [[adjunction formula]] ''D''~~ has ~~a canonical linebundle <math>\mathcal{O}_D(2K)</math>, therefore <math>K\|_D</math> is a [[special divisor]]~~ and ~~the [[Clifford's theorem on special divisors\|Clifford inequality]] applies, which gives~~
	~~:<math> h^0(K\|_D) - 1 \le \frac{1}{2}\mathrm{deg}_D(K) = \frac{1}{2}K^2~~.~~\, </math>~~

	In ~~general, essentially the same argument applies using a more general version of~~ the ~~Clifford inequality~~ for ~~local complete intersections with~~ a ~~dualising line bundle and 1 dimensional sections in the trivial line bundle~~. ~~These conditions are satisfied~~ for the ~~curve ''D'' by the adjunction formula and~~ the ~~fact that ''D''~~ is ~~numerically connected~~.

	~~==References==~~
	* {{Citation \| last1=Barth \| first1=Wolf P. \| last2=Hulek \| first2=Klaus \| last3=Peters \| first3=Chris A.~~M. \| last4=Van de Ven \| first4=Antonius \| title=Compact Complex Surfaces \| publisher= Springer~~-~~Verlag, Berlin \| series=Ergebnisse der Mathematik und ihrer Grenzgebiete~~. ~~3. Folge. \| isbn=978-3-540-00832-3 \| id={{MathSciNet \| id = 2030225}} \| year=2004 \| volume=4}}~~
	*{{Citation\| last1=Liedtke \|first1=Christian\|title=Algebraic Surfaces of general type with small c<~~sub~~>1<~~/sub~~>~~<sup>2</sup> in positive characteristic\|journal=Nagoya Math~~. ~~J.\|volume=191\|year=2008\|pages=111–134\|~~
	~~url=~~http://~~projecteuclid~~.~~org~~/~~DPubS/Repository/1~~.~~0/Disseminate?view=body&id=pdf_1&handle=euclid~~.~~nmj/1221656783 }}~~
	*{{Citation \|doi=10.~~1007/BF02106598 \|last1=Noether \|first1 = Max\| title=Zur Theorie der eindeutigen Entsprechungen algebraischer Gebilde\|journal=Math~~. ~~Ann~~.~~\|volume=8 \|issue=4\| year=1875\|pages=495–533}}~~

	~~[[Category:Inequalities]]~~
	~~[[Category:Algebraic surfaces]]~~

en>ChrisGualtieri: Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB

2013-12-20T06:40:22Z

Remove stub template(s). Page is start class or higher. Also check for and do General Fixes + Checkwiki fixes using AWB

← Older revision		Revision as of 08:40, 20 December 2013
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	~~Google... 1 of~~ the ~~Internets MOST effective search engines recently announced~~ a ~~new Totally free service~~ that ~~tends to make it less difficult~~ for ~~you to boost your internet sites coverage within Googles index by way~~ of ~~its NEW collaborative crawling technique~~.~~<br><br>~~		In mathematics, the '''Noether inequality''', named after [[Max Noether]], is a property of [[compact space\|compact]] minimal [[complex surface]]s that restricts the topological type of the underlying topological [[4-manifold]]. It holds more generally for minimal projective surfaces of general type over an algebraically closed field.

	On the ~~down side although, Google does not "Guarantee" instant inclusion, But, is nevertheless very good news for these~~ of ~~you who have just completed developing~~ a ~~site and are ready to submit to the prime search engines.~~<br><br>So, ~~what"s this NEW service referred to as that Google just released?~~<br><br>~~It"s named "Google Sitemaps".~~<br><br>~~The genuine cool issue about "Google Sitemaps" IS it accepts RSS~~(~~Actual Easy Syndication~~) ~~and Atom feeds with~~ the ~~.XML extension as 1~~ of the ~~accepted formats for submitting your internet sites Sitemap for inclusion in Googles index.~~<br><br>~~This is just a single~~ of the four ~~formats Google accepts and~~ is the ~~one that I am going to use for~~ the ~~rest of this article to show you how to create your own .XML sitemap of your site for inclusion~~ in ~~Google.~~<br><br>~~It really is actual effortless to do so don~~&~~quot~~;~~t worry about it becoming to complicated~~.<br><br>So, ~~with that stated, lets go to Step #1.~~<br><br>~~Step #1. Discover further on our related use with by visiting~~ [~~http://about.me/moveacttym tell us what you think~~]~~. I learned about [http:~~/~~/national-student-loan-consolidation.com/showthread.php?tid=33868 patent pending] by searching Google. Sign up for "Google Sitemaps".~~<br><br>~~This step~~ is ~~easy, basically stick to~~ the ~~hyperlink offer below and sign up for a Free~~ of ~~charge account~~. ~~It should only take you a handful of minutes to do~~, ~~then when your done, come back right here and go to Step #two.~~<br><br>~~Go to: #2. Produce your .XML sitemap.~~<br><br>~~In this step your going to create a .XML sitemap for your internet site using a Totally free on~~-~~line sitemap generator.~~<br><br>~~This Free of charge on~~-~~line generator will build you a sitemap up to 500 pages~~. ~~If you have a larger site~~ with ~~thousands of webpages you are going to have to use a paid service and~~/~~or software to create yours.~~<br><br>~~Go to~~: ~~your essential info on~~ the ~~form offered.~~<br><br>~~Step #3. Upload your generated .XML sitemap to your server.~~<br><br>~~Once your generated sitemap is completed~~(~~your generated .XML file need to be referred to as website~~.~~xml~~.gz) ~~it really is now time to upload it to your server utilizing your FTP(File Transfer Protocol) system~~.<br><br>~~This is quite critical~~ that ~~you do this ahead of submitting it to Google Sitemaps for inclusion.~~<br><br>~~Step #4~~. ~~Submit your NEW .XML sitemap to "Google Sitemaps"~~.<br>~~<br>Now that you have generated your "Sitemap" for your internet site and uploaded it~~ to ~~your server all you have~~ to ~~do now is~~ to go to ~~"Google Sitemaps" and submit your sitemap URL.~~<br><br>(ex. ~~it.~~<br>~~<br>Now, based on Google, your sitemap must get downloaded within the subsequent 24 hours, so be positive to check and see~~ that ~~it did and that there wasn"t any issues or errors~~.<br><br>~~Also, your going to want to be able to track your indexed webpages inside the search engines, especially Google, so you can see how items are going~~, ~~so I"ll leave you with a single last resource that will do this for you.~~<br><br>~~Go to: you have to do~~ is ~~enter your URL~~ and ~~click "GO".~~<br><br>~~The results page will list~~ the ~~benefits from Google~~ and ~~9 other common search engines~~.~~<br><br>Properly~~, ~~there you have it~~... "4 ~~Rapid and Straightforward Steps To Receiving Your Web site IN By means~~ of ~~Googles "Back Door""..~~<br><br>~~In case you loved this post and you want to receive much more information concerning health insurance providers ([~~http://~~www~~.~~getjealous~~.~~com~~/~~judiciousaxle1805 getjealous~~.~~com~~]~~) kindly visit our own web-page.~~		==Formulation of the inequality==
			Let ''X'' be a smooth [[Minimal model (birational geometry)\|minimal]] [[Algebraic variety#Projective varieties\|projective]] [[surface of general type]] defined over an [[algebraically closed field]] (or a smooth minimal compact complex surface of general type) with canonical divisor ''K'' = −''c''<sub>1</sub>(''X''), and let ''p''<sub>g</sub> = ''h''<sup>0</sup>(''K'') be the dimension of the space of holomorphic two forms, then
			:<math> p_g \le \frac{1}{2} c_1(X)^2 + 2. </math>

			For complex surfaces, an alternative formulation expresses this inequality in terms of topological invariants of the underlying real oriented four manifold. Since a surface of general type is a [[Kaehler manifold\|Kähler]] surface, the dimension of the maximal positive subspace in intersection form on the second cohomology is given by ''b''<sub>+</sub> = 1 + 2''p''<sub>g</sub>. Moreover by the [[Hirzebruch signature theorem]] ''c''<sub>1</sub><sup>2</sup> (''X'') = 2''e'' + 3''σ'', where ''e'' = ''c''<sub>2</sub>(''X'') is the topological [[Euler characteristic]] and ''σ'' = ''b''<sub>+</sub> − ''b''<sub>−</sub> is the signature of the [[intersection form]]. Therefore the Noether inequality can also be expressed as
			:<math> b_+ \le 2 e + 3 \sigma + 5 \, </math>

			or equivalently using ''e'' = 2 – 2 ''b''<sub>1</sub> + ''b''<sub>+</sub> + ''b''<sub>-</sub>
			:<math> b_- + 4 b_1 \le 4b_+ + 9. \, </math>

			Combining the Noether inequality with the [[Noether formula]] 12χ=''c''<sub>1</sub><sup>2</sup>+''c''<sub>2</sub> gives
			:<math> 5 c_1(X)^2 - c_2(X) + 36 \ge 12q </math>
			where ''q'' is the [[irregularity of a surface]], which leads to
			a slightly weaker inequality, which is also often called the Noether inequality:
			:<math> 5 c_1(X)^2 - c_2(X) + 36 \ge 0 \quad (c_1^2(X)\text{ even}) </math>
			:<math> 5 c_1(X)^2 - c_2(X) + 30 \ge 0 \quad (c_1^2(X)\text{ odd}). </math>

			Surfaces where equality holds (i.e. on the Noether line) are called [[Surfaces of general type#Horikawa surfaces\|Horikawa surfaces]].

			==Proof sketch==
			It follows from the minimal general type condition that ''K''<sup>2</sup> > 0. We may thus assume that ''p''<sub>g</sub> > 1, since the inequality is otherwise automatic. In particular, we may assume there is an effective divisor ''D'' representing ''K''. We then have an exact sequence
			:<math> 0 \to H^0(\mathcal{O}_X) \to H^0(K) \to H^0( K\|_D) \to H^1(\mathcal{O}_X) \to </math>

			so <math> p_g - 1 \le h^0(K\|_D). </math>

			Assume that ''D'' is smooth. By the [[adjunction formula]] ''D'' has a canonical linebundle <math>\mathcal{O}_D(2K)</math>, therefore <math>K\|_D</math> is a [[special divisor]] and the [[Clifford's theorem on special divisors\|Clifford inequality]] applies, which gives
			:<math> h^0(K\|_D) - 1 \le \frac{1}{2}\mathrm{deg}_D(K) = \frac{1}{2}K^2.\, </math>

			In general, essentially the same argument applies using a more general version of the Clifford inequality for local complete intersections with a dualising line bundle and 1 dimensional sections in the trivial line bundle. These conditions are satisfied for the curve ''D'' by the adjunction formula and the fact that ''D'' is numerically connected.

			==References==
			* {{Citation \| last1=Barth \| first1=Wolf P. \| last2=Hulek \| first2=Klaus \| last3=Peters \| first3=Chris A.M. \| last4=Van de Ven \| first4=Antonius \| title=Compact Complex Surfaces \| publisher= Springer-Verlag, Berlin \| series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. \| isbn=978-3-540-00832-3 \| id={{MathSciNet \| id = 2030225}} \| year=2004 \| volume=4}}
			*{{Citation\| last1=Liedtke \|first1=Christian\|title=Algebraic Surfaces of general type with small c<sub>1</sub><sup>2</sup> in positive characteristic\|journal=Nagoya Math. J.\|volume=191\|year=2008\|pages=111–134\|
			url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.nmj/1221656783 }}
			*{{Citation \|doi=10.1007/BF02106598 \|last1=Noether \|first1 = Max\| title=Zur Theorie der eindeutigen Entsprechungen algebraischer Gebilde\|journal=Math. Ann.\|volume=8 \|issue=4\| year=1875\|pages=495–533}}

			[[Category:Inequalities]]
			[[Category:Algebraic surfaces]]

128.250.55.245: /* Monochromatic waves */ Removed mostly-irrelevant "main article" link

2012-07-31T04:01:02Z

Monochromatic waves: Removed mostly-irrelevant "main article" link

New page

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On the down side although, Google does not "Guarantee" instant inclusion, But, is nevertheless very good news for these of you who have just completed developing a site and are ready to submit to the prime search engines. So, what"s this NEW service referred to as that Google just released? It"s named "Google Sitemaps". The genuine cool issue about "Google Sitemaps" IS it accepts RSS(Actual Easy Syndication) and Atom feeds with the .XML extension as 1 of the accepted formats for submitting your internet sites Sitemap for inclusion in Googles index. This is just a single of the four formats Google accepts and is the one that I am going to use for the rest of this article to show you how to create your own .XML sitemap of your site for inclusion in Google. It really is actual effortless to do so don"t worry about it becoming to complicated. So, with that stated, lets go to Step #1. Step #1. Discover further on our related use with by visiting [http://about.me/moveacttym tell us what you think]. I learned about [http://national-student-loan-consolidation.com/showthread.php?tid=33868 patent pending] by searching Google. Sign up for "Google Sitemaps". This step is easy, basically stick to the hyperlink offer below and sign up for a Free of charge account. It should only take you a handful of minutes to do, then when your done, come back right here and go to Step #two. Go to: #2. Produce your .XML sitemap. In this step your going to create a .XML sitemap for your internet site using a Totally free on-line sitemap generator. This Free of charge on-line generator will build you a sitemap up to 500 pages. If you have a larger site with thousands of webpages you are going to have to use a paid service and/or software to create yours. Go to: your essential info on the form offered. Step #3. Upload your generated .XML sitemap to your server. Once your generated sitemap is completed(your generated .XML file need to be referred to as website.xml.gz) it really is now time to upload it to your server utilizing your FTP(File Transfer Protocol) system. This is quite critical that you do this ahead of submitting it to Google Sitemaps for inclusion. Step #4. Submit your NEW .XML sitemap to "Google Sitemaps". Now that you have generated your "Sitemap" for your internet site and uploaded it to your server all you have to do now is to go to "Google Sitemaps" and submit your sitemap URL. (ex. it. Now, based on Google, your sitemap must get downloaded within the subsequent 24 hours, so be positive to check and see that it did and that there wasn"t any issues or errors. Also, your going to want to be able to track your indexed webpages inside the search engines, especially Google, so you can see how items are going, so I"ll leave you with a single last resource that will do this for you. Go to: you have to do is enter your URL and click "GO". The results page will list the benefits from Google and 9 other common search engines. Properly, there you have it... "4 Rapid and Straightforward Steps To Receiving Your Web site IN By means of Googles "Back Door"".. In case you loved this post and you want to receive much more information concerning health insurance providers ([http://www.getjealous.com/judiciousaxle1805 getjealous.com]) kindly visit our own web-page.