Graham number - Revision history

en>Ronz: original research, not a reliable source, refspam

2014-07-02T20:18:37Z

original research, not a reliable source, refspam

← Older revision		Revision as of 22:18, 2 July 2014
Line 1:		Line 1:
	~~I am Oscar~~ and ~~I completely dig that title~~. ~~For many years he~~'s been ~~working~~ as a ~~receptionist. North Dakota is her beginning location but she will have to move one day~~ or ~~another~~. ~~Doing ceramics~~ is ~~what her family members and her appreciate~~.<br><br>~~My blog post:~~ [http://~~www~~.~~ba8ba~~.~~com~~/~~space~~.~~php?uid=35900&do=blog&id=33357~~ http://www.~~ba8ba~~.com/~~space.php~~?~~uid~~=~~35900~~&do=~~blog~~&id=~~33357~~]

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en>Dimo414: Added quote from The Intelligent Investor expanding on the origin of this calculation, removed citations that were minimally informative / unrelated.

2014-02-05T04:02:10Z

Added quote from The Intelligent Investor expanding on the origin of this calculation, removed citations that were minimally informative / unrelated.

← Older revision		Revision as of 06:02, 5 February 2014
Line 1:		Line 1:
	~~{{Unreferenced\| date=August 2012}}~~		I am Oscar and I completely dig that title. For many years he's been working as a receptionist. North Dakota is her beginning location but she will have to move one day or another. Doing ceramics is what her family members and her appreciate.<br><br>My blog post: [http://www.ba8ba.com/space.php?uid=35900&do=blog&id=33357 http://www.ba8ba.com/space.php?uid=35900&do=blog&id=33357]
	~~In [[abstract algebra]], if ''~~I'' and ~~''J'' are [[ideal (ring theory)\|ideals]] of a commutative [[ring (mathematics)\|ring]] ''R'', their '''ideal quotient''' (''~~I'~~' : ''J'') is the set~~

	~~:<math>(I : J) = \{r \in R \| rJ \subset I\}</math>~~

	~~Then (''I'' : ''J'') is itself an ideal in ''R''. The ideal quotient is viewed~~ as a ~~quotient because <math>IJ \subset K</math> if and only if <math>I \subset K : J</math>~~. ~~The ideal quotient~~ is ~~useful for calculating [[primary decomposition]]s. It also arises in the description of the [[Complement (set theory)#Relative complement\|set difference]] in [[algebraic geometry]] (see below).~~

	~~(''I'' : ''J'') is sometimes referred~~ to ~~as a '''colon ideal''' because of the notation~~. ~~There~~ is ~~an unrelated notion of the inverse of an ideal, known as a [[fractional ideal]] which is defined for Dedekind rings~~.

	~~==Properties==~~
	~~The ideal quotient satisfies the following properties:~~
	*<math>(I :J)=\mathrm{Ann}_R((J+I)/I)<~~/math~~> as <~~math~~>~~R</math>-modules, where <math>\mathrm{Ann}_R(M)</math> denotes the~~ [~~[annihilator (ring theory)\|annihilator]] of <math>M</math> as an <math>R</math>-module.~~
	*<math>J \subset I \Rightarrow I : J = R</math>
	*<math>I : R = I</math>
	*<math>R : I = R</math>
	*<math>I : ~~(J + K) = (I : J) \cap (I : K)<~~/~~math>~~
	*<math>I : (r) = \frac{1}{r}(I \cap (r))</~~math> (as long as ''R'' is an integral domain)~~

	~~==Calculating the quotient==~~
	The above properties can be used to calculate the quotient of ideals in a polynomial ring given their generators. For example, if ''I'' = (''f''<sub>1</sub>, ''f''<sub>2</sub>, ''f''<sub>3</sub>) and ''J'' = (''g''<sub>1</sub>, ''g''<sub>2</sub>) are ideals in ''k''[''x''<sub>1</sub>, ...~~, ''x''<sub>''n''<~~/~~sub>], then~~
	~~:<math>I : J = (I : (g_1)) \cap (I : (g_2)) = \left(\frac{1}{g_1}(I \cap (g_1))\right) \cap \left(\frac{1}{g_2}(I \cap (g_2))\right)</math>~~

	~~Then elimination theory can be used to calculate the intersection of ''I'' with (''g''<sub>1</sub>) and (''g''<sub>2</sub>):~~
	~~:<math>I \cap (g_1) = tI + (1-t)(g_1) \cap k[x_1, \dots, x_n], \quad I \cap (g_2) = tI + (1-t)(g_1) \cap k[x_1, \dots, x_n]</math>~~

	~~Calculate a [[Gröbner basis]] for ''tI'' + (1-''t'')(''g''<sub>1</sub>) with respect to lexicographic order. Then the basis functions which have no ''t'' in them generate <math>I \cap (g_1)</math>~~.

	==~~Geometric interpretation~~==
	~~The ideal quotient corresponds to [[Complement (set theory)#Relative complement\|set difference]] in [[algebraic geometry]]. More precisely,~~
	*If ''W'' is an affine variety and ''V'' is a subset of the affine space (not necessarily a variety), then
	:~~''I''(''V'') : ''I''(''W'') = ''I''(''V'' \ ''W''),~~

	~~where ''I'' denotes the taking of the ideal associated to a subset.~~
	*If ''I'' and ''J'' are ideals in ''k''[''x''<sub>1</~~sub>, .~~..~~, ''x''<sub>''n''<~~/~~sub>], then~~
	~~:''Z''(''I'' : ''J'') = cl(''Z''(''I'') \ ''Z''(''J''))~~

	~~where "cl" denotes the [[Zariski topology\|Zariski]] [[Closure (topology)\|closure]], and ''Z'' denotes the taking of the variety defined by the ideal ''I''~~.

	==~~References~~==
	~~{{Reflist}}~~
	~~Viviana Ene, Jürgen Herzog: 'Gröbner Bases in Commutative Algebra', AMS Graduate Studies in Mathematics, Vol 130 (AMS 2012)~~

	~~M.F.Atiyah, I.G.MacDonald: 'Introduction to Commutative Algebra', Addison-Wesley 1969.~~
	~~[[Category:Ideals]~~]

92.19.239.125 at 11:53, 23 December 2013

2013-12-23T11:53:36Z

← Older revision		Revision as of 13:53, 23 December 2013
Line 1:		Line 1:
	~~Many folks have this habit~~ of ~~doing all~~ the ~~stuff by themselves, regardless of how important or simple they are! These persons won~~'~~t let others interfere~~ in ~~their affairs~~. ~~While this stance can work inside other regions~~ of ~~existence, it~~'~~s not the best way~~ to ~~reply when you need to fix~~ the ~~Windows registry~~. There ~~are some jobs such~~ as ~~removing spywares, virus plus obsolete registry entries, that are best left to pro softwares~~. ~~In this article~~ I ~~may tell we why it is actually critical to fix Windows registry NOW!~~<br><br>~~We require to know certain effortless and cheap methods which will resolve~~ the ~~issue~~ of ~~the computer plus speed it up. The earlier you fix it, the less damage the computer gets~~. I ~~may tell regarding several useful techniques that may aid we to speed up you computer.~~<br><br>~~If you compare registry products you need a rapidly acting registry cleaning. It~~'~~s no wise spending hours plus~~ the ~~PC waiting for a registry cleaning~~ to ~~complete its task. We want~~ the ~~cleaner to complete its task inside minutes~~.<br><br>~~Always see with it which you have installed antivirus~~, ~~anti-spyware~~ and ~~anti-adware programs plus have them updated on a regular basis~~. ~~This can help stop windows XP running slow~~.<br><br>~~These are the results which the [http~~://~~bestregistrycleanerfix.com~~/~~tune~~-~~up-utilities tuneup utilities] found: 622 incorrect registry entries~~, 45,~~810 junk files~~, 15,~~643 unprotected confidentiality files~~, ~~8,462 bad Active X products that were not blocked, 16 performance features which were not optimized, and 4 updates which the computer needed.~~<br><br>~~The main reason why I couldn~~'t ~~create my PC run quicker was the program registry plus it being fragmented~~. ~~So software~~ to ~~defragment or clean the registry are required~~. ~~Such software are called registry cleaners. Like all different software~~, ~~there are paid ones plus free ones with their advantages~~ and ~~disadvantages. To choose between your two~~ is the ~~user~~'~~s choice~~.<br><br>~~The System File Checker (SFC) will help in resolving error 1721 as it~~, ~~by its nature, scans the system files for corruption and replaces them with their authentic versions~~. ~~This requires you to have the Windows Installation DVD ROM for continuing~~.<br><br>~~There are many businesses that provide~~ the ~~service~~ of ~~troubleshooting~~ the ~~PC each time we call them~~, ~~all you must do is signal up with them plus for a small fee~~, ~~you have the machine constantly functioning well plus serve you greater~~.		{{Unreferenced\| date=August 2012}}
			In [[abstract algebra]], if ''I'' and ''J'' are [[ideal (ring theory)\|ideals]] of a commutative [[ring (mathematics)\|ring]] ''R'', their '''ideal quotient''' (''I'' : ''J'') is the set

			:<math>(I : J) = \{r \in R \| rJ \subset I\}</math>

			Then (''I'' : ''J'') is itself an ideal in ''R''. The ideal quotient is viewed as a quotient because <math>IJ \subset K</math> if and only if <math>I \subset K : J</math>. The ideal quotient is useful for calculating [[primary decomposition]]s. It also arises in the description of the [[Complement (set theory)#Relative complement\|set difference]] in [[algebraic geometry]] (see below).

			(''I'' : ''J'') is sometimes referred to as a '''colon ideal''' because of the notation. There is an unrelated notion of the inverse of an ideal, known as a [[fractional ideal]] which is defined for Dedekind rings.

			==Properties==
			The ideal quotient satisfies the following properties:
			*<math>(I :J)=\mathrm{Ann}_R((J+I)/I)</math> as <math>R</math>-modules, where <math>\mathrm{Ann}_R(M)</math> denotes the [[annihilator (ring theory)\|annihilator]] of <math>M</math> as an <math>R</math>-module.
			*<math>J \subset I \Rightarrow I : J = R</math>
			*<math>I : R = I</math>
			*<math>R : I = R</math>
			*<math>I : (J + K) = (I : J) \cap (I : K)</math>
			*<math>I : (r) = \frac{1}{r}(I \cap (r))</math> (as long as ''R'' is an integral domain)

			==Calculating the quotient==
			The above properties can be used to calculate the quotient of ideals in a polynomial ring given their generators. For example, if ''I'' = (''f''<sub>1</sub>, ''f''<sub>2</sub>, ''f''<sub>3</sub>) and ''J'' = (''g''<sub>1</sub>, ''g''<sub>2</sub>) are ideals in ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>], then
			:<math>I : J = (I : (g_1)) \cap (I : (g_2)) = \left(\frac{1}{g_1}(I \cap (g_1))\right) \cap \left(\frac{1}{g_2}(I \cap (g_2))\right)</math>

			Then elimination theory can be used to calculate the intersection of ''I'' with (''g''<sub>1</sub>) and (''g''<sub>2</sub>):
			:<math>I \cap (g_1) = tI + (1-t)(g_1) \cap k[x_1, \dots, x_n], \quad I \cap (g_2) = tI + (1-t)(g_1) \cap k[x_1, \dots, x_n]</math>

			Calculate a [[Gröbner basis]] for ''tI'' + (1-''t'')(''g''<sub>1</sub>) with respect to lexicographic order. Then the basis functions which have no ''t'' in them generate <math>I \cap (g_1)</math>.

			==Geometric interpretation==
			The ideal quotient corresponds to [[Complement (set theory)#Relative complement\|set difference]] in [[algebraic geometry]]. More precisely,
			*If ''W'' is an affine variety and ''V'' is a subset of the affine space (not necessarily a variety), then
			:''I''(''V'') : ''I''(''W'') = ''I''(''V'' \ ''W''),

			where ''I'' denotes the taking of the ideal associated to a subset.
			*If ''I'' and ''J'' are ideals in ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>], then
			:''Z''(''I'' : ''J'') = cl(''Z''(''I'') \ ''Z''(''J''))

			where "cl" denotes the [[Zariski topology\|Zariski]] [[Closure (topology)\|closure]], and ''Z'' denotes the taking of the variety defined by the ideal ''I''.

			==References==
			{{Reflist}}
			Viviana Ene, Jürgen Herzog: 'Gröbner Bases in Commutative Algebra', AMS Graduate Studies in Mathematics, Vol 130 (AMS 2012)

			M.F.Atiyah, I.G.MacDonald: 'Introduction to Commutative Algebra', Addison-Wesley 1969.
			[[Category:Ideals]]

27.61.45.3 at 17:22, 7 July 2012

2012-07-07T17:22:36Z

New page

Many folks have this habit of doing all the stuff by themselves, regardless of how important or simple they are! These persons won't let others interfere in their affairs. While this stance can work inside other regions of existence, it's not the best way to reply when you need to fix the Windows registry. There are some jobs such as removing spywares, virus plus obsolete registry entries, that are best left to pro softwares. In this article I may tell we why it is actually critical to fix Windows registry NOW! We require to know certain effortless and cheap methods which will resolve the issue of the computer plus speed it up. The earlier you fix it, the less damage the computer gets. I may tell regarding several useful techniques that may aid we to speed up you computer. If you compare registry products you need a rapidly acting registry cleaning. It's no wise spending hours plus the PC waiting for a registry cleaning to complete its task. We want the cleaner to complete its task inside minutes. Always see with it which you have installed antivirus, anti-spyware and anti-adware programs plus have them updated on a regular basis. This can help stop windows XP running slow. These are the results which the [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities] found: 622 incorrect registry entries, 45,810 junk files, 15,643 unprotected confidentiality files, 8,462 bad Active X products that were not blocked, 16 performance features which were not optimized, and 4 updates which the computer needed. The main reason why I couldn't create my PC run quicker was the program registry plus it being fragmented. So software to defragment or clean the registry are required. Such software are called registry cleaners. Like all different software, there are paid ones plus free ones with their advantages and disadvantages. To choose between your two is the user's choice. The System File Checker (SFC) will help in resolving error 1721 as it, by its nature, scans the system files for corruption and replaces them with their authentic versions. This requires you to have the Windows Installation DVD ROM for continuing. There are many businesses that provide the service of troubleshooting the PC each time we call them, all you must do is signal up with them plus for a small fee, you have the machine constantly functioning well plus serve you greater.