Hexagonal number - Revision history

en>Onmywaybackhome: Fixed definition using phrase adapted from description of pentagonal numbers.

2014-04-16T11:22:33Z

Fixed definition using phrase adapted from description of pentagonal numbers.

← Older revision		Revision as of 13:22, 16 April 2014
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	~~{{Otheruses4\|free algebras in ring theory\|the more general free algebras in [[universal algebra]]\|free object}}~~		Howdy. The [http://Imageshack.us/photos/author%27s author's] name is Eusebio remember, though , he never really cherished that name. In his professional life he can be a people manager. He's always loved living located in Guam and he features everything that he prerequisites there. To drive is one of a person's things he loves mainly. He's been working by his website for various time now. Check one out here: http://[http://Answers.yahoo.com/search/search_result?p=prometeu.net&submit-go=Search+Y!+Answers prometeu.net]<br><br>Feel free to surf to my web blog :: [http://prometeu.net Clash Of Clans Hack No Survey No Jailbreak]
	In [[mathematics]], especially in the area of [[abstract algebra]] known as [[ring theory]], a '''free algebra''' is the noncommutative analogue of a [[polynomial ring]] (which may be regarded as a '''free commutative algebra''').

	~~==Definition==~~
	~~For ''R'' a~~ [~~[commutative ring]], the free ([[associative]], [[unital algebra\|unital]]) [[algebra (ring theory)\|algebra]] on ''n'' [[indeterminate_(variable)\|indeterminate]]s {''X''<sub>1<~~/~~sub>,~~.~~..,''X<sub>n</sub>''} is the [[free module\|free ''R''-module]] with a basis consisting of all [[Word_(mathematics)\|words]] over the alphabet {''X''<sub>1<~~/~~sub>,...,''X<sub>n<~~/~~sub>~~'~~'} (including the empty word, which is the unity of the free algebra). This ''R''-module becomes an [[algebra (ring theory)\|''R''-algebra]~~] ~~by defining a multiplication as follows: the product of two basis elements is the [[concatenation]] of the corresponding words:~~

	~~:<math>\left(X_{i_1}X_{i_2} \cdots X_{i_m}\right) \cdot \left(X_{j_1}X_{j_2} \cdots X_{j_n}\right) = X_{i_1}X_{i_2} \cdots X_{i_m}X_{j_1}X_{j_2} \cdots X_{j_n},</math>~~

	~~and the product of two arbitrary elements is thus uniquely determined (because the multiplication in an ''R''-algebra must be ''R''-bilinear). This ''R''-algebra~~ is ~~denoted ''R''⟨''X''<sub>1</sub>~~,~~...~~,~~''X<sub>n</sub>''⟩~~. ~~This construction~~ can ~~easily~~ be ~~generalized to an arbitrary set ''X'' of indeterminates~~.

	~~In short, for an arbitrary set <math>X=\{X_i\,;\; i\in I\}</math>, the~~ '~~''free ([[associative]], [[unital algebra\|unital]]) ''R''-[[algebra (ring theory)\|algebra]] on ''X''''' is~~
	~~:<math>R\langle X\rangle:=\bigoplus_{w\~~in ~~X^\ast}R w</math>~~
	~~with the ''R''-bilinear multiplication~~ that is concatenation on words, where ''X''* denotes the [[free monoid]] on ''X'' (i.e. words on the letters ''X''<sub>i</sub>), <math>\oplus</math> denotes the external [[Direct sum of modules\|direct sum]], and ''Rw'' denotes the [[free module\|free ''R''-module]] on 1 element, the word ''w''.

	~~For example, in ''R''⟨''X''<sub>1</sub>,''X''<sub>2</sub>,''X''<sub>3</sub>,''X''<sub>4</sub>⟩, for scalars ''α,β,γ,δ'' ∈''R'', a concrete example of a product of two elements~~ is ~~<math>(\alpha X_1X_2^2 + \beta X_2X_3)\cdot(\gamma X_2X_1 + \delta X_1^4X_4) = \alpha\gamma X_1X_2^3X_1 + \alpha\delta X_1X_2^2X_1^4X_4 + \beta\gamma X_2X_3X_2X_1 + \beta\delta X_2X_3X_1^4X_4</math>.~~

	~~The non-commutative polynomial ring may be identified with the [[monoid ring]] over ''R'' of the [[free monoid]]~~ of ~~all finite words in the ''X''<sub>''i''</sub>.~~

	~~==Contrast with Polynomials==~~
	~~Since the words over the alphabet {''X''<sub>1</sub>, ...,''X<sub>n</sub>''} form~~ a ~~basis of~~ '~~'R''⟨''X''<sub>1</sub>,..~~.,'~~'X<sub>n</sub>''⟩, it is clear that any element of ''R''⟨''X''<sub>1</sub>,~~ .~~..,''X<sub>n</sub>''⟩ can be uniquely written in the form~~:

	:~~<math>\sum\limits_{i_1,i_2, \cdots ,i_k\in\left\lbrace 1,2, \cdots ,n\right\rbrace} a_{i_1,i_2, \cdots ,i_k} X_{i_1} X_{i_2} \cdots X_{i_k},<~~/~~math>~~

	~~where <math>a_{i_1,i_2,...,i_k}<~~/~~math> are elements of ''R'' and all but finitely many of these elements are zero. This explains why the elements of ''R''⟨''X''<sub>1</sub>,...,''X<sub>n<~~/~~sub>''⟩ are often denoted as "non-commutative polynomials" in the "variables" (or "indeterminates") ''X''<sub>1<~~/~~sub>,.~~..~~,''X<sub>n<~~/~~sub>''; the elements <math> a_{i_1,i_2,...,i_k}<~~/~~math> are said to be "coefficients" of these polynomials, and the ''R''-algebra ''R''⟨''X''<sub>1</sub>,..~~.~~,''X<sub>n</sub>''⟩ is called the "non~~-~~commutative polynomial algebra over ''R'' in ''n'' indeterminates"~~. ~~Note that unlike in an actual polynomial ring, the variables do not [[commutative operation\|commute]~~]~~. For example ''X''<sub>1</sub>''X''<sub>2</sub> does not equal ''X''<sub>2</sub>''X''~~<~~sub~~>1<~~/sub~~>.

	More generally, one can construct the free algebra ''R''⟨''E''⟩ on any set ''E'' of [[generating set\|generators]]. Since rings may be regarded as '''Z'''-algebras, a '''free ring''' on ''E'' can be defined as the free algebra '''Z'''⟨''E''⟩.

	Over a [[field (mathematics)\|field]], the free algebra on ''n'' indeterminates can be constructed as the [[tensor algebra]] on an ''n''-dimensional [[vector space]]. For a more general coefficient ring, the same construction works if we take the [[free module]] on ''n'' [[generating set\|generators]].

	~~The construction of the free algebra on ''E'' is [[functor]]ial in nature and satisfies an appropriate [[universal property]]. The~~ free ~~algebra functor is [[left adjoint]]~~ to ~~the [[forgetful functor]] from the category of ''R''-algebras~~ to ~~the [[category of sets]].~~

	~~Free algebras over [[division ring]]s are [[free ideal ring]]s.~~

	~~==See also==~~

	*[[Cofree coalgebra]]
	*[[Tensor algebra]]
	*[[Free object]]
	*[[Noncommutative ring]]

	~~==References==~~
	* {{cite book \| last1=Berstel \| first1=Jean \| last2=Reutenauer \| first2=Christophe \| title=Noncommutative rational series with applications \| series=Encyclopedia of Mathematics and Its Applications \| volume=137 \| location=Cambridge \| publisher=[[Cambridge University Press]] \| year=2011 \| isbn=978-0-521-19022-0 \| zbl=1250.68007 }}
	* {{springer\|id=f/f041520\|author=L.A. Bokut'\|title=Free associative algebra}}

	~~[[Category~~:~~Algebras]]~~
	~~[[Category~~:~~Ring theory]]~~
	[[~~Category~~:~~Free algebraic structures]~~]

en>PrimeHunter: /* See also */ remove Centered triangular number, unclear why this should be singled out among centered polygonal numbers

2013-12-29T18:49:05Z

See also: remove Centered triangular number, unclear why this should be singled out among centered polygonal numbers

← Older revision		Revision as of 20:49, 29 December 2013
Line 1:		Line 1:
	~~Do you wish you could give~~ the ~~best massage? Do you know how to give good massages to help your loved ones relax? If your answer to these questions~~ is ~~yes, this article will~~ be ~~very helpful~~. ~~Here is more~~ on ~~incall massage london~~ (~~[http:~~/~~/Jawabanbagus~~.~~com~~/~~index.php?qa=6283&qa_1=great-tips-for-obtaining-~~the-~~maximum-enjoyment-from-massage try this web-site~~]) ~~look into~~ the ~~web site~~. ~~This article has gathered some of the most helpful information on massage therapy that will help you get started~~.<br><br>~~When you go to get a massage~~, ~~make sure you are open-minded about~~ the ~~whole process~~. ~~If you have never gotten~~ a ~~massage before, you might think some~~ of the ~~methods used are strange. Don't let this stop you from enjoying~~ the ~~massage. Just calm down~~ and ~~allow~~ the ~~masseuse to do what they do best~~.<br><br>~~As you progress in the massage~~, ~~be sure to use your thumbs, as well as your palms and fingers~~. ~~Your thumbs~~, ~~because they~~'~~re so strong,~~ can ~~also stimulate muscle therapy. Remember not~~ to ~~press too hard when massaging~~.<br><br>~~Different kinds of massage require different amounts of pressure. If~~ the ~~person you are providing a massage to has many knots~~, ~~then you should slowly add pressure~~ on ~~them~~ in ~~order to relieve their tension~~. ~~Maintaining consistent pressure will help to decrease tension~~. ~~This is one of~~ the ~~main principals of deep tissue massage.~~<br><br>~~If you~~'~~ve got tons of shoulder tension~~, ~~give~~ the ~~bear hug technique a try~~. ~~Put your arms around your body~~ in ~~the shape of an~~ X~~. Put one hand on each shoulder and rub. This can help give you a quick boost of energy and helps with anxiety.~~<br><br>~~Don~~'~~t take massages~~ for ~~granted. It can help reduce your pain level~~, ~~minimize your stress and rejuvenate you. It does not matter what your ailments are~~, ~~you should go to~~ a ~~professional massage therapist to experience it for yourself~~.<br><br>~~Use slow movements if you want your massage to be calm and soothing~~. ~~When applying pressure~~ with ~~your fingers~~, ~~provide support with your other fingers so that you avoid your thumbs from wearing out~~. ~~Use your weight too~~.<br><br>~~Don~~'~~t preoccupy yourself with worries about what~~ a ~~massage therapist will see at a massage appointment~~. ~~The less you wear~~, ~~the deeper the massage~~ is. ~~Massage therapists know how to skillfully drape sheets so that you stay covered~~. ~~So take a breath and relax~~, ~~and don~~'~~t worry about your physique~~ in ~~front of your masseuse.~~<br><br>~~Most hard-working athletes~~ are ~~aware~~ of ~~sports massages~~. This ~~is also good for whoever likes to exercise~~. ~~The purpose of sports massage is to condition~~ the ~~muscles to prevent injury rather than to relax and reduce stress~~.<br><br>~~A deep tissue type of massage is~~ the ~~best way~~ to ~~go for any type~~ of ~~injury~~. ~~The motions used in deep tissue massages cause friction that goes against the grain of the muscle~~. ~~This can relieve muscles which plague you often~~, ~~whether from posture issues or frequent injury.~~<br><br>~~You should offer a shower to someone whose body you oiled while giving them a massage. This helps remove~~ the ~~oil and also soothe their body~~. ~~It can also help their skin~~, ~~keeping pores from becoming clogged and forming blemishes~~.<br><br>~~You will really~~ be ~~able to impress your loved ones with~~ the ~~tips you just learned~~. ~~People go into this~~ field ~~for many reasons~~, ~~but being able to do this for~~ a ~~friend to improve his life makes it all worth while~~. ~~Use~~ the ~~tips you~~'~~ve learned until they become easy~~ to ~~implement. Teach~~ the ~~skills~~ to ~~your partner~~. ~~You~~'~~ll both be glad you did.~~		{{Otheruses4\|free algebras in ring theory\|the more general free algebras in [[universal algebra]]\|free object}}
			In [[mathematics]], especially in the area of [[abstract algebra]] known as [[ring theory]], a '''free algebra''' is the noncommutative analogue of a [[polynomial ring]] (which may be regarded as a '''free commutative algebra''').

			==Definition==
			For ''R'' a [[commutative ring]], the free ([[associative]], [[unital algebra\|unital]]) [[algebra (ring theory)\|algebra]] on ''n'' [[indeterminate_(variable)\|indeterminate]]s {''X''<sub>1</sub>,...,''X<sub>n</sub>''} is the [[free module\|free ''R''-module]] with a basis consisting of all [[Word_(mathematics)\|words]] over the alphabet {''X''<sub>1</sub>,...,''X<sub>n</sub>''} (including the empty word, which is the unity of the free algebra). This ''R''-module becomes an [[algebra (ring theory)\|''R''-algebra]] by defining a multiplication as follows: the product of two basis elements is the [[concatenation]] of the corresponding words:

			:<math>\left(X_{i_1}X_{i_2} \cdots X_{i_m}\right) \cdot \left(X_{j_1}X_{j_2} \cdots X_{j_n}\right) = X_{i_1}X_{i_2} \cdots X_{i_m}X_{j_1}X_{j_2} \cdots X_{j_n},</math>

			and the product of two arbitrary elements is thus uniquely determined (because the multiplication in an ''R''-algebra must be ''R''-bilinear). This ''R''-algebra is denoted ''R''⟨''X''<sub>1</sub>,...,''X<sub>n</sub>''⟩. This construction can easily be generalized to an arbitrary set ''X'' of indeterminates.

			In short, for an arbitrary set <math>X=\{X_i\,;\; i\in I\}</math>, the '''free ([[associative]], [[unital algebra\|unital]]) ''R''-[[algebra (ring theory)\|algebra]] on ''X''''' is
			:<math>R\langle X\rangle:=\bigoplus_{w\in X^\ast}R w</math>
			with the ''R''-bilinear multiplication that is concatenation on words, where ''X''* denotes the [[free monoid]] on ''X'' (i.e. words on the letters ''X''<sub>i</sub>), <math>\oplus</math> denotes the external [[Direct sum of modules\|direct sum]], and ''Rw'' denotes the [[free module\|free ''R''-module]] on 1 element, the word ''w''.

			For example, in ''R''⟨''X''<sub>1</sub>,''X''<sub>2</sub>,''X''<sub>3</sub>,''X''<sub>4</sub>⟩, for scalars ''α,β,γ,δ'' ∈''R'', a concrete example of a product of two elements is <math>(\alpha X_1X_2^2 + \beta X_2X_3)\cdot(\gamma X_2X_1 + \delta X_1^4X_4) = \alpha\gamma X_1X_2^3X_1 + \alpha\delta X_1X_2^2X_1^4X_4 + \beta\gamma X_2X_3X_2X_1 + \beta\delta X_2X_3X_1^4X_4</math>.

			The non-commutative polynomial ring may be identified with the [[monoid ring]] over ''R'' of the [[free monoid]] of all finite words in the ''X''<sub>''i''</sub>.

			==Contrast with Polynomials==
			Since the words over the alphabet {''X''<sub>1</sub>, ...,''X<sub>n</sub>''} form a basis of ''R''⟨''X''<sub>1</sub>,...,''X<sub>n</sub>''⟩, it is clear that any element of ''R''⟨''X''<sub>1</sub>, ...,''X<sub>n</sub>''⟩ can be uniquely written in the form:

			:<math>\sum\limits_{i_1,i_2, \cdots ,i_k\in\left\lbrace 1,2, \cdots ,n\right\rbrace} a_{i_1,i_2, \cdots ,i_k} X_{i_1} X_{i_2} \cdots X_{i_k},</math>

			where <math>a_{i_1,i_2,...,i_k}</math> are elements of ''R'' and all but finitely many of these elements are zero. This explains why the elements of ''R''⟨''X''<sub>1</sub>,...,''X<sub>n</sub>''⟩ are often denoted as "non-commutative polynomials" in the "variables" (or "indeterminates") ''X''<sub>1</sub>,...,''X<sub>n</sub>''; the elements <math> a_{i_1,i_2,...,i_k}</math> are said to be "coefficients" of these polynomials, and the ''R''-algebra ''R''⟨''X''<sub>1</sub>,...,''X<sub>n</sub>''⟩ is called the "non-commutative polynomial algebra over ''R'' in ''n'' indeterminates". Note that unlike in an actual polynomial ring, the variables do not [[commutative operation\|commute]]. For example ''X''<sub>1</sub>''X''<sub>2</sub> does not equal ''X''<sub>2</sub>''X''<sub>1</sub>.

			More generally, one can construct the free algebra ''R''⟨''E''⟩ on any set ''E'' of [[generating set\|generators]]. Since rings may be regarded as '''Z'''-algebras, a '''free ring''' on ''E'' can be defined as the free algebra '''Z'''⟨''E''⟩.

			Over a [[field (mathematics)\|field]], the free algebra on ''n'' indeterminates can be constructed as the [[tensor algebra]] on an ''n''-dimensional [[vector space]]. For a more general coefficient ring, the same construction works if we take the [[free module]] on ''n'' [[generating set\|generators]].

			The construction of the free algebra on ''E'' is [[functor]]ial in nature and satisfies an appropriate [[universal property]]. The free algebra functor is [[left adjoint]] to the [[forgetful functor]] from the category of ''R''-algebras to the [[category of sets]].

			Free algebras over [[division ring]]s are [[free ideal ring]]s.

			==See also==

			*[[Cofree coalgebra]]
			*[[Tensor algebra]]
			*[[Free object]]
			*[[Noncommutative ring]]

			==References==
			* {{cite book \| last1=Berstel \| first1=Jean \| last2=Reutenauer \| first2=Christophe \| title=Noncommutative rational series with applications \| series=Encyclopedia of Mathematics and Its Applications \| volume=137 \| location=Cambridge \| publisher=[[Cambridge University Press]] \| year=2011 \| isbn=978-0-521-19022-0 \| zbl=1250.68007 }}
			* {{springer\|id=f/f041520\|author=L.A. Bokut'\|title=Free associative algebra}}

			[[Category:Algebras]]
			[[Category:Ring theory]]
			[[Category:Free algebraic structures]]

en>Benmachine: Undid revision 456217640 by Bender235 (talk) - a Wiener sausage is actually a thing, but I suspect it's not what is meant here.

2012-08-31T17:42:57Z

Undid revision 456217640 by Bender235 (talk) - a Wiener sausage is actually a thing, but I suspect it's not what is meant here.

New page

Do you wish you could give the best massage? Do you know how to give good massages to help your loved ones relax? If your answer to these questions is yes, this article will be very helpful. Here is more on incall massage london ([http://Jawabanbagus.com/index.php?qa=6283&qa_1=great-tips-for-obtaining-the-maximum-enjoyment-from-massage try this web-site]) look into the web site. This article has gathered some of the most helpful information on massage therapy that will help you get started. When you go to get a massage, make sure you are open-minded about the whole process. If you have never gotten a massage before, you might think some of the methods used are strange. Don't let this stop you from enjoying the massage. Just calm down and allow the masseuse to do what they do best. As you progress in the massage, be sure to use your thumbs, as well as your palms and fingers. Your thumbs, because they're so strong, can also stimulate muscle therapy. Remember not to press too hard when massaging. Different kinds of massage require different amounts of pressure. If the person you are providing a massage to has many knots, then you should slowly add pressure on them in order to relieve their tension. Maintaining consistent pressure will help to decrease tension. This is one of the main principals of deep tissue massage. If you've got tons of shoulder tension, give the bear hug technique a try. Put your arms around your body in the shape of an X. Put one hand on each shoulder and rub. This can help give you a quick boost of energy and helps with anxiety. Don't take massages for granted. It can help reduce your pain level, minimize your stress and rejuvenate you. It does not matter what your ailments are, you should go to a professional massage therapist to experience it for yourself. Use slow movements if you want your massage to be calm and soothing. When applying pressure with your fingers, provide support with your other fingers so that you avoid your thumbs from wearing out. Use your weight too. Don't preoccupy yourself with worries about what a massage therapist will see at a massage appointment. The less you wear, the deeper the massage is. Massage therapists know how to skillfully drape sheets so that you stay covered. So take a breath and relax, and don't worry about your physique in front of your masseuse. Most hard-working athletes are aware of sports massages. This is also good for whoever likes to exercise. The purpose of sports massage is to condition the muscles to prevent injury rather than to relax and reduce stress. A deep tissue type of massage is the best way to go for any type of injury. The motions used in deep tissue massages cause friction that goes against the grain of the muscle. This can relieve muscles which plague you often, whether from posture issues or frequent injury. You should offer a shower to someone whose body you oiled while giving them a massage. This helps remove the oil and also soothe their body. It can also help their skin, keeping pores from becoming clogged and forming blemishes. You will really be able to impress your loved ones with the tips you just learned. People go into this field for many reasons, but being able to do this for a friend to improve his life makes it all worth while. Use the tips you've learned until they become easy to implement. Teach the skills to your partner. You'll both be glad you did.