Euler class - Revision history

en>Enyokoyama: /* Relations to other invariants */ add link fundamental homology class -> fundamental class

2015-01-12T02:42:47Z

Relations to other invariants: add link fundamental homology class -> fundamental class

← Older revision		Revision as of 04:42, 12 January 2015
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	~~'''Quantum calculus'''~~, ~~sometimes called~~ '~~''calculus without limits''', is equivalent to traditional [[infinitesimal calculus]] without the notion of [[Limit of~~ a ~~function\|limits]]. It defines "q-calculus" and "h-calculus". h ostensibly stands for [[Planck~~'~~s constant]] while~~ '~~'q'' stands for quantum. The two parameters are related by the formula~~		I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. For a whilst I've been in Mississippi but now I'm considering other options. My day occupation is an info officer but I've already applied for another 1. To play lacross is the thing I adore most of all.<br><br>Take a look at my webpage: [http://myoceancounty.net/groups/apply-these-guidelines-when-gardening-and-grow/ free tarot readings]

	~~:<math>q = e^{i h} = e^{2 \pi i \hbar} \,</math>~~

	~~where <math>\scriptstyle \hbar = \frac{h}{2 \pi} \,</math> is the [[reduced Planck constant]]~~.

	~~==Differentiation==~~

	~~In the q-calculus and h-calculus, [[differential of a function\|differentials]] of functions are defined as~~

	~~:<math>d_q(f(x)) = f(qx) - f(x) \,</math>~~

	~~and~~

	~~:<math>d_h(f(x)) = f(x + h) - f(x) \,</math>~~

	~~respectively. [[Derivative]]s of functions are then defined as fractions by the [[q-derivative]]~~

	~~:<math>D_q(f(x)) = \frac{d_q(f(x))}{d_q(x)} = \frac{f(qx) - f(x)}{(q - 1)x}</math>~~

	~~and by~~

	~~:<math>D_h(f(x)) = \frac{d_h(f(x))}{d_h(x)} = \frac{f(x + h) - f(x)}{h}</math>~~

	~~In the [[Limit of a function\|limit]], as h goes to 0, or equivalently as q goes to 1, these expressions take on the form of the derivative of classical calculus.~~

	~~==Integration==~~

	~~===q-integral===~~
	~~A function F(x) is a q-antiderivative of f(x) if D<sub>q</sub>F(x)=f(x). The q-antiderivative (or q-integral)~~ is ~~denoted by <math>\int f(x)d_qx</math> and~~ an ~~expression~~ for ~~F(x) can be found from the formula~~
	~~<math>\int f(x)d_qx = (~~1~~-q)\sum_{j=0}^\infty xq^jf(xq^j)</math> which~~ is ~~called~~ the ~~[[Jackson integral]]~~ of ~~f(x)~~. ~~For 0 < q < 1, the series converges to a function F(x) on an interval (0,A] if \|f(x)x^α\| is bounded on the interval (0,A] for some 0 <= α~~ < 1.

	~~The q-integral is a [[Riemann-Stieltjes integral]] with respect to a [[step function]] having infinitely many points of increase at the points q<sup~~>j<~~/sup~~>~~, with the jump~~ at ~~the point q<sup>j</sup> being q<sup>j</sup>. If we call this step function g<sub>q</sub>(t) then dg<sub>q</sub>(t) = d<sub>q</sub>t.<ref>~~[http://~~www.mat.uc~~.~~pt/preprints~~/ps/~~p0432.pdf FUNCTIONS q~~-~~ORTHOGONAL WITH RESPECT TO THEIR OWN ZEROS], LUIS DANIEL ABREU, Pre-Publicacoes do Departamento de Matematica Universidade de Coimbra, Preprint Number 04–32</ref>~~

	~~===h~~-~~integral===~~
	~~A function F(x) is an h~~-~~antiderivative of f(x) if D<sub>h</sub>F(x)=f(x). The h~~-~~antiderivative (or h~~-~~integral) is denoted by <math>\int f(x)d_hx</math>. If a~~ and ~~b differ by an integer multiple of h then the definite integral<math>\int_a^b f(x)d_hx</math> is given by a [[Riemann sum]] of f(x) on the interval [a,b] partitioned into subintervals of width h.~~

	~~== Example ==~~
	~~The derivative of the function <math>x^n</math> (for some positive integer <math>n</math>) in the classical calculus is <math>nx^{n~~-~~1}<~~/~~math>. The corresponding expressions in q-calculus and h-calculus are~~

	~~:<math>D_q(x^n) = \frac{q^n - 1}{q - 1} x^{n - 1} = [n]_q\ x^{n - 1}</math>~~

	~~with the [[q-bracket]]~~

	~~:<math>[n]_q = \frac{q^n - 1}{q - 1}</math>~~

	~~and~~

	~~:<math>D_h(x^n) = x^{n - 1} + h x^{n - 2} + \cdots + h^{n - 1}</math>~~

	~~respectively. The expression <math>[n]_q x^{n - 1}</math> is then the q-calculus analogue of the simple power rule for~~
	~~positive integral powers. In this sense, the function <math>x^n</math> is still ''nice'' in the q-calculus, but rather~~
	~~ugly in the h-calculus – the h-calculus analog of <math>x^n</math> is instead the [[falling factorial]], <math>(x)_n := x(x-1)\cdots(x-n+1).</math>~~
	One may proceed further and develop, for example, equivalent notions of [[Q-derivative\|Taylor expansion]], et cetera, and even arrive at q-calculus analogues for all of the usual functions one would want to have, such as an analogue for the [[sine]] function whose q-derivative is the appropriate analogue for the [[cosine]].

	~~== History ==~~
	The h-calculus is just the [[calculus of finite differences]], which had been studied by [[George Boole]] and others, and has proven useful in a number of fields, among them [[combinatorics]] and [[fluid mechanics]]. The q-calculus, while dating in a sense back to [[Leonhard Euler]] and [[Carl Gustav Jacobi]], is only recently beginning to see more usefulness in [[quantum mechanics]], having an intimate connection with commutativity relations and [[Lie algebra]].

	~~== See also ==~~
	* [[Noncommutative geometry]]
	* [[Quantum differential calculus]]
	* [[Time scale calculus]]
	* [[q-analog]]

	~~==References==~~
	~~{{reflist}}~~
	~~<!-- this section is for references, sources used to write a part of the article or cited in the article to justify a statement. Supplementary reading should go into "further reading" -->~~
	* F. H. Jackson (1908), "On q-functions and a certain difference operator", ''Trans. Roy. Soc. Edin.'', '''46''' 253-281.

	* Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914 , ISBN 0470274530 , ISBN 978-0470274538

	*[[Victor Kac]], [[Pokman Cheung]], Quantum calculus'', Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8

	~~[[Category:Quantum mechanics\| ]]~~
	~~[[Category:Differential calculus\| ]~~]

en>Bvarxj: /* Relations to other invariants */ corrected misprint

2013-12-25T18:15:14Z

Relations to other invariants: corrected misprint

← Older revision		Revision as of 20:15, 25 December 2013
Line 1:		Line 1:
	~~Alyson~~ is ~~what my husband loves~~ to ~~contact me but I don~~'~~t like when people use my full title~~. ~~To climb~~ is ~~something she would never give up~~. ~~I've usually cherished living in Mississippi~~. ~~She~~ functions as a ~~travel agent but quickly she'll be~~ on ~~her own~~.<br><br>~~Feel free~~ to ~~surf~~ to ~~my webpage;~~ [http://~~help~~.~~ksu~~.~~edu~~.sa/~~node~~/~~65129 psychic phone~~]		'''Quantum calculus''', sometimes called '''calculus without limits''', is equivalent to traditional [[infinitesimal calculus]] without the notion of [[Limit of a function\|limits]]. It defines "q-calculus" and "h-calculus". h ostensibly stands for [[Planck's constant]] while ''q'' stands for quantum. The two parameters are related by the formula

			:<math>q = e^{i h} = e^{2 \pi i \hbar} \,</math>

			where <math>\scriptstyle \hbar = \frac{h}{2 \pi} \,</math> is the [[reduced Planck constant]].

			==Differentiation==

			In the q-calculus and h-calculus, [[differential of a function\|differentials]] of functions are defined as

			:<math>d_q(f(x)) = f(qx) - f(x) \,</math>

			and

			:<math>d_h(f(x)) = f(x + h) - f(x) \,</math>

			respectively. [[Derivative]]s of functions are then defined as fractions by the [[q-derivative]]

			:<math>D_q(f(x)) = \frac{d_q(f(x))}{d_q(x)} = \frac{f(qx) - f(x)}{(q - 1)x}</math>

			and by

			:<math>D_h(f(x)) = \frac{d_h(f(x))}{d_h(x)} = \frac{f(x + h) - f(x)}{h}</math>

			In the [[Limit of a function\|limit]], as h goes to 0, or equivalently as q goes to 1, these expressions take on the form of the derivative of classical calculus.

			==Integration==

			===q-integral===
			A function F(x) is a q-antiderivative of f(x) if D<sub>q</sub>F(x)=f(x). The q-antiderivative (or q-integral) is denoted by <math>\int f(x)d_qx</math> and an expression for F(x) can be found from the formula
			<math>\int f(x)d_qx = (1-q)\sum_{j=0}^\infty xq^jf(xq^j)</math> which is called the [[Jackson integral]] of f(x). For 0 < q < 1, the series converges to a function F(x) on an interval (0,A] if \|f(x)x^α\| is bounded on the interval (0,A] for some 0 <= α < 1.

			The q-integral is a [[Riemann-Stieltjes integral]] with respect to a [[step function]] having infinitely many points of increase at the points q<sup>j</sup>, with the jump at the point q<sup>j</sup> being q<sup>j</sup>. If we call this step function g<sub>q</sub>(t) then dg<sub>q</sub>(t) = d<sub>q</sub>t.<ref>[http://www.mat.uc.pt/preprints/ps/p0432.pdf FUNCTIONS q-ORTHOGONAL WITH RESPECT TO THEIR OWN ZEROS], LUIS DANIEL ABREU, Pre-Publicacoes do Departamento de Matematica Universidade de Coimbra, Preprint Number 04–32</ref>

			===h-integral===
			A function F(x) is an h-antiderivative of f(x) if D<sub>h</sub>F(x)=f(x). The h-antiderivative (or h-integral) is denoted by <math>\int f(x)d_hx</math>. If a and b differ by an integer multiple of h then the definite integral<math>\int_a^b f(x)d_hx</math> is given by a [[Riemann sum]] of f(x) on the interval [a,b] partitioned into subintervals of width h.

			== Example ==
			The derivative of the function <math>x^n</math> (for some positive integer <math>n</math>) in the classical calculus is <math>nx^{n-1}</math>. The corresponding expressions in q-calculus and h-calculus are

			:<math>D_q(x^n) = \frac{q^n - 1}{q - 1} x^{n - 1} = [n]_q\ x^{n - 1}</math>

			with the [[q-bracket]]

			:<math>[n]_q = \frac{q^n - 1}{q - 1}</math>

			and

			:<math>D_h(x^n) = x^{n - 1} + h x^{n - 2} + \cdots + h^{n - 1}</math>

			respectively. The expression <math>[n]_q x^{n - 1}</math> is then the q-calculus analogue of the simple power rule for
			positive integral powers. In this sense, the function <math>x^n</math> is still ''nice'' in the q-calculus, but rather
			ugly in the h-calculus – the h-calculus analog of <math>x^n</math> is instead the [[falling factorial]], <math>(x)_n := x(x-1)\cdots(x-n+1).</math>
			One may proceed further and develop, for example, equivalent notions of [[Q-derivative\|Taylor expansion]], et cetera, and even arrive at q-calculus analogues for all of the usual functions one would want to have, such as an analogue for the [[sine]] function whose q-derivative is the appropriate analogue for the [[cosine]].

			== History ==
			The h-calculus is just the [[calculus of finite differences]], which had been studied by [[George Boole]] and others, and has proven useful in a number of fields, among them [[combinatorics]] and [[fluid mechanics]]. The q-calculus, while dating in a sense back to [[Leonhard Euler]] and [[Carl Gustav Jacobi]], is only recently beginning to see more usefulness in [[quantum mechanics]], having an intimate connection with commutativity relations and [[Lie algebra]].

			== See also ==
			* [[Noncommutative geometry]]
			* [[Quantum differential calculus]]
			* [[Time scale calculus]]
			* [[q-analog]]

			==References==
			{{reflist}}
			<!-- this section is for references, sources used to write a part of the article or cited in the article to justify a statement. Supplementary reading should go into "further reading" -->
			* F. H. Jackson (1908), "On q-functions and a certain difference operator", ''Trans. Roy. Soc. Edin.'', '''46''' 253-281.

			* Exton, H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, 1983, ISBN 0853124914 , ISBN 0470274530 , ISBN 978-0470274538

			*[[Victor Kac]], [[Pokman Cheung]], Quantum calculus'', Universitext, Springer-Verlag, 2002. ISBN 0-387-95341-8

			[[Category:Quantum mechanics\| ]]
			[[Category:Differential calculus\| ]]

en>AnomieBOT: Dating maintenance tags: {{Why}}

2012-08-17T22:57:24Z

Dating maintenance tags: {{Why}}

New page

Alyson is what my husband loves to contact me but I don't like when people use my full title. To climb is something she would never give up. I've usually cherished living in Mississippi. She functions as a travel agent but quickly she'll be on her own.<br><br>Feel free to surf to my webpage; [http://help.ksu.edu.sa/node/65129 psychic phone]