Pyramid (geometry) - Revision history

en>Tomruen: 4-polytope

2014-12-31T20:49:25Z

4-polytope

← Older revision		Revision as of 22:49, 31 December 2014
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	~~They contact me Emilia~~. ~~For~~ a ~~whilst~~ she's ~~been in South Dakota~~. One of the extremely ~~very~~ best ~~things~~ in the ~~world~~ for me is to do aerobics and I'~~ve been doing~~ it ~~for quite a while. For many years he's been working as a receptionist~~.<br><br>my ~~weblog:~~ [http://www.~~animecontent~~.~~com~~/~~user~~/~~D2456 www.animecontent.com~~]		Greetings! I am Myrtle Shroyer. Minnesota is exactly where he's been living for many years. She is a librarian but she's usually needed her personal company. One of the extremely best issues in the globe for me is to do aerobics and now I'm attempting to make cash with it.<br><br>Here is my website - over the counter std test ([http://www.smylestream.org/groups/solid-advice-in-relation-to-candida/ Going In this article])

en>MrFawwaz: Is it allowed to redirect ?

2014-02-25T05:40:54Z

Is it allowed to redirect ?

← Older revision		Revision as of 07:40, 25 February 2014
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	~~[[Image:MathematicalMeans~~.~~svg\|thumb\|right\|A geometric construction of the Quadratic mean and the Pythagorean means (of two numbers ''~~a'~~' and ''b''). Harmonic mean denoted by ''H'', Geometric by ''G'', Arithmetic by ''A'' and Quadratic mean (also known as [[Root mean square]]) denoted by ''Q''.]]~~		They contact me Emilia. For a whilst she's been in South Dakota. One of the extremely very best things in the world for me is to do aerobics and I've been doing it for quite a while. For many years he's been working as a receptionist.<br><br>my weblog: [http://www.animecontent.com/user/D2456 www.animecontent.com]
	~~[[Image:Comparison_Pythagorean_means~~.~~svg\|thumb\|right\|Comparison~~ of the ~~arithmetic, geometric and harmonic means of a pair of numbers. The vertical dashed lines are [[asymptote]]s for the harmonic means.]]~~
	~~In mathematics, the three classical '''Pythagorean means''' are the [[arithmetic mean]] (''A''), the [[geometric mean]] (''G''), and the [[harmonic mean]] (''H''). They are defined by:~~

	* <math> A(x_1, \ldots, x_n) = \frac{1}{n}(x_1 + \cdots + x_n) </math>
	* <math> G(x_1, \ldots, x_n) = \sqrt[n]{x_1 \cdots x_n} </math>
	* <math> H(x_1, \ldots, x_n) = \frac{n}{\frac{1}{x_1} + \cdots + \frac{1}{x_n}} </math>

	~~Each mean has~~ the ~~following properties:~~
	* Value preservation: <math> M(x,x, \ldots,x) = x </math>
	* First order [[homogeneous function\|homogeneity]]: <math> M(bx_1, \ldots, bx_n) = b M(x_1, \ldots, x_n) </math>
	* Invariance under exchange: <math> M(\ldots, x_i, \ldots, x_j, \ldots ) = M(\ldots, x_j, \ldots, x_i, \ldots) </math> for ~~any <math>i</math> and <math>j</math>.~~
	* Averaging: <math> \min(x_1,\ldots,x_n) \leq M(x_1,\ldots,x_n) \leq \max(x_1,\ldots,x_n)</math>

	These means were studied with proportions by [[Pythagoreans]] and later generations of Greek mathematicians (Thomas Heath, History of Ancient Greek Mathematics) because of their importance in geometry and music.

	~~There~~ is ~~an ordering~~ to ~~these means (if all of the <math> x_i </math> are positive), along with the [[quadratic mean]] <math>Q=\sqrt{\frac{x_1^2+x_2^2+ \cdots + x_n^2}{n}}</math>:~~

	~~: <math> \min \leq H \leq G \leq A \leq Q \leq \max </math>~~

	~~with equality holding if and only if the <math> x_i </math> are all equal. This is a generalization of the [[inequality of arithmetic and geometric means]]~~ and a ~~special case of an inequality for [[generalized mean]]s~~. ~~This inequality sequence can be proved for the <math>n=2</math> case for the numbers '~~'a'' and ''b'' using a sequence of [[right triangle]]s (''x'', ''y'', ''z'') with [[hypotenuse]] ''z'' and the [[Pythagorean theorem]], which states that <math>x^2 + y^2 = z^2</math> and implies that <math>z > x</math> and <math>z > y</math>. The right triangles are<ref>Kung, Sidney H., "The Harmonic mean—geometric mean—arithmetic mean—root mean square inequality II," in Roger B. Nelsen, ''Proofs Without Words'', [[The Mathematical Association of America]], 1993, p. 54.<~~/ref>~~

	~~:<math>\left(\frac{b-a}{b+a}\sqrt{ab}, \frac{2ab}{a+b}, \sqrt{ab}\right) = \left(\frac{b-a}{b+a}\sqrt{ab}, H(a,b), G(a,b)\right),</math>~~

	~~showing that <math>H(a,b) < G(a,b)</math>;~~

	~~:<math>\left(\frac{b-a}{2}, \sqrt{ab}, \frac{a+b}{2}\right) = \left(\frac{b-a}{2}, G(a,b), A(a,b)\right),</math~~>

	~~showing that~~ <~~math~~>~~G(a,b) < A(a,b)</math>;~~

	~~and~~

	:~~<math>\left(\frac{b-a}{2}, \frac{a+b}{2}, \sqrt{\frac{a^2+b^2}{2}}\,\right) = \left(\frac{b-a}{2},A(a,b), Q(a,b)\right),</math>~~

	~~showing that <math>A(a,b) < Q(a,b)</math>.~~

	~~==See also==~~
	* [[Arithmetic-geometric mean]]
	* [[Average]]
	* [[Generalized mean]]

	~~==References==~~

	~~{{reflist}}~~

	~~==External links==~~
	*{{MathWorld\|urlname=PythagoreanMeans\|title=Pythagorean Means\|author=Cantrell, David W.}}
	* Nice comparison of Pythagorean means with emphasis on the [http://www.~~cse~~.~~unsw.edu.au~~/~~~teachadmin~~/~~info/harmonic3~~.~~html harmonic mean]~~.

	~~[[Category:Means]~~]

en>Tomruen: /* Polyhedral pyramid */

2014-01-25T05:06:43Z

Polyhedral pyramid

← Older revision		Revision as of 07:06, 25 January 2014
Line 1:		Line 1:
	~~The author is recognized~~ by the ~~title~~ of ~~Figures Lint~~. ~~Puerto Rico is where std testing at home he'~~s ~~usually home std test been residing but she needs to transfer because of her family members~~. ~~His spouse doesn~~'~~t like it~~ the ~~way he does but what he truly likes performing is to do aerobics~~ and he'~~s been performing it for fairly a while~~. ~~[http~~://~~www.hotporn123.com~~/~~blog~~/~~154316 std testing at home~~] ~~He used to [http~~://~~www~~.~~blaze16~~.~~com~~/~~blog/255682 over~~ the ~~counter std test~~] ~~be [http~~://~~Www~~.~~Mayoclinic~~.~~org~~/~~diseases-conditions/sexually-transmitted-diseases-stds/in-depth/std-testing/art-20046019 unemployed~~] ~~but now~~ ~~over~~ the [~~http:~~//~~Www~~.~~Rensco~~.~~com/publichealth_ff_trichomoniasis~~.~~asp counter std~~] ~~test he is~~ a ~~meter reader.~~<br><br>~~my web page~~ :: ~~at home std test~~ ([http://www.~~revleft~~.~~com~~/vb/~~member~~.~~php?u=160656 click this over here now~~])		[[Image:MathematicalMeans.svg\|thumb\|right\|A geometric construction of the Quadratic mean and the Pythagorean means (of two numbers ''a'' and ''b''). Harmonic mean denoted by ''H'', Geometric by ''G'', Arithmetic by ''A'' and Quadratic mean (also known as [[Root mean square]]) denoted by ''Q''.]]
			[[Image:Comparison_Pythagorean_means.svg\|thumb\|right\|Comparison of the arithmetic, geometric and harmonic means of a pair of numbers. The vertical dashed lines are [[asymptote]]s for the harmonic means.]]
			In mathematics, the three classical '''Pythagorean means''' are the [[arithmetic mean]] (''A''), the [[geometric mean]] (''G''), and the [[harmonic mean]] (''H''). They are defined by:

			* <math> A(x_1, \ldots, x_n) = \frac{1}{n}(x_1 + \cdots + x_n) </math>
			* <math> G(x_1, \ldots, x_n) = \sqrt[n]{x_1 \cdots x_n} </math>
			* <math> H(x_1, \ldots, x_n) = \frac{n}{\frac{1}{x_1} + \cdots + \frac{1}{x_n}} </math>

			Each mean has the following properties:
			* Value preservation: <math> M(x,x, \ldots,x) = x </math>
			* First order [[homogeneous function\|homogeneity]]: <math> M(bx_1, \ldots, bx_n) = b M(x_1, \ldots, x_n) </math>
			* Invariance under exchange: <math> M(\ldots, x_i, \ldots, x_j, \ldots ) = M(\ldots, x_j, \ldots, x_i, \ldots) </math> for any <math>i</math> and <math>j</math>.
			* Averaging: <math> \min(x_1,\ldots,x_n) \leq M(x_1,\ldots,x_n) \leq \max(x_1,\ldots,x_n)</math>

			These means were studied with proportions by [[Pythagoreans]] and later generations of Greek mathematicians (Thomas Heath, History of Ancient Greek Mathematics) because of their importance in geometry and music.

			There is an ordering to these means (if all of the <math> x_i </math> are positive), along with the [[quadratic mean]] <math>Q=\sqrt{\frac{x_1^2+x_2^2+ \cdots + x_n^2}{n}}</math>:

			: <math> \min \leq H \leq G \leq A \leq Q \leq \max </math>

			with equality holding if and only if the <math> x_i </math> are all equal. This is a generalization of the [[inequality of arithmetic and geometric means]] and a special case of an inequality for [[generalized mean]]s. This inequality sequence can be proved for the <math>n=2</math> case for the numbers ''a'' and ''b'' using a sequence of [[right triangle]]s (''x'', ''y'', ''z'') with [[hypotenuse]] ''z'' and the [[Pythagorean theorem]], which states that <math>x^2 + y^2 = z^2</math> and implies that <math>z > x</math> and <math>z > y</math>. The right triangles are<ref>Kung, Sidney H., "The Harmonic mean—geometric mean—arithmetic mean—root mean square inequality II," in Roger B. Nelsen, ''Proofs Without Words'', [[The Mathematical Association of America]], 1993, p. 54.</ref>

			:<math>\left(\frac{b-a}{b+a}\sqrt{ab}, \frac{2ab}{a+b}, \sqrt{ab}\right) = \left(\frac{b-a}{b+a}\sqrt{ab}, H(a,b), G(a,b)\right),</math>

			showing that <math>H(a,b) < G(a,b)</math>;

			:<math>\left(\frac{b-a}{2}, \sqrt{ab}, \frac{a+b}{2}\right) = \left(\frac{b-a}{2}, G(a,b), A(a,b)\right),</math>

			showing that <math>G(a,b) < A(a,b)</math>;

			and

			:<math>\left(\frac{b-a}{2}, \frac{a+b}{2}, \sqrt{\frac{a^2+b^2}{2}}\,\right) = \left(\frac{b-a}{2},A(a,b), Q(a,b)\right),</math>

			showing that <math>A(a,b) < Q(a,b)</math>.

			==See also==
			* [[Arithmetic-geometric mean]]
			* [[Average]]
			* [[Generalized mean]]

			==References==

			{{reflist}}

			==External links==
			*{{MathWorld\|urlname=PythagoreanMeans\|title=Pythagorean Means\|author=Cantrell, David W.}}
			* Nice comparison of Pythagorean means with emphasis on the [http://www.cse.unsw.edu.au/~teachadmin/info/harmonic3.html harmonic mean].

			[[Category:Means]]

en>Magioladitis: Removed empty comments and/or general fixes, removed: using AWB (8252)

2012-08-14T10:09:15Z

Removed empty comments and/or general fixes, removed:  using AWB (8252)

New page

The author is recognized by the title of Figures Lint. Puerto Rico is where std testing at home he's usually home std test been residing but she needs to transfer because of her family members. His spouse doesn't like it the way he does but what he truly likes performing is to do aerobics and he's been performing it for fairly a while. [http://www.hotporn123.com/blog/154316 std testing at home] He used to [http://www.blaze16.com/blog/255682 over the counter std test] be [http://Www.Mayoclinic.org/diseases-conditions/sexually-transmitted-diseases-stds/in-depth/std-testing/art-20046019 unemployed] but now over the [http://Www.Rensco.com/publichealth_ff_trichomoniasis.asp counter std] test he is a meter reader.<br><br>my web page :: at home std test ([http://www.revleft.com/vb/member.php?u=160656 click this over here now])