Difference between revisions of "Template:AotD"

Revision as of 23:05, 13 December 2007

Rearrangement Inequality

The Rearrangement Inequality states that, if $A=\{a_1,a_2,\cdots,a_n\}$ is a permutation of a finite set (in fact, multiset) of real numbers and $B=\{b_1,b_2,\cdots,b_n\}$ is a permutation of another finite set of real numbers, the quantity $a_1b_1+a_2b_2+\cdots+a_nb_n$ is maximized when ${A}$ and ${B}$ are similarly sorted (that is, if $a_k$ is greater than or equal to exactly ${i}$ of the other members of $A$ , then ${b_k}$ is also greater than or equal to exactly ${i}$ of the other members of $B$ ). Conversely, $a_1b_1+a_2b_2+\cdots+a_nb_n$ is minimized when $A$ and $B$ are oppositely sorted (that is, if $a_k$ is less than or equal

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 <blockquote style="display:table;background:#eeeeee;padding:10px;" class="toccolours">
-===[[Trigonometric identities]]===
+===[[Rearrangement Inequality]]===
-'''Trigonometric identities''' are used to manipulate [[trigonometry]] [[equation]]s in certain ways.  Here is a list of them:
+The '''Rearrangement Inequality''' states that, if <math>A=\{a_1,a_2,\cdots,a_n\}</math> is a [[permutation]] of a [[finite]] [[set]] (in fact, [[multiset]]) of [[real number]]s and <math>B=\{b_1,b_2,\cdots,b_n\}</math> is a permutation of another finite set of real numbers, the quantity <math>a_1b_1+a_2b_2+\cdots+a_nb_n</math> is maximized when <math>{A}</math> and <math>{B} </math> are similarly sorted (that is, if <math>a_k</math> is greater than or equal to exactly <math>{i}</math> of the other members of <math>A</math>, then <math> {b_k} </math> is also greater than or equal to exactly <math>{i}</math> of the other members of <math>B</math>).  Conversely, <math>a_1b_1+a_2b_2+\cdots+a_nb_n</math> is minimized when <math>A</math> and <math>B</math> are oppositely sorted (that is, if <math>a_k</math> is less than or equal
-The six basic trigonometric functions can be defined using [[Trigonometric identities|[more]]]
 </blockquote>

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Difference between revisions of "Template:AotD"

Revision as of 23:05, 13 December 2007

Rearrangement Inequality