View source for cnansari

==== valsi ====
cnansari
==== type ====
fu'ivla
==== creator ====
[[personal/krtisfranks|krtisfranks]]
==== time entered ====
Sun Apr  1 05:01:25 2018
 
== English ==
=== Definition #70583 - Preferred ===
 
==== definition ====
x<sub>1</sub> is the mean-value theorem mean/forward-difference-quotient mean of the elements of (multi)set x<sub>2</sub> (1-element or 2-element set) under/for function x<sub>3</sub>.
==== notes ====

Let the sumti of <span class="MATH"><i>x</i><sub>3</sub></span> be a real-valued, univariate function <span class="MATH"><i>f</i></span> which is defined and continuous on at least the closed interval [ MATH
 $min(x_2), max(x_2)$
 
<span class="MATH"><i>min</i>(<i>x</i><sub>2</sub>), <i>max</i>(<i>x</i><sub>2</sub>)</span>] and differentiable on at least the open interval  MATH
 $(min(x_2), max(x_2))$
 
<span class="MATH">(<i>min</i>(<i>x</i><sub>2</sub>), <i>max</i>(<i>x</i><sub>2</sub>))</span>, such that the derivative <span class="MATH"><i>f'</i></span> is injective. Then  MATH
 $x_1 = (f')^<a class="undefined" href="../dict/(-1)?bg=1;langidarg=2">(-1)</a> ((f(max(x_2)) - f(min(x_2)))/(max(x_2)-min(x_2)))$
 
<span class="MATH"><i>x</i><sub>1</sub> = (<i>f'</i>)<sup>(-1)</sup>((<i>f</i> (<i>max</i>(<i>x</i><sub>2</sub>)) - <i>f</i> (<i>min</i>(<i>x</i><sub>2</sub>)))/(<i>max</i>(<i>x</i><sub>2</sub>) - <i>min</i>(<i>x</i><sub>2</sub>)))</span> if <span class="MATH"><i>x</i><sub>2</sub></span> has cardinality 2; if <span class="MATH"><i>x</i><sub>2</sub></span> has cardinality 1, then <span class="MATH"><i>x</i><sub>1</sub></span> is equal to the sole element of that set; other repairs may be necessary, depending on <span class="MATH"><i>f</i></span>: for example, <span class="MATH"><i>f</i> = <i>log</i></span> has <span class="MATH"><i>x</i><sub>1</sub> = 0</span> if 0 is an element of <span class="MATH"><i>x</i><sub>2</sub></span>. If <span class="MATH"><i>x</i><sub>3</sub></span> is filled with the monic, single-term polynomial of (integer) degree <span class="MATH"><i>p</i></span> such that <span class="MATH"><i>p</i> &gt; 1</span>, then <span class="MATH"><i>x</i><sub>1</sub></span> will be the principal (p-1)th root of the (1/p)-scaled weighted (p,1)-Lehmer mean ([[cnanlime|cnanlime]]) of the elements of <span class="MATH"><i>x</i><sub>2</sub></span> wherein the weight of the lesser of them is -1 (and the other weight is +1) unless <span class="MATH"><i>x</i><sub>2</sub></span> is a singleton (in which case the weights are both +1). Usually, <span class="MATH"><i>x</i><sub>1</sub></span> is the value which is guaranteed to exist for appropriate function <span class="MATH"><i>x</i><sub>2</sub></span> by the Mean-Value Theorem.

==== gloss words ====

<ul><li>[[natlang/en/forward-difference-quotient mean|forward-difference-quotient mean]]</li><li>[[natlang/en/logarithm mean|logarithm mean]]</li><li>[[natlang/en/log mean|log mean]]</li><li>[[natlang/en/MVT mean|MVT mean ; Mean-Value Theorem mean]]</li></ul>

==== created by ====
[[personal/krtisfranks|krtisfranks]]
==== vote information ====
1
     
==== time ====
Sun Apr  1 05:59:29 2018



<br/><font size="+1">Examples</font>
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=== Etymology ===