View source for fi'au

==== valsi ====
fi'au
==== type ====
experimental cmavo
==== creator ====
[[personal/krtisfranks|krtisfranks]]
==== time entered ====
Mon Apr  2 06:25:33 2018
 
== English ==
=== Definition #70585 - Preferred ===
 
==== definition ====

mekso operator: continued fraction, Kettenbruch notation; for ordered input  MATH
 $(X_1, X_2, X_3, X_4)$
 
<span class="MATH">(<i>X</i><sub>1</sub>, <i>X</i><sub>2</sub>, <i>X</i><sub>3</sub>, <i>X</i><sub>4</sub>)</span>, where: <span class="MATH"><i>X</i><sub>1</sub></span> is an ordered pair of functions and <span class="MATH"><i>X</i><sub>2</sub></span> is a free or dummy variable/input/index which ranges through set <span class="MATH"><i>X</i><sub>3</sub></span> in order(ing) <span class="MATH"><i>X</i><sub>4</sub></span>, the result is  MATH
 $K_{(X_2, (X_3, X_4))} (X_1(X_2))$
 
<span class="MATH"><i>K</i><sub>(X<sub>2</sub>,(X<sub>3</sub>, X<sub>4</sub>))</sub>(<i>X</i><sub>1</sub>(<i>X</i><sub>2</sub>))</span> for Kettenbruch notation <span class="MATH"><i>K</i></span>.

==== notes ====

<span class="MATH"><i>X</i><sub>1</sub></span> will be an ordered pair of functions defined on <span class="MATH"><i>X</i><sub>3</sub></span> and with input <span class="MATH"><i>X</i><sub>2</sub></span>. Let  MATH
 $X_1 = (a, b)$
 
<span class="MATH"><i>X</i><sub>1</sub> = (<i>a</i>, <i>b</i>)</span> where '<span class="MATH"><i>a</i></span>' and <span class="MATH"><i>b</i></span> are each functions (defined on X_3), <span class="MATH"><i>X</i><sub>3</sub></span> be the set of all nonnegative integers which are strictly less than some positive integer or positive infinity (denoted by "n+1") as determined by the context/problem, and <span class="MATH"><i>X</i><sub>4</sub></span> be the standard ordering on <span class="MATH"><i>X</i><sub>3</sub></span>; then the output will be: a(0) + b(0)/(a(1) + b(1)/(a(2) + b(2)/(.../(a(n-1) + b(n-1)/(a(n) + b(n)))))), where division has higher operator priority than addition and where, if n is positive infinity, then the expression never terminates and convergence of subfractions is necessary for well-definition; the inputs of '<span class="MATH"><i>a</i></span>' and <span class="MATH"><i>b</i></span> technically can be ordered tuples so long as they symbolically include <span class="MATH"><i>X</i><sub>2</sub></span> and all other parameters are specified.� In general,  MATH
 $a(min(X_3))$
 
<span class="MATH"><i>a</i>(<i>min</i>(<i>X</i><sub>3</sub>))</span>, for <span class="MATH"><i>min</i></span> being defined according to the ordering <span class="MATH"><i>X</i><sub>4</sub></span>, will be the formally 'integer'-part of the expression (even if it evaluates to a fraction) - in other words, it is the number which is entirely outside of the nested fractions. This operator is intended to be maximally general; some definition for <span class="MATH"><i>X</i><sub>3</sub></span> being the empty set can be devised, or '<span class="MATH"><i>a</i></span>' or <span class="MATH"><i>b</i></span> could be function-valued or otherwise exotic-valued functions (so long as their outputs have sum, division, and closure properties), etc. See also: "[[se'au|se'au]]". If diphthong "eu" is ever accepted into Lojban, this definition would preferably be moved to "[[fi'eu|fi'eu]]" (because the "eu" would also be etymologically tied to "Kettenbruch", and the "au" diphthong is more basic and should be reserved for higher-priority words).

==== gloss words ====

<ul><li>[[natlang/en/continued fraction|continued fraction ; operator/big operator K]]</li><li>[[natlang/en/Kettenbruch|Kettenbruch ; operator/big operator K]]</li></ul>

==== created by ====
[[personal/krtisfranks|krtisfranks]]
==== vote information ====
1
     
==== time ====
Mon Apr  2 06:27:46 2018

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=== Etymology ===