"Add"-operator "Mul"-operator "Pow"-operator. def. h b. def

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1 Opertors A sort review of opertors. Te isussions out tetrtion le me to two impressions. ) It my e etter to see opertors using prmeters, inste of two, s it is ommon use upte 4 ) Sering for noter onsistent onept for ontinuous frtionl opertions, it seeme to require, tt someow te se-prmeter for tetrtion soul e tougt s "imprinte" in te opertor, wi, wit tis "imprint", will ten e pplie wit frtionl itertion-ontrol. So tt essentilly we o not work wit funtion of te se-prmeter only, ut wit funtion of te opertor in onnetion wit te se-prmeter. In oter wors: neiter te seprmeter lone nor te strt opertor itself n e frtione witout respet to te oter - t lest is it so wit tetrtion. Here I trie to re-generlize tis iffiult to unerstn ie lso to te ommon opertors ition n multiplition. Te ommon exponentition omes out to not to require speil opertor t ll, so I isuss in ft only tree opertors n teir itertions, inste of four. Anoter very nie property of tis opertor-onept is finlly, tt it n one-to-one e trnslte to mtrix-opertions, were te frtionl itertes re ten expresse y te frtionl powers of mtries, wi re typil for speifi opertor n moifie y te se prmeter. See more out tis t te ppenix. Bsi inition we ve numers n vrile-nmes, opertor-symols y inition. Lter we my ine /inlue te usul funtion nmes. Te opertors ve prmeters: strt-opern in exmple ""), se-opern in exmple ""), itertor-opern in exmple ""). For onveniene we pt some very si nottions: ) te unry /- sign for numers n vriles, ) te inry "/-" - sign for numers n vriles ) te strt-opern wit "A"-opertor my e omitte, if zero 4) te strt-opern wit "Mul"-opertor my e omitte, if one "A"-opertor "Mul"-opertor "Pow"-opertor ) * * * * ) Allowing te unry minus we ve )?? * * * ) Te ierry of opertors ours most smootly, if we reurse n expression into te itertoropern, see elow.

2 Te si ie for eveloping n expression is: * we egin wit te strt-opern s te initil intermeite expression, * ten te opertor pplies te se-opern wit its speifi opertion * s mny to te intermeite expression s te itertor-opern ittes. Te opertions n e ontente wen one or ll of te tree operns re reple y new instne of n expression. Most interesting ere is for te eginning te ontention of te sme type of opertion. I in't onsier opertor-preeenes in etils. In first glne it seems, tt y te onstrution tings re utomtilly in inite orer, ut, for instne, I in't tink out ontention of opertors of ifferent type yet. Some very si remrks: "Strt" n "se" n e internge, if itertor "Strt" n "se" n e internge, if itertor -not possile- "itertor" n "se" n e exnge If <>: not possile If <>: not possile Tus lso "top" n "own"- itertion n e exnge

3 Horizontl itertion Synttilly susequent expression uses te urrent expression s its own strt-opern. Te orer of evlution is priniplly from te most elementry position ) ) * ) ) Some primitive forms of te expression reurse re expressile in te iger opertion, ut tis nnot ine te full rnge for te iger opertors, so tis re not te initions for te ierry:

4 Left-own-itertion Repling te se-opern y new expression. Tere is urrently no nottion for n opertor of tis type of itertion. Note, tt in effet we rete te ylotomi polynomil y tis opertion, wen pplie to te ""-opertor or in te exponent, wen pplie to te "Mul"-opertor). ) ) )?? Reursions wit primitive expressions * sine se- n itertion-prmeter re exngle, tis is lso vli for left-up-itertion??

5 Left-up-iterting repling te itertor-opern: tis re te rtionles for te "ierry-of-opertor"-initions Te initions in tis tle re not in use: * )? * oo, lim <? Primitive forms wit strt- or en?) point in te reursion serve s initions for te opertorierry in te most onsistent wy: *

6 Reltion to mtrix-opertors Te opertors re one-to-one expressile s mtrix-formule ting on forml powerseries, n te expressions re ll extensile to ontinuous itertion. Te mtries ontin te oeffiients for te powerseries, wi re evlute wit te prmeter-vetor Vx) oring to te mtrixmultiplition-rules. Te itertor-opern ours s exponent of te opertor-mtrix; n sine tese mtries ve eiter essile eigensystems or mtrix-logritms, we n use ny omplex vlue for te exponent/itertoropern. "A" "Mul" "Pow" V)~ * P~ V)~ V)~* V)V*)~ V)~ * B V ) ~ V)~ * P~ V*)~ V)~* V) V* )~ V)~ * B V{,}^^) ~ V)~ * P~ ) V*)~ * * {, }^ ^ Here te Vx) terms re tougt s olumn-vetors olvetorx,x,x,), wi implements te prmeter of te forml powerseries-expression wen expne from te pplie mtrixmultiplition. Te ~-symol mens "trnspose". A tiny -prefix elres tis s igonl-mtrix. "A": P is te lower tringulr mtrix of inomil-oeffiients or "Psl"-mtrix). Te eigensystem of P is egenerte; ut it s n exeptionl simple mtrix-logritm, y wi ten generl power n e esily ompute wen just multiplie wit te -prmeter. "Mul" is espeilly simple, sine te opertor is simply igonl-mtrix itself n generl powers of igonl-mtrix re ine y just pplying te powers to its slr igonl-elements. "Pow" uses te B -mtrix, s ine in my postings n rtiles I usully enote it s B s -mtrix wit te prmeter s). For te prmeter tere is onventionlly te rnge e -e < < e /e, n for tis rnge non-egenerte eigen-eomposition oul e sown to e vli. Te extension for to te generl omplex omin is ssume to e possile, ut not yet fully estlise. However, te eigensystem-eomposition exiits te reltion to te "fixpoint"-onept. Assume te eigensystem-eomposition B W D W - or W - B D W - Now ssume t lest) one eigenvlue k D[k,k] orere to te topmost position in D, so k Ten using te first row of W - only we ve W - [,] * B * W - [,] n te first row in W - reflets te "fixpoint"-onept, sine te rowvetor W - [,] is invrint uner trnsformtion y B. Te oter rows of W - my e lle "pseuo"-fixpoints, sine tey re only slr multiples uner tis trnsformtion oring to te slr sling ftor k.wi is te k't eigenvlue). For te infinite imensionl se we ve tus n infinite set of pseuo)-fixpoints rowvetors of oeffiients for forml powerseries) for te speifi opertor uner onsiertion n tis seems ten to e suffiient to uniquely ine te mtemtil rter of su mtrixexpressile opertors.

7 Inverse opertions Tere re two ovious inverses of te "pow"- itertion. Given onstnt z, we my sk eiter for te top-left vlue, given lso te se, or we my sk for te ottom-left-vlue, given. ) How mny possily frtionl) o I ve to pply te opertor wit se t te strting vlue until I re z? x z ) Wi se-opertor, - repetely pplie strte t, les to my given vlue z? x z Exmple: given z6, Exmple: given z6, 6 x x x x x 6 x often lle "slog" often lle "tetr-root" Te esrie mtrix-opertion is est suite for nlysis of ), sine most nturlly we el wit fixe se, n isuss te mount of itertion, wi is neee to rrive t ertin output strting wit ertin input orizontl strt-prmeter). For ) we urrently ve only te possiility to fin te se y itertively ppling te regul flsi or relte proeures for interpoltion. Horizontl ontention of terms wit te sme se is speil simple lgeri opertion ition) on te itertor-prmeter, so generl frtionl itertes of ny rel n e reue to one step of integer-tetrtion [] n one step of frtionl-tetrtion wit te frtionl eigt-prmeter <{}<. [ ] { }

8 In ete in te internet-newsgroup news://si.mt te position from te view of te tetr-roots were onsiere: ^^n^^/) / ^^n/). Tis prolem my e isplye witin tis seme s ) / n n ontinuous tetrtion ws isre from tis oservtion. I've urrently no goo ie out ritmetis in te exponent wit ifferent ses, ut my e, proper rules n e stte. In te tetrtion-forum tis prolem seems to ve een resse in te tres roun "se-nge", n were mostly pose y Jy Fox. Te prolem, s stte in ^^n^^/) / ^^n/) in te urrent view of tis rtile, impliitely involves se-nge, for wi I in't evelop smoot rules so fr. But s oserve, ritmetil opertions of tis type in te itertor-prmeter n smootly e esrie using te )-version ut wi, tully, oes not fit te prolem s stte sine it uses onstnt, given se-prmeter): / ut / n n / n ) n ) n n ) n / n / n / n / In terms of, for instne, ynmil systems tis looks like te following iotomy: View of slog-ener If I ve si esription of te rteristis Bs of ertin system, ow mny possily frtionl ) o I ve to pply it to rrive from strting onition to te finl sttus? If I ve iterte te rteristi Bs) of system to re n intermeite sttus, n ten pply it -/, ten I ve te sme sttus, s if I pplie it 5 to te initil sttus. View of tetr-root-ener If I look t te strting onition n te finl sttus, wi rteristi Bs for my system o I nee, to rrive t te finl sttus y x possily frtionl) itertions? I ve iterte te rteristi Bs of system to re n intermeite sttus. Ten I etermine te rteristi Bt, wi woul llow to proee from te initil onition to te finl sttus in only steps inste. Tus Bt soul ve te mening of Bs^^/) But ten te rteristi Bt is not te rteristi Bs. An iterting Bs 5 from te initil stte is not te sme s iterting Bt one time. Tis is te inerent wekness of ontinuous tetrtion. At te moment I feel not le to mke onluing remrk. It is still not ler to me, ow te ovious prolems wit opertions, lgerilly relting se- n itertor-prmeter, n e esrie n even less, e solve. Te onventionl inry nottion for te tetrtion-opertor suggests, tt ) see news://news.t-online.e:9/ @tprx or ttp://groups.google.s/group/si.mt/msg/e8659 see ttp://mt.eretrnre.org/tetrtionforum/sowtre.pp?ti4&pi4#pi4

9 soul e n equlity. But it seems, tis is tus merely nottion-prolem. I tink, wit my seme ere one s tools to point out te ore of tis prolem more preisely tn wit te ommon inryopertor ^^ in formule like ^^ n possily to proee to n greement etween te onurring views of te forml funtionlity of te opertor.

10 Some rules * *k k k k * ) ) k k Itertion ownwys ) * ) ) Itertion upwys some rules oo Gottfrie Helms