Quantization in Dynamic Smarandache Multi-Space

Transcription

1 Quatzato Dyamc Smaradache Mult-Space Fu Yuhua Cha Offshore Ol Research Ceter, Beg, 7, Cha (E-mal: ) Abstract: Dscussg the applcatos of Dyamc Smaradache Mult-Space (DSMS) Theory. Supposg for the dfferet dyamc spaces ( s a dyamc postve teger ad the fucto of tme) the dfferet equatos have bee establshed, as these dfferet dyamc spaces sythesze the DSMS, ad they are mutually affected, some ew coupled equatos eed to establsh the DSMS to replace some equatos the orgal dyamc spaces, as well as supply other equatos to process the cotact, boudary codtos ad so o. For the ufed processg of all equatos the DSMS, ths paper proposes to ru the quatzato processg to all the varables ad all the equatos ad establsh the ufed varatoal prcple of quatzato wth the collocato method based o the method of weghted resduals, ad smultaeously solve all the equatos the DSMS wth the optmzato method. Thus by usg the ufed varatoal prcple of quatzato the DSMS ad the fractal quatzato method, wll pave the way for the ufed processg of the theory of relatvty ad the quatum mechacs, ad the ufed processg of the four foudatoal teractos. Fally the coupled soluto for the problem of relatvty ad quatum mechacs s dscussed. Key words: Dyamc Smaradache Mult-Space (DSMS), coupled equato, collocato method, ufed varatoal prcple of quatzato, fractal quatzato, four foudatoal teractos, ufed processg Itroducto I ths paper we wll propose the structure of Dyamc Smaradache Mult-Space (DSMS) frstly, the dscuss the problems of the ew coupled equatos the DSMS, ufed processg all the equatos the DSMS, rug the quatzato processg to all the varables ad equatos wth the collocato method based o the method of weghted resduals, varable quatzato method, fractal quatzato method, the ufed varatoal prcple of quatzato, the coupled soluto for the problem of relatvty ad quatum mechacs, ad so o. Smaradache Mult-Space ad Dyamc Smaradache Mult-Space The oto of Smaradache Mult-Space was proposed by Smaradache 969 [-3]. A Smaradache mult-space s a uo of sets or spaces M, M, M. M = M U M U U M where s a teger, ad. For the reaso that s a costat, so the Smaradache Mult-Space s a statc Mult-Space. But may practcal problems, for the cotuous chage of the umber ad structure of the sets or spaces, the dyamc mult-space should be cosdered. We defe that a Dyamc Smaradache Mult-Space (DSMS) s a uo of ( sets or spaces M (, M (, M (.

2 M( = M ( U M ( U U M ( where( s tegers, ad the fucto of tme t(. Ufed varatoal prcple sgle space I the referece [4], a ufed varatoal prcple of flud mechacs was preseted for the sgle flud space.. Optmzato method of weghted resduals Optmzato method of weghted resduals (OMWR) has bee used for solvg some problems mechacs, physcs ad astroomy [5], t ca be stated brefly as follows. Supposg that we have the operator equato F = doma Boudary codto B = o boudary S The the fuctoal Π defed by OMWR reads F) d W A ( B) ds = A m S ( 3 where A (F) s a o-egatve fuctoal about F ts value beg = f F = A ( F) 4 > f F m deotes mmum ad ts value should be equal to zero. It was troduced the referece [6]. W s postve weghted umber, ad may cases t ca be tae as W =. The way for choosg W geeral cases s show the referece [7]. I Eq.(4), f the cases of A ( F) = F, the t gves the Least Square Method. I the referece [7], A ( F) = F F 4 F F e F max ad the le have also bee dscussed. Obvously the codto of zero mmum value of fuctoal Π Eq.(3) s equvalet to the soluto of Eqs.() ad (), the proof s show the referece [4]. It should be oted that, f the doma cossts of soltary pots P ~ P, the boudary S cossts of m soltary pots Q ~ Qm (such as the quatzato problems we wll dscuss ths paper), the the tegral Eq.(3) should be replaced by the sum W A F( P )) W W A ( B( Q )) = m ( m 3 I addto, the establshmet of fuctoal Π OMWR s very easy, ad the mmum value of Π s ow advace (equal to zero). As for fdg the mmum value of Π, two methods ca be used. Oe s solvg the

3 equatos Π '= gve by the statoary codto. Aother s usg the optmzato methods used the refereces [4-8] such as steepest descet method, searchg method.. A ufed varatoal prcple of flud mechacs (UPFM) Basc equatos of flud mechacs ad boudary codto read Cotuty equato F = doma 5 Equato of moto G = doma 6 Eergy equato H = doma 7 Costtutve equato I = doma 8 Equato of state J = doma 9 Boudary codto B = o boudary S Accordg to OMWR, the UPFM the followg form ca be obtaed A ( G) d W A ( H d W A F) d W ) ( 3 A ( J ) d W6 A ( B) ds = W4 A ( I) d W5 m S where W s sutable postve weghted umbers, A (F) A (G) ad the le are o-egatve fuctoals defed by Eq.(4). By usg ths method, the referece [8] ufed processg varous water gravty wave theores, the referece [4] accordg to the OMWR for soltary doma or soltary pot, the soluto of hydrodyamcs equato for the soltary doma or for the soltary pot (pot soluto) s developed. I the solvg process, the compatblty wth other doma or other pot does ot eed to be cosdered. 3 Ufed varatoal prcple the DSMS All the equatos should be cosdered the DSMS are as follows The equatos that are establshed the orgal ( sets or spaces, ad stll correct the DSMS F = doma = ~ ( The boudary codtos that are establshed the orgal ( sets or spaces, ad stll correct the DSMS B = o boudary S = ~ ( 3 As the ( dfferet dyamc sets or spaces sythesze the DSMS, ad they are mutually affected, some ew coupled equatos eed to establsh the DSMS to replace some equatos the orgal dyamc sets or spaces, as well as supply other equatos to process the cotact, boudary codtos ad so o. Here the cotact codto deotes the codto should be satsfed the case that two sets or spaces have partal commo elemets. As establshg the ew coupled equatos the DSMS, the exstg coupled equatos physcs ca be referred, such as the pressure-velocty coupled equato, 3

4 temperature-stress coupled equato, ad so o. For the sae of coveece, all the ew coupled equatos, the cotact ad boudary codtos ad so o wll be wrtte the ufed form as follows ( whch doma or boudary) C = for = ~ m( 4 Accordg to OMWR, the ufed varatoal prcple the DSMS s as follows ( W A ( F ) d W A ( B) ds m( ( W3 A ( C ) d = m 5 S may be It also should be oted that, f the doma cossts of '( soltary pots P ~ P ', the boudary S cossts of m '( soltary pots Q ~ Q m ', the doma cossts of ( soltary pots P ~ P, the the tegral Eq.(5) should be replaced by the sum, ad the ufed varatoal prcple of quatzato may be obtaed ( W '( W A ( F ( P )) ' ' ( W m'( W A ( B ( Q " " )) m( 3 ( W W A ( C ( P )) = m 5 ' ' 4 arable quatzato ad equato quatzato The varable quatzato cossts two methods: average value method ad represetatve value method. The average value method meas that the value of a whole terval s represeted by the average value of ths terval. For example, a spaceshp avgates alog a straght le. Its speed orgally s cotual. But we tae the cotual fve le segmets, ad tae the average speed of each le segmet, moreover the speed of etre le segmet s represeted by the average speed. The the speed has ot bee cotual, we may th that the speed has bee carred o the quatzato processg (smlarly, other parameters such as eergy, temperature ad so o, may be carred o the quatzato processg). The represetatve value method meas that the value of a whole terval s represeted by the value of a represetatve pot that s chose sutably ths terval. For the equato quatzato, the represetatve value method ca be used oly, because the soluto of equato s ot ow advace. The fte dfferece method ad the fte elemet method are all the typcal 4

5 represetatve value method for the equato quatzato. Obvously, quatzato methods for varable ad equato are sutable for all the cotuty problems. 5 Fractal quatzato The quatzato realzed by fractal method, could be reached through tag certa varables fractal formula for the tegers. The fractal formula reads C N = 6 D r Now, the fractal formula, we carry o the quatzato processg to value of N, amely tae the value of N for the arrage sequece umber, ad N =,, 3. For example, the solar system for the orbtal moto average veloctes of the e plaets (ut: m/s), tag the characterstc dmeso r for some plaet orbtal moto average velocty, tag the value of N for the seral umber accordg to the sze of the orbtal moto average velocty, frstly cosderg the case of Mercury r=47.89, the we have N= (Mercury's orbtal moto average velocty s the greates, thereupo we have the coordates pot (47.89, ), accordg to aalogzes other 8 plaets coordates pots are as follows: (35.3, ), (9.79, 3), (4.3, 4), (3.6, 5), (9.64, 6), (6.8, 7), (5.43, 8), (4.74, 9). The above 9 coordates pots may be plotted o the double logarthmc coordates, the we may obta 8 straght-le segmets. I order to adapt the applcato of Smaradache geometrc ad eutrosophc methods, here we do ot fd the fttg curve of these 8 straght-le segmets wth least squares method, but for the 8 straght les, usg the coordates pots data to determe accurately ther fractal parameters (costat C ad fractal dmeso D ). For example, accordg to Mercury's coordates (47.89, ) ad eus's coordates (35.3, ), may obta the fractal parameters for the frst straght-le segmet: C =53.684, D = The fractal formula for the frst straght-le segmet reads: N =. Ths formula r may be used as the extrapolato formula to forecast the orbtal moto average velocty of ext plaet (Earth) by substtutg N =3 to ths formula ad solvg the value of r. Smlarly, all the forecastg results for other plaets may be reached. By usg the st straght-le segmet, the forecastg result of the orbtal moto average velocty of ext plaet (Earth) reads: =9.7, the error s.7%. By usg the d straght-le segmet, the forecastg result of ext plaet (Mars) reads: =6.55, the error s.%. By usg the 3rd straght-le segmet, the forecastg result of ext plaet (Jupter) reads: =.49, the error s 59.9%. By usg the 4th straght-le segmet, the forecastg result of ext plaet (Satur) reads: =7.9, the error s 8.%. By usg the 5th straght-le segmet, the forecastg result of ext plaet (Uraus) reads: =7.46, the error s 9.5%. By usg the 6th straght-le segmet, the forecastg result of ext plaet (Neptue) reads: =5.4, the error s 7.9%. 5

6 By usg the 7th straght-le segmet, the forecastg result of ext plaet (Pluto) reads: =4.45, the error s 6.8%. By usg the 8th straght-le segmet, the forecastg result of ext plaet (teth plae reads: =4., the error s determacy, because the teth plaet s ot dscovered. I addto, the referece [9], the cocept of varable dmeso fractal was troduced. I whch the fractal dmeso D s a varable stead of a costat, for example D = a ar ar... a r 7 I some cases, for the sae of coveece, the fractal formula ca be wrtte as the followg form l N = l C D l r 8 Substtutg Eq.7 to Eq.8, the form easy to hadle the fractal quatzato ca be obtaed l N = l C ( a a r a r... a r ) l r 9 Namely F = l C ( a ar ar... a r ) l r l N = 6 Applcato of ufed varatoal prcple of quatzato ad fractal quatzato Now we dscuss the ufed processg for the problems of specal relatvty ad quatum mechacs. Cosderg the ufed processg for the problems of the wavelegth of Balmer seres atomc spectrum of hydroge ad the ultmate speed expermet. Wth the method of quatum mechacs, the wavelegth of Balmer seres atomc spectrum of hydroge may be obtaed theoretcally 4 λ ( ) = 9. = 3,4, 5 4 From table we ca see that comparg wth the expermetal data λ 3), λ (4), (5), the results of Eq.() have some small errors. ( λ I order to obta the better results, we choose the pots wth = 3,4, 5 as the represetatve pots, usg the fractal quatzato method ad supposg C λ ( ) = = 3,4, 5 D The cocrete form wll be decded wth varatoal prcple of quatzato. Table. Wavelegth of Balmer seres atomc spectrum of hydroge 6

7 alue of Expermetal value of λ Quatum mechacs method aratoal prcple of quatzatoo-coupled soluto aratoal prcple of quatzatocoupled soluto I specal relatvty, as dscussg the ultmate speed, t gves E v ( E ) = c [ ( ) ] 3 m c From table we ca see that comparg wth the Bertozz expermetal value of v, the results of Eq.(3) also have some small errors. I order to obta the better results, usg the fractal quatzato method ad supposg C v ( E ) = E =.,.8, E D Table. Bertozz ultmate speed expermet alue of eergy E Expermetal value of v Specal relatvty aratoal prcple of quatzatoo-coupled soluto aratoal prcple of quatzatocoupled soluto I varatoal prcple 5, adoptg Least Square Method, ad the weghted umber beg, the we have Π 5 Π = m where [l λ ( ) l λ( )] Π = 3,4,5 = [l v ( E ) l v ( E )] E =.,.8,4.7 Because the coupled equato of quatum mechacs ad specal relatvty has ot bee establshed, there s ot such term Eq.(5). Now we dscuss the soluto of Eq.(5). Frst s the o-coupled soluto, amely the quatum mechacs soluto ad the specal relatvty soluto wll have ot ay relatos. I Eqs.() ad (4), supposg 7

8 D = a a D = b b E The the soluto of Eq.(5) s decded wth the followg equatos Π C = Π a = Π b = 6 Solvg these equatos, the o-coupled expressos of fractal quatzato for the wavelegth ad ultmate speed are as follows λ ( ) = = 3,4, v ( E ) = E =.,.8,4. E E From Eqs.(7) ad (8), we ca see that C = C a = b. The results of fractal formulas Eqs.(7) ad (8) are show Table ad table, these results ad the expermetal results are completely same. Secodly dscussg the coupled soluto, we tae all the costat terms the quatum mechacs soluto to be equal to all these the specal relatvty soluto. Namely Eqs.() ad (4) we should have C = C a = b D = a a a D = b b E b E Solvg Eqs.(6), the coupled expressos of fractal quatzato for the wavelegth ad ultmate speed are as follows λ ( ) = = 3,4, v ( E ) = E =.,.8,4. E E E The results of fractal formulas Eqs.(9) ad (3) are show Table ad table, these results ad the expermetal results are completely same. From the above results we may see that, for the questos of two completely dfferet domas of quatum mechacs ad specal relatvty, the extremely smlar solutos ca be obtaed wth the method preseted ths paper. Moreover, although the coupled equato of quatum mechacs ad specal relatvty has ot bee establshed, the 8

9 coupled soluto for the quatum mechacs ad the theory of relatvty really ca be obtaed. 7 Further dscusso The ufed varatoal prcple of quatzato DSMS ad the fractal quatzato method, smlarly ca be used for the ufed processg of the questos dfferet domas. Thus wll pave the way for the ufed processg of the theory of relatvty ad the quatum mechacs, ad the ufed processg of the four foudatoal teractos. For example, we may dscuss the smplest stuato, amely the equatos for descrbg the four foudatoal teractos read: F = = ~ 4, structurg ther acto domas as a DSMS, supposg that all the coupled equatos ad supplemetary cotact ad boudary codtos read: C =, the the varatoal prcple for the ufed processg of the four foudatoal teractos may be wrtte as follows 4 = W F d W C d = m The further topcs are how to derve the coupled equatos DSMS, as well as supplemet sutable cotact ad boudary codtos ad so o. Whe these equatos have ot bee establshed, the coupled soluto for dfferet domas may be obtaed frstly, order to dscover the commo groud of them, ad create the codtos for fally dervg the coupled equatos ad so o. Refereces F. Smaradache. Mxed oeucldea geometres. eprt arxv: math/9. /. F. Smaradache. A Ufyg Feld Logcs. Neutrosopy: Neutrosophc Probablty, Set, ad Logc. Amerca research Press. Rehoboth L. F. Mao. Smaradache mult-space theory. Hexs. Phoex. AZ, 6. 4 Fu Yuhua. A Ufed aratoal Prcple of Flud Mechacs ad Applcato o Soltary Subdoma or Pot. Cha Ocea Egeerg, 994, ol.8, No., Fu Yuhua. The applcato of optmum MWR mechacs, physcs ad astroomy, : The advace ad egeerg applcato of the method of weghted resdual, Wuha Uversty Press, 99. ( Chese) 6 Fu Yuhua. New soluto for problem advace of Mercury s perhelo. Acta Astroomca Sca, 3(4). ( Chese) 7 Fu Yuhua. A ew way for solvg boudary layer equatos, Cha Offshore Ol Gas (Egeerg), (6). ( Chese) 8 Fu Yuhua. Ufed Water Gravty Wave Theory ad Improved Lear Wave. Cha Ocea Egeerg, 6(). 9 Fu Yuhua. arable dmeso fractal flud mechacs. Proceedgs of secod computatoal hydrodyamcs coferece. Wuha, 993~7. ( Chese) 9