Average Number of Real Zeros of Random Fractional Polynomial-II
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1 Average Number of Real Zeros of Radom Fractioal Polyomial-II K Kadambavaam, PG ad Research Departmet of Mathematics, Sri Vasavi College, Erode, Tamiladu, Idia M Sudharai, Departmet of Mathematics, Velalar college of Egieerig ad Techology, Erode, Tamiladu, Idia Abstract - Let a0, a, be a sequece of idepedet ad idetically distributed stadard ormal radom variables I this paper, the average umber of real zeros of the radom fractioal polyomial Further it is proved that the average umber of real zeros (0 0 EN is asymptotic to log Keywords: Radom variables, Radom fractioal polyomial, Normal distributio, for large values of is obtaied INTRODUCTION Mathematical models are idispesable i may areas of Sciece ad Egieerig For bacterial growth the models usually tae the form of differetial equatios or a system of differetial equatios Differetial equatios have bee etesively studied, both from the aalytical ad umerical view poits However, there are may situatios, where the equatios with radom coefficiets are better suited i describig the behavior of the quatities of iterest Radomess i the coefficiets may arise, because of errors i the observed or measured data, variability i eperimet ad empirical coditios, ucertaities (variables that caot be measured, missig data, etc or plaily because of lac of owledge Differetial equatios where some or all the coefficiets are cosidered as radom variables or that icorporate stochastic effects have bee icreasigly used i the last few decades to deal with errors ad ucertaities ad to represet the growig field of great scietific iterests [6, 7, 9] Fractioal calculus is a brach of mathematics that grows out of the traditioal defiitios of the itegral calculus ad derivative operators I the same way fractioal epoets is a outgrowth of epoets with iteger value [] I a letter to L Hopital i 695, Leibiz raised the followig questio [7], Ca the meaig of derivatives with iteger order be geeralized to derivatives with o iteger orders? L Hopital posted the questio to Leibiz, what would be the result if the order will be Leibiz replied as: ʻIt will lead to a parado, from which oe day useful cosequeces will be drawʼfrom these words fractioal calculus has bee iitiated Radom fractioal algebraic polyomials arise i the study of differetial equatios with radom coefficiets[5,0,] Cosider the polyomial, a a a a ( ( ( 0 If a s ( = 0,,, (- are all idepedet real radom variables, the the above polyomial becomes a radom polyomial Set, f ( a, (0 The average umber of real zeros of this polyomial is deoted by 0, the polyomial ( becomes, 0 0 f ( a a a a (0 ( EN I particular if wwwijertorg 368 (This wor is licesed uder a Creative Commos Attributio 0 Iteratioal Licese
2 Whe, that is for the polyomial the the umber of real zeros, is estimated as EN = log, the above polyomial ( taes the form,where a s (=0,,,(- are idepedet ad stadard ormal variables, 0 i Kac[3] This relatio is ow as Kac s result Whe The average umber of real zeros of this fractioal polyomial the iterval (0, is estimated i [] as EN = log Average umber of real zeros of various radom polyomials is discussed i [] I this paper, the fractioal polyomial of the form, give i ( has bee discussed The mai theorem proved i this article is stated below THEOREM ( If the radom variables a s ( = 0,,, (- are all idepedet stadard ormal variables,the average umber of real zeros, EN of the fractioal polyomial ( is give by the followig formula, EN [ ( ] 3 d, (3 0 ( ( ad its asymptotic relatio is give by EN log ( ad the iterval estimatio is, log EN log 0 (5 where EN deotes the average umber of real zeros of the fractioal polyomial (The followig theorem ( is eeded to prove the theorem ( completely THEOREM ( For the radom fractioal polyomial, (0 0, the umber of real zeros i (0,, ad i (, are equal Proof: Cosider the radom fractioal polyomial, where a a a a a 0 a0 a a a 3 0 a0 y a y a y a 0, 3 y y So, as varies from 0 to, the rage of y is from to Therefore the umber of real zeros i the iterval (0, is same as that of the iterval (, By Kac s formula [3], for the radom polyomial PROOF OF THE MAIN THEOREM ( 0 the umber of real zeros i the iterval (a,b is give by the equatio,, wwwijertorg 369 (This wor is licesed uder a Creative Commos Attributio 0 Iteratioal Licese
3 b ( AC B EN( a, b d ( A I the preset case, the fractioal polyomial where B = A C Let a, for 0 is cosidered 0 AC B ( 0 0 ad (3 0 6 Equivaletly, A B ( ( ( C ( ad ( ( ( ( ( Substitutig the values of A, B ad C, give i the system of equatios ( i the equatio ( yields, ( 3 6 ( ad by a aalytical computatio i the equatio (, gives (5 EN (0, [ ( ] d (6 3 0 ( ( Employig the theorem (, EN (0, EN (0, is of the form [ ( ] 3 0 ( ( d (7 wwwijertorg 370 (This wor is licesed uder a Creative Commos Attributio 0 Iteratioal Licese
4 Cosider EN (0, where h ( 3 ASYMPTOTIC VALUE OF EN [ h ( ] 3, 0 ( The h ( > h ( <, ( (, ad d (3 (3 h ( ( (33 Suppose f( is differetiable o (0,, applyig the mea value theorem for f( o (, gives, f ( f ( f ( ( for,θ (0, (3 Set f (, 5 h ( (, for < θ< h ( ( (, as θ h ( ( (, sice <, for [0,] ( [ h ( ], 3 3 for [0,] (35 ( ( O the other had, h ( 0, as, so h ( 0, as The ad [ h ( ] 3 0 h ( [ h ( ] 3 [ h ( ] ( ( d 3 ( (, for [0, ], (36 d ( ( ( ( Evaluatio of the first itegral i the right had side of (37 yields, d (37 wwwijertorg 37 (This wor is licesed uder a Creative Commos Attributio 0 Iteratioal Licese
5 ( d [log( log ] ( ( The secod itegral i the right had side of (37 teds to zero, whe The the iequatio (37 becomes, [ h ( ] 3 0 ( From the equatio (3, d [log( log ] (39 EN (0, log log( (30 For large values of, EN (0, log 0 (3 To obtai the lower estimate, let ε ad δ be two real quatities with 0 < (ε, δ <, h ( 0, whe So, for sufficietly large, [ h ( ] 3 0 ( h ( h ( (3 d ( [ h ( ] 3 0 ( [ ], d (33 for [0,( ] Usig the relatio (3, [ h ( ] 3 0 ( d ( [ h ( ] 3 0 ( d ( ( 3 0 ( d (3 Maig suitable trasformatio, [ h ( ] 3 0 ( d ( ( log (35 That is, EN (0, ( ( log (36 From the iequatios (3 ad (36 the asymptotic formula EN (0, log (37 is obtaied Equatio (37 represets the asymptotic value of the real zeros of the radom fractioal polyomial for 0 < < This proves the theorem wwwijertorg 37 (This wor is licesed uder a Creative Commos Attributio 0 Iteratioal Licese
6 A GENERALISATION OF THE RESULTS ON RANDOM AND FRACTIONAL POLYNOMIALS I this sectio, a geeralisatio of the Kac s formula [3], ad the theorem ( o radom fractioal polyomial is proved Cosider the algebraic equatio, ( a a a a 0, ( ( 0 where the a s ( = 0,, (- are idepedet radom variables assumig real values oly The By Kac s formula [3], where, The from [3], Equivaletly, A B C Let f ( a, (0 0 b ( AC B EN( a, b d ( A a AC B (3 A var[ f ( ] ( B cov[ f ( f ( ] (5 C Var[ f ( ] (6 A B C 0 ad (7 0 0 ( ( ( ad (8 ( ( ( ( 3 ( Substitutig the values of A, B ad C, give i the system of equatios (8 i the equatio (3 yields, ( Substitutig the value of Δ obtaied i the equatio (9 to the equatio ( yields, EN (, ( ( ( ( Whe α =, equatio (0 coicides the relatio derived i Kac s formula [3] EN (9 d (0 ( ( d ( ( ( wwwijertorg 373 (This wor is licesed uder a Creative Commos Attributio 0 Iteratioal Licese
7 That is for the polyomial,where a s ( = 0,,,(- are idepedet ad ormally distributed radom variables with mea 0 ad variace, Kac[3] estimated the average umber of real zeros i (, as log But whe, the equatio (0 becomes, ( ( EN(, ( ( This result coicides with the result derived i [] REFERENCES [] Adam Loverro, 00, ʻFractioal Calculus: History, Defiitios ad Applicatios For the Egieerʼ [] Bharucha-Reid AT ad Sambadham M, 986, ʻRadom Polyomials, Academic Press, Orlado, [3] KacM,93, ʻO the Average Number Of Real Zeros Of a Radom Algebraic Equatio, BullAmemathSoc, 9,(3-30 [] Kadambavaam K, Sudharai M, Oct 0, ʻAverage Number Of Real Zeros Of Radom Fractioal Polyomialʼ, Joural of Global Research i Mathematical Archives, [5] Vol,No 0,(67-75 [6] AAKilbas, HMSrivastava, ad JJ Trujillo Theory ad Applicatios of Fractioal Differetial Equatios, North-Hollad Mathematics Studies, 006 [7] Necel Tobias ad Floria Rupp,03,ʻRadom Differetial Equatios i Scietific Computigʼ,De Gruyter Ope,Warsaw,Polad [8] Osedal B, 003, ʻStochastic Differetial Equatios,Sith Ed, [9] Spriger -Verlag, Heidelberg [0] Podluby I, 999, ʻFractioal Differetial Equatios, ʻMathematics i Sciece Ad Egieerig V98ʼ,Academic Press [] Soog T, 99 ʻProbabilistic Modelig ad Aalysis i Sciece Ad Egieerig,Wiley Newyor [] Tomas Kisela, Fractioal Differetial Equatios ad Their Applicatios, Diploma Thesis, [3] Bro Uiversity of Techology, Bro 008 [] Yog Zhou, Basic Theory of Fractioal Differetial Equatios, World Sciece Publicatios, 0 wwwijertorg 37 (This wor is licesed uder a Creative Commos Attributio 0 Iteratioal Licese
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