On the Riemann-Siegel formula
|
|
- Edwina Montgomery
- 6 years ago
- Views:
Transcription
1 On he Riemann-Siegel formula A. Kuznesov Dep. of Mahemaical Sciences Universiy of New Brunswick Sain John, NB, EL L5, Canada Curren version: June 6, 7 Absrac In his aricle we derive a generalizaion of he Riemann-Siegel asympoic formula for he Riemann zea funcion. By subracing he singulariies closes o he criical poin we obain a significan reducion of he error erm a he expense of a few evaluaions of he error funcion. We illusrae he efficiency of his mehod by comparing i o he classical Riemann-Siegel formula. Keywords: Riemann zea funcion, Riemann-Siegel formula, asympoic expansion, incomplee Gamma funcion Published in Proc. R. Soc. A 6, doi:.98/rspa The auhor would like o hank anonymous referees for many helpful commens.
2 Inroducion The Riemann-Siegel RS asympoic formula is a very efficien mehod o compue he Riemann zea funcion ζ + i for large and i has been used exensively in he las seveny years o compue he nonrivial zeros [ ] of he zea funcion see Odlyzko 99. The RS formula conains he main sum of N = erms and he asympoic correcion erms which allow reducion of he error. The funcion being approximaed is acually Z, which is defined as Z = ζ + i e iθ, where θ = Im { ln Γ + } i lnπ. The funcion Z is real for real his follows from he funcional equaion for ζ and Z = ζ + i. Below we presen he RS formula: π Z = N n= cosθ logn n + N π [ m j= π j C j + O ] m, where he firs hree correcion erms are C = Ψτ = cos π τ τ 6, cosπτ and τ = series: π C = 96π Ψ τ, C = 8π Ψ 6 τ + 6π Ψ τ, N. To compue θ for large one can use he asympoic expansion derived from he Sirling θ = ln π π In he las fifeen years, several new asympoic expansions for he Riemann zea funcion have appeared. In Berry & Keaing 99 he auhors use he Cauchy inegral formula o derive he asympoic expansion for Z, where he leading erm Z, K is Z, K = Re expi[θ lnn] ξn, Erfc, n n QK, where ξn, = lnn θ, Q K, = K iθ and K is a free real parameer. Noe ha he leading erm is similar o he main sum of he RS, wih he cuoff a n = N being smoohed by he complemenary error funcion, hus he approximaion o Z is a smooh funcion, unlike he RS. Auhors show ha increasing K gives a beer accuracy and ha for suiable K he leading erm Z, K always gives beer accuracy han he RS main sum. As K increases o infiniy Z also conains a leas he firs correcion erm C. Higher correcion erms Z j, K also provide a significan increase in accuracy. In Paris 99 he auhor uses he Poisson summaion formula and he uniform asympoic expansion for he incomplee Gamma funcion o derive an asympoic expansion for Z. This expansion also involves he complemenary error funcion, alhough in a differen manner compared o he Berry & Keaing 99 approximaion. The free parameers in his approximaion can be chosen o decrease he error significanly hough again a he expense of compuing error funcions, see Paris 99 for deails.
3 In his aricle we propose anoher mehod of approximaing Z for large. The resul is a generalizaion of he RS formula wih an addiional free parameer δ: in essence we replace δ highes erms in he main sum by δ erms involving he incomplee Gamma funcion and he funcion Ψτ in he correcion erms is replaced by a similar funcion Ψτ, δ. This approximaion is obained by removing δ poles of he inegrand around he saionary poin and hen performing asympoic expansion. We find ha he form of he correcion erms is he same as in RS, and when δ = we recover he classical RS formula as a special case. Combined wih he asympoic expansion for he incomplee Gamma funcion derived in Temme 979 see also Dunser e al. 998 we obain a simple and efficien mehod of reducing he error in he RS formula see secion for he numerical resuls. Derivaion of he asympoic formula We sar wih he following inegral represenaion for he Riemann zea funcion, which Riemann used o prove he funcional equaion see Tichmarsh 986, page 7: π s s Γ ζs = Υs + Υ s, where Υs = e πis s π s s Γ πi+e πi R e iw π w s sinh w dw. 5 Noe ha he inegral in equaion 5 converges absoluely for all complex s. Throughou his aricle we assume ha [ s = + i and ha is large and posiive. To find he saionary poin of he inegral in 5 we solve ] d iw + i lnw =, hus w = π and he saionary poins are ±i π. We can no move he conour dw π of inegraion o w = i π because of he branch poin a w =, hus we choose he saionary poin w = i π. Remark : Noe ha equaion w = π has wo real soluions when is negaive, however we do no obain new asympoic formulas. We canno move he conour of inegraion o w = π because of he branch poin a, and if we move he conour of inegraion o w = π we would in fac obain he same asympoic expansion as by choosing posiive and w = i π. In order o obain he asympoic represenaion for he inegral in equaion 5 we need o move he conour of inegraion o pass hrough he saionary poin w = i π, expand funcion e iw π w s in Taylor series around w and inegrae erm by erm. However he funcion {sinh w } always has poles near he criical poin, and his will affec he accuracy of he approximaion. Thus we do he following: we fix an ineger number δ and subrac δ singulariies of {sinh w } around he criical poin. Thus we define a funcion F N,δ w as F N,δ w = sinh w N+δ n=n+ δ n w πin. 6
4 Now we move he conour of inegraion o w and we obain he following decomposiion for Υs: Υs = Υ s + Υ s + Υ s = 7 N δ [ ] = e πis s π s s e iw π w s Γ πi Res w=πin sinh + w n= N+δ + e πis s π s s Γ n e iw π w s w πin dw + n=n+ δ αi+r + e πis s π s s Γ e iw π w s F N,δ wdw. Firs we simplify Υ s by compuing he residues: Υ s = π s s Γ w +e πi R N δ Nex, we express Υ s in erms of he incomplee Gamma funcions Υ s = π s Γ N+δ s n=n+ δ n s. 8 n= n s [ Q s ; πin + signnq s ; πin], 9 where Q a; x = Γa,x is he normalized incomplee Gamma funcion see Gradsheyn & Ryzhik. If Γx n = he erm in he sum should be replaced by π s πi iθ+ e s /Γ s. Equaion 9 was derived using he following inegral ɛi+r e x x s x + bi ] dx = πie b + πis ˆb ib [ˆbQ s s ; b Q s; b, where he argumen of ib is beween π; π] and ˆb = signreb. To obain we separae x + bi ino real and imaginary pars and use Gradsheyn & Ryzhik o evaluae each inegral. Now we have o approximae Υ s. Firs we perform a change of variables, w = πix + w, and obain Υ s = s s e i s Γ πi where funcion g, x is defined as g, x = e x ix + x +i = e x i R e x g, xf N,δ πix + w dx, ix+ +i ln + x i Expanding ln + x i in a Taylor series we find ha for every x g, x as and we obain he following asympoic expansion g, x = e x i x x i x +O = + x + x i + 8 x 5 x + 8 x6 i + O. =.
5 As a nex sep we subsiue ino and inegrae erm by erm. To achieve his we need o compue funcions f n τ, δ = e x x n F N,δ πix + w dx, where τ = N. We follow he seps of he π R derivaion of he RS formula in Tichmarsh 986 and compue he exponenial generaing funcion for f n τ, δ f n τ, δ u n = e x +xu F N,δ πix + w dx = n! n = R R e x +xu sinh πix + w dx τ + u, δ πi = N πie u Ψ where he funcion Ψτ, δ is defined as πi e 8 sinπτ e Ψτ, πiτ τ δ = cosπτ and Φx = π k= δ N+δ n=n+ δ n R e x +xu πix + w πin dx = δ k e [sign ] πiτ k k + Φ π τ k, i x e d is he error funcion see Gradsheyn & Ryzhik. The firs inegral in is one of he Mordell inegrals see Ramanujan 95, Mordell 9, Kuznesov 6 and can be compued using he mehod described in Tichmarsh 986. The second inegral can be found in Gradsheyn & Ryzhik. Taking n-h derivaive of boh sides of we obain f n τ, δ = N πi 8π n e πin s Γ [ n ] k= n! k!n k! πik Ψn k τ, δ. 5 Now we use 5, and o obain he following asympoic formula for Υ s: Υ s = N s s [ e i S + π S + π where he correcion erms are S = Ψ τ, δ S = Ψ τ, δ 96π S = i Ψ τ, δ + Ψ τ, δ + Ψ 6 τ, δ. 96π 6π 8π Finally, we combine, 7, 8, 9 and 6 o obain he following expression for Z: N δ Z = + Re k= { N+δ ] S + O, 6 cosθ lnn n + 7 n=n+ δ + N π Re iθ ln n e [ Q + i; πin + signnq } + i; πin] + n { e iθ i πe [ S + π 5 S + π ]} S + O.
6 We have also used he fac ha π s Γ s = e Γ iθ which follows from he definiion of θ see. s The correcion erms in he above equaion 7 can be simplified if we use he asympoic formula o rewrie he facor e iθ πe i as e iθ i = e πi 8 i 8 7i 576 πe { Thus we inroduce he new funcion Ψτ, δ = Re e πi 8 Ψτ, δ }, use o simplify he expression for Ψτ, δ and presen all he resuls in he following heorem: [ Theorem. Le be a real posiive number. Define N = δ. Then N δ Z = + Re k= { N+δ π ] and τ = π N. Fix an ineger number cosθ lnn n + 8 n=n+ δ + N π iθ ln n e [ Q + i; πin + signnq } + i; πin] + n [ m j= where he firs hree correcion erms are: and Ψτ, δ = δ cos π τ τ 6 cosπτ π j S j + O ] m, S = Ψ τ, δ 9 S = 96π Ψ τ, δ S = 8π Ψ 6 τ, δ + 6π Ψ τ, δ. cosπδτ + { δ k Re k= δ e πi 8 πiτ k Φ Here Qa, z = Γa,z Γa is he normalized incomplee Gamma funcion and Φx = π funcion. If δ > N he n = erm in he second sum in equaion 8 should be replaced by π i τ k }. x e d is he error. π e πi 8 π + i Γ +. i Remark : Noe ha he correcion erms 9 have he same form as in he classical RS formula see equaion. This can be explained as follows: he coefficiens in fron of derivaives of Ψτ, δ are obained from he expansion of g, x and hus do no depend on δ. However, when δ = equaion 8 mus give us he classical RS formula, hus hese coefficiens mus be he same for all δ. 6
7 Remark : Noe ha using he asympoic expansion of g, x is no he only way o approximae he erm Υ s. One could prove ha if a consan a saisfies < a, hen g, x can be expanded in convergen series in he Hermie polynomials H n ax. In his way we would obain a convergen asympoic series for Z, where he correcion erms would also involve linear combinaions of he derivaives of Ψτ, δ. However we found ha such an expansion is cerainly more complicaed and is no as accurae as 8. Remark : When δ + we see ha he firs sum in 8 vanishes, funcion Ψ τ, δ his follows from equaion and he fac ha F N,δ w, hus 8 reduces o he following expansion: Z = Re { e iθ n n s Q s, πin π e πis s Γ s }, s = + i, which was used in Paris & Cang 997 o derive an asympoic expansion for ζ + i. Numerical resuls Before we can use formula 8 we need o be able o compue efficienly he incomplee Gamma funcion Q σ + i; πin, where σ =, and N + δ n N + δ. In applicaions, especially when is large, we will be ineresed in he case when δ N. Bu hen we find ha πin, hus we need o approximae σ+ i Qa, z in he region z. Forunaely we have an excellen approximaion o Qa, z in his region derived a in Temme 979, see also Dunser e al Here we presen an approximaion o Qa, z of order a 5 which is enough for our purposes for all he deails see Temme 979. Proposiion. Define µ = z a and η = µ ln + µ = µ µ µ 7 5 µ Then Qa, z = Erfc a η + e a η πa [ c + c a + O a ], where coefficiens c, c are given by c = η + µ = µ η c = η + µ + µ µ = µ 7 5 µ +... µ η 5 7 µ +... Remark 5: Noe ha coefficiens c k have removable singulariies a µ =, hus when is close o πn and parameers µ and η are close o we have o use he second se of equaions in, which do no involve subracing large numbers. Below we presen he numerical resuls. An approximaion o Z given by equaion8 wih m correcion erms and fixed δ will be denoed as RS[m, δ]. The classical RS formula will be denoed as RS[m, ]. As we will see, for differen choices of parameers m and δ hese approximaions have differen shapes of he error Z m,δ Z as a funcion of : someimes he error is smaller for τ while for oher choices of m and δ he error is smaller a he endpoins τ and τ. Thus i is hard o compare he efficiency of approximaion RS[m, δ] a a single poin and insead we will presen he error graphically for he range of. 7
8 Firs, we compare RS[, ] wih RS[, ], see figure. We find ha for < < 7 approximaion RS[, ] is acually beer and for < < 6 boh of hese approximaions have comparable accuracy. For even larger RS[, ] becomes a beer approximaion, since i has an error erm of he order O 7 while RS[, ] is O. Bu we find ha even a large we could increase δ and make RS[, δ] as good as RS[, ]: for example a 5 < < 7 we find ha δ = 6 is enough for his purpose. x RS: m= RS: m=, δ= x x 7 RS: m= RS: m=, δ= x Figure : The error for [, 7] N and [, 6] 5 N 9. 8 x 5 RS: m=, δ= RS: m=, δ= 6 6 x x 7 RS: m=, δ= RS: m=, δ= x x x Figure : The error for [5, 7] 9 N 5. m = and δ increases from o. 8
9 RS: m=, δ= RS: m=, δ= x 5 6 x 6 RS: m=, δ= RS: m=, δ= x x RS: m=, δ= RS: m=, δ= x x RS: m=, δ= RS: m=, δ= x Figure : The error for [5, ] 5 N 8. m increases from o, δ = in he op row and δ = in he boom row. Second, we examine he effecs of increasing δ while keeping m = fixed see figure in he region 5 < < 7. We see ha increasing δ by decreases he error roughly by a facor of. Also noe ha he shape of he error he shape of he graph of he nex correcion erm becomes more linear as δ increases. This is easy o explain if we remember ha δ is he number of subraced singulariies, hus for larger δ funcion Ψτ, δ and is derivaives become less oscillaory. This fac means ha insead of compuing Ψτ, δ and is derivaives we can efficienly approximae he correcion erm by jus a few erms of is expansion in a Taylor series or Chebyshev polynomials. Finally, we examine he effecs of increasing he number of correcion erms, while keeping δ fixed see figure. In he op boom row we plo he absolue error of RS[m, ] RS[m, ] when m increases from o. Noe he decrease of 5 orders of magniude as m goes from o in he second row δ =, while in he op row δ = we have a decrease of orders of magniude. Afer his fas iniial decrease we gain roughly order of magniude in he boom row and orders in he op row. This seems o be a general rend: he error decreases faser wih he increase in m for small δ compared o large δ. Noe again ha in he boom 9
10 row δ = he correcion erms are smooher compared o he op row δ = x Figure : The effec of using a smooh cuoff funcion. [, 7] N, m = and δ = Conclusion In his aricle we presen a generalizaion of he Riemann-Siegel asympoic formula, which allows o obain a significan increase in he accuracy wihou a lo of exra compuaional effor. This approximaion has an exra free ineger parameer δ, which corresponds o half of he number of he singulariies removed around he criical poin. In general increasing δ resuls in he decrease of he error. In his aricle we compared his new approximaion scheme o he classical RS formula, and found ha he new formula consisenly gives beer accuracy even for small δ. We did no compare our approximaion o oher resuls, such as approximaions by Berry & Keaing 99 or by Paris 99: while i seems ha our approximaion can achieve he same accuracy, he quesion is a wha compuaional cos. To answer his quesion one would have o opimize approximaion schemes. Noe ha in our scheme here are several hings ha can be done o reduce he compuaional cos o jus δ evaluaions of he error funcion:. In equaion 8 we have wo incomplee gamma funcions Q σ + i; πin wih σ =,, bu when is large we can approximae hem wih jus a few erms of he Taylor series around σ =.. The almos linear form of he graphs of correcion erms see figures and suggess an approximaion by polynomials: one could use jus a few erms of eiher he Taylor series around τ = or he Chebyshev series o approximae S j. An advanage of he Berry & Keaing 99 asympoic formula is ha he approximaion [ ] erms are smooh funcions of ; in he approximaion by Paris 99 one can also choose N and hus have a smooh π approximaion a he ransiion poins. There is also a simple way o do i in our approximaion: insead of compleely removing δ singulariies around he criical poin =, we can assign o every singulariy a π
11 weigh, depending on he disance from he criical poin. For example, Theorem could be obained using he following weighs: if he singulariy is wihin he disance of πδ from he criical poin, i is assigned he weigh of and is removed compleely, oherwise he weigh is see equaion 6. Noe ha his scheme is equivalen o using a sharp cuoff funcion wih jumps a ±πδ see figure, and hese jumps creae disconinuiies in he RS formula. However, one could also use a smooh cuoff funcion, where he weigh of each singulariy is if i is close o he criical poin and he weigh would decrease smoohly o as he disance o increases see figure. The resul of using a smooh cuoff funcion is ha he error a he ransiion poins πn is cerainly smaller, bu a he same ime he compuaional complexiy is increased: noe ha on figure he sharp cuoff conains jus ransiion poins n = and n =, while he smooh cuoff also conains n = and we will need more evaluaions of he error funcion. Anoher undesirable feaure of using he smooh cuoff is ha he correcion erms become dependen on and no jus on τ as wih he sharp cuoff. References [] Berry, M.V. & Keaing, J.P. 99 A new asympoic represenaion for ζ + i and quanum specral deerminans. Proc. Roy. Soc. London, A7, No. 899, 5-7. [] Dunser, T.M., Paris, R.B. & Cang S. 998 On he high-order coefficiens in he uniform asympoic expansion for he incomplee gamma funcion. Mehods Appl. Anal. 5, -7. [] Gradsheyn I.S. & Ryzhik I.M. Tables of inegrals, series and producs. 6h edn, Academic Press. [] Kuznesov A. 7 Inegral represenaions for he Dirichle L-funcions and heir expansions in Meixner- Pollaczek polynomials and rising facorials. Inegral Transforms and Special Funcions o appear. [5] Mordell L.J. 9 The definie inegral -6. e a +b d and he analyic heory of numbers. Aca Mah. 6, e c +d [6] Odlyzko A.M. 99 Analyic compuaions in number heory. Mahemaics of Compuaion 9-99: A Half-Cenury of Compuaional Mahemaics, W. Gauschi ed., Amer. Mah. Soc., Proc. Symp. Appl. Mah. 8, pp [7] Paris R.B. 99 An asympoic represenaion for he Riemann zea funcion on he criical line. Proc. Roy. Soc. London, A6, No. 98, 99, [8] Paris R.B. & Cang S. 997 An asympoic represenaion for ζ + i. Mehods Appl. Anal.,,997, 9-7. [9] Ramanujan S. 95 Some definie inegrals conneced wih Gauss sums. Messenger of Mahemaics, XLIV, [] Temme N.M. 979 The asympoic expansion of he incomplee gamma funcions. SIAM J.Mah.Anal., [] Tichmarsh E.C. 986 The heory of he Riemann zea-funcion. nd edn, Oxford Universiy Press.
Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle
Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationSOLUTIONS TO ECE 3084
SOLUTIONS TO ECE 384 PROBLEM 2.. For each sysem below, specify wheher or no i is: (i) memoryless; (ii) causal; (iii) inverible; (iv) linear; (v) ime invarian; Explain your reasoning. If he propery is no
More informationVehicle Arrival Models : Headway
Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where
More informationdt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
More informationMATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence
MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories
More informationHarmonic oscillator in quantum mechanics
Harmonic oscillaor in quanum mechanics PHYS400, Deparmen of Physics, Universiy of onnecicu hp://www.phys.uconn.edu/phys400/ Las modified: May, 05 Dimensionless Schrödinger s equaion in quanum mechanics
More informationChapter 6. Systems of First Order Linear Differential Equations
Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh
More informationLAPLACE TRANSFORM AND TRANSFER FUNCTION
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions
More informationThe Asymptotic Behavior of Nonoscillatory Solutions of Some Nonlinear Dynamic Equations on Time Scales
Advances in Dynamical Sysems and Applicaions. ISSN 0973-5321 Volume 1 Number 1 (2006, pp. 103 112 c Research India Publicaions hp://www.ripublicaion.com/adsa.hm The Asympoic Behavior of Nonoscillaory Soluions
More informationMath 333 Problem Set #2 Solution 14 February 2003
Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial
More information( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationMATH 128A, SUMMER 2009, FINAL EXAM SOLUTION
MATH 28A, SUMME 2009, FINAL EXAM SOLUTION BENJAMIN JOHNSON () (8 poins) [Lagrange Inerpolaion] (a) (4 poins) Le f be a funcion defined a some real numbers x 0,..., x n. Give a defining equaion for he Lagrange
More informationd 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3
and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or
More informationMath 334 Fall 2011 Homework 11 Solutions
Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then
More information6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF s-shifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
More informationAverage Number of Lattice Points in a Disk
Average Number of Laice Poins in a Disk Sujay Jayakar Rober S. Sricharz Absrac The difference beween he number of laice poins in a disk of radius /π and he area of he disk /4π is equal o he error in he
More informationNotes for Lecture 17-18
U.C. Berkeley CS278: Compuaional Complexiy Handou N7-8 Professor Luca Trevisan April 3-8, 2008 Noes for Lecure 7-8 In hese wo lecures we prove he firs half of he PCP Theorem, he Amplificaion Lemma, up
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More informationINDEX. Transient analysis 1 Initial Conditions 1
INDEX Secion Page Transien analysis 1 Iniial Condiions 1 Please inform me of your opinion of he relaive emphasis of he review maerial by simply making commens on his page and sending i o me a: Frank Mera
More informationVariational Iteration Method for Solving System of Fractional Order Ordinary Differential Equations
IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 1, Issue 6 Ver. II (Nov - Dec. 214), PP 48-54 Variaional Ieraion Mehod for Solving Sysem of Fracional Order Ordinary Differenial
More information15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
More informationSMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15.
SMT Calculus Tes Soluions February 5,. Le f() = and le g() =. Compue f ()g (). Answer: 5 Soluion: We noe ha f () = and g () = 6. Then f ()g () =. Plugging in = we ge f ()g () = 6 = 3 5 = 5.. There is a
More informationChallenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k
Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationThe Miki-type identity for the Apostol-Bernoulli numbers
Annales Mahemaicae e Informaicae 46 6 pp. 97 4 hp://ami.ef.hu The Mii-ype ideniy for he Aposol-Bernoulli numbers Orli Herscovici, Toufi Mansour Deparmen of Mahemaics, Universiy of Haifa, 3498838 Haifa,
More informationSingle and Double Pendulum Models
Single and Double Pendulum Models Mah 596 Projec Summary Spring 2016 Jarod Har 1 Overview Differen ypes of pendulums are used o model many phenomena in various disciplines. In paricular, single and double
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationFinal Spring 2007
.615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o
More informationOrdinary differential equations. Phys 750 Lecture 7
Ordinary differenial equaions Phys 750 Lecure 7 Ordinary Differenial Equaions Mos physical laws are expressed as differenial equaions These come in hree flavours: iniial-value problems boundary-value problems
More information23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes
Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion
More informationA Note on the Equivalence of Fractional Relaxation Equations to Differential Equations with Varying Coefficients
mahemaics Aricle A Noe on he Equivalence of Fracional Relaxaion Equaions o Differenial Equaions wih Varying Coefficiens Francesco Mainardi Deparmen of Physics and Asronomy, Universiy of Bologna, and he
More informationGeneralized Chebyshev polynomials
Generalized Chebyshev polynomials Clemene Cesarano Faculy of Engineering, Inernaional Telemaic Universiy UNINETTUNO Corso Viorio Emanuele II, 39 86 Roma, Ialy email: c.cesarano@unineunouniversiy.ne ABSTRACT
More informationMatrix Versions of Some Refinements of the Arithmetic-Geometric Mean Inequality
Marix Versions of Some Refinemens of he Arihmeic-Geomeric Mean Inequaliy Bao Qi Feng and Andrew Tonge Absrac. We esablish marix versions of refinemens due o Alzer ], Carwrigh and Field 4], and Mercer 5]
More information2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
More informationDesigning Information Devices and Systems I Spring 2019 Lecture Notes Note 17
EES 16A Designing Informaion Devices and Sysems I Spring 019 Lecure Noes Noe 17 17.1 apaciive ouchscreen In he las noe, we saw ha a capacior consiss of wo pieces on conducive maerial separaed by a nonconducive
More informationChapter 7 Response of First-order RL and RC Circuits
Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial
More informationThe equation to any straight line can be expressed in the form:
Sring Graphs Par 1 Answers 1 TI-Nspire Invesigaion Suden min Aims Deermine a series of equaions of sraigh lines o form a paern similar o ha formed by he cables on he Jerusalem Chords Bridge. Deermine he
More informationRobotics I. April 11, The kinematics of a 3R spatial robot is specified by the Denavit-Hartenberg parameters in Tab. 1.
Roboics I April 11, 017 Exercise 1 he kinemaics of a 3R spaial robo is specified by he Denavi-Harenberg parameers in ab 1 i α i d i a i θ i 1 π/ L 1 0 1 0 0 L 3 0 0 L 3 3 able 1: able of DH parameers of
More informationNumerical Dispersion
eview of Linear Numerical Sabiliy Numerical Dispersion n he previous lecure, we considered he linear numerical sabiliy of boh advecion and diffusion erms when approimaed wih several spaial and emporal
More informationOrdinary dierential equations
Chaper 5 Ordinary dierenial equaions Conens 5.1 Iniial value problem........................... 31 5. Forward Euler's mehod......................... 3 5.3 Runge-Kua mehods.......................... 36
More informationChapter 3 Boundary Value Problem
Chaper 3 Boundary Value Problem A boundary value problem (BVP) is a problem, ypically an ODE or a PDE, which has values assigned on he physical boundary of he domain in which he problem is specified. Le
More informationSELBERG S CENTRAL LIMIT THEOREM ON THE CRITICAL LINE AND THE LERCH ZETA-FUNCTION. II
SELBERG S CENRAL LIMI HEOREM ON HE CRIICAL LINE AND HE LERCH ZEA-FUNCION. II ANDRIUS GRIGUIS Deparmen of Mahemaics Informaics Vilnius Universiy, Naugarduko 4 035 Vilnius, Lihuania rius.griguis@mif.vu.l
More information) were both constant and we brought them from under the integral.
YIELD-PER-RECRUIT (coninued The yield-per-recrui model applies o a cohor, bu we saw in he Age Disribuions lecure ha he properies of a cohor do no apply in general o a collecion of cohors, which is wha
More informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationMatlab and Python programming: how to get started
Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signals & Sysems Prof. Mark Fowler Noe Se #1 C-T Sysems: Convoluion Represenaion Reading Assignmen: Secion 2.6 of Kamen and Heck 1/11 Course Flow Diagram The arrows here show concepual flow beween
More informationSimulation-Solving Dynamic Models ABE 5646 Week 2, Spring 2010
Simulaion-Solving Dynamic Models ABE 5646 Week 2, Spring 2010 Week Descripion Reading Maerial 2 Compuer Simulaion of Dynamic Models Finie Difference, coninuous saes, discree ime Simple Mehods Euler Trapezoid
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and
More information8. Basic RL and RC Circuits
8. Basic L and C Circuis This chaper deals wih he soluions of he responses of L and C circuis The analysis of C and L circuis leads o a linear differenial equaion This chaper covers he following opics
More informationA New Perturbative Approach in Nonlinear Singularity Analysis
Journal of Mahemaics and Saisics 7 (: 49-54, ISSN 549-644 Science Publicaions A New Perurbaive Approach in Nonlinear Singulariy Analysis Ta-Leung Yee Deparmen of Mahemaics and Informaion Technology, The
More informationTHE BERNOULLI NUMBERS. t k. = lim. = lim = 1, d t B 1 = lim. 1+e t te t = lim t 0 (e t 1) 2. = lim = 1 2.
THE BERNOULLI NUMBERS The Bernoulli numbers are defined here by he exponenial generaing funcion ( e The firs one is easy o compue: (2 and (3 B 0 lim 0 e lim, 0 e ( d B lim 0 d e +e e lim 0 (e 2 lim 0 2(e
More informationCHAPTER 2 Signals And Spectra
CHAPER Signals And Specra Properies of Signals and Noise In communicaion sysems he received waveform is usually caegorized ino he desired par conaining he informaion, and he undesired par. he desired par
More informationPENALIZED LEAST SQUARES AND PENALIZED LIKELIHOOD
PENALIZED LEAST SQUARES AND PENALIZED LIKELIHOOD HAN XIAO 1. Penalized Leas Squares Lasso solves he following opimizaion problem, ˆβ lasso = arg max β R p+1 1 N y i β 0 N x ij β j β j (1.1) for some 0.
More informationProperties Of Solutions To A Generalized Liénard Equation With Forcing Term
Applied Mahemaics E-Noes, 8(28), 4-44 c ISSN 67-25 Available free a mirror sies of hp://www.mah.nhu.edu.w/ amen/ Properies Of Soluions To A Generalized Liénard Equaion Wih Forcing Term Allan Kroopnick
More informationChapter 8 The Complete Response of RL and RC Circuits
Chaper 8 The Complee Response of RL and RC Circuis Seoul Naional Universiy Deparmen of Elecrical and Compuer Engineering Wha is Firs Order Circuis? Circuis ha conain only one inducor or only one capacior
More informationSTATE-SPACE MODELLING. A mass balance across the tank gives:
B. Lennox and N.F. Thornhill, 9, Sae Space Modelling, IChemE Process Managemen and Conrol Subjec Group Newsleer STE-SPACE MODELLING Inroducion: Over he pas decade or so here has been an ever increasing
More informationWeyl sequences: Asymptotic distributions of the partition lengths
ACTA ARITHMETICA LXXXVIII.4 (999 Weyl sequences: Asympoic disribuions of he pariion lenghs by Anaoly Zhigljavsky (Cardiff and Iskander Aliev (Warszawa. Inroducion: Saemen of he problem and formulaion of
More informationdy dx = xey (a) y(0) = 2 (b) y(1) = 2.5 SOLUTION: See next page
Assignmen 1 MATH 2270 SOLUTION Please wrie ou complee soluions for each of he following 6 problems (one more will sill be added). You may, of course, consul wih your classmaes, he exbook or oher resources,
More informationUndetermined coefficients for local fractional differential equations
Available online a www.isr-publicaions.com/jmcs J. Mah. Compuer Sci. 16 (2016), 140 146 Research Aricle Undeermined coefficiens for local fracional differenial equaions Roshdi Khalil a,, Mohammed Al Horani
More informationKEY. Math 334 Midterm III Winter 2008 section 002 Instructor: Scott Glasgow
KEY Mah 334 Miderm III Winer 008 secion 00 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering
More informationNature Neuroscience: doi: /nn Supplementary Figure 1. Spike-count autocorrelations in time.
Supplemenary Figure 1 Spike-coun auocorrelaions in ime. Normalized auocorrelaion marices are shown for each area in a daase. The marix shows he mean correlaion of he spike coun in each ime bin wih he spike
More informationChapter 7: Solving Trig Equations
Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions
More informationOn some Properties of Conjugate Fourier-Stieltjes Series
Bullein of TICMI ol. 8, No., 24, 22 29 On some Properies of Conjugae Fourier-Sieljes Series Shalva Zviadadze I. Javakhishvili Tbilisi Sae Universiy, 3 Universiy S., 86, Tbilisi, Georgia (Received January
More informationSolutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
More informationarxiv:math/ v1 [math.nt] 3 Nov 2005
arxiv:mah/0511092v1 [mah.nt] 3 Nov 2005 A NOTE ON S AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION D. A. GOLDSTON AND S. M. GONEK Absrac. Le πs denoe he argumen of he Riemann zea-funcion a he poin 1 + i. Assuming
More informationShort Introduction to Fractional Calculus
. Shor Inroducion o Fracional Calculus Mauro Bologna Deparameno de Física, Faculad de Ciencias Universidad de Tarapacá, Arica, Chile email: mbologna@ua.cl Absrac In he pas few years fracional calculus
More informationL07. KALMAN FILTERING FOR NON-LINEAR SYSTEMS. NA568 Mobile Robotics: Methods & Algorithms
L07. KALMAN FILTERING FOR NON-LINEAR SYSTEMS NA568 Mobile Roboics: Mehods & Algorihms Today s Topic Quick review on (Linear) Kalman Filer Kalman Filering for Non-Linear Sysems Exended Kalman Filer (EKF)
More informationImproved Approximate Solutions for Nonlinear Evolutions Equations in Mathematical Physics Using the Reduced Differential Transform Method
Journal of Applied Mahemaics & Bioinformaics, vol., no., 01, 1-14 ISSN: 179-660 (prin), 179-699 (online) Scienpress Ld, 01 Improved Approimae Soluions for Nonlinear Evoluions Equaions in Mahemaical Physics
More informationMorning Time: 1 hour 30 minutes Additional materials (enclosed):
ADVANCED GCE 78/0 MATHEMATICS (MEI) Differenial Equaions THURSDAY JANUARY 008 Morning Time: hour 30 minues Addiional maerials (enclosed): None Addiional maerials (required): Answer Bookle (8 pages) Graph
More information1 Review of Zero-Sum Games
COS 5: heoreical Machine Learning Lecurer: Rob Schapire Lecure #23 Scribe: Eugene Brevdo April 30, 2008 Review of Zero-Sum Games Las ime we inroduced a mahemaical model for wo player zero-sum games. Any
More informationMA 214 Calculus IV (Spring 2016) Section 2. Homework Assignment 1 Solutions
MA 14 Calculus IV (Spring 016) Secion Homework Assignmen 1 Soluions 1 Boyce and DiPrima, p 40, Problem 10 (c) Soluion: In sandard form he given firs-order linear ODE is: An inegraing facor is given by
More informationMath 315: Linear Algebra Solutions to Assignment 6
Mah 35: Linear Algebra s o Assignmen 6 # Which of he following ses of vecors are bases for R 2? {2,, 3, }, {4,, 7, 8}, {,,, 3}, {3, 9, 4, 2}. Explain your answer. To generae he whole R 2, wo linearly independen
More informationMATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.
MATH 3B: MIDTERM REVIEW JOE HUGHES. Inegraion by Pars. Evaluae 3 e. Soluion: Firs make he subsiuion u =. Then =, hence 3 e = e = ue u Now inegrae by pars o ge ue u = ue u e u + C and subsiue he definiion
More informationA Shooting Method for A Node Generation Algorithm
A Shooing Mehod for A Node Generaion Algorihm Hiroaki Nishikawa W.M.Keck Foundaion Laboraory for Compuaional Fluid Dynamics Deparmen of Aerospace Engineering, Universiy of Michigan, Ann Arbor, Michigan
More informationAn Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
More informationWEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x
WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile
More informationOn a Fractional Stochastic Landau-Ginzburg Equation
Applied Mahemaical Sciences, Vol. 4, 1, no. 7, 317-35 On a Fracional Sochasic Landau-Ginzburg Equaion Nguyen Tien Dung Deparmen of Mahemaics, FPT Universiy 15B Pham Hung Sree, Hanoi, Vienam dungn@fp.edu.vn
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationChapter 4. Truncation Errors
Chaper 4. Truncaion Errors and he Taylor Series Truncaion Errors and he Taylor Series Non-elemenary funcions such as rigonomeric, eponenial, and ohers are epressed in an approimae fashion using Taylor
More information20. Applications of the Genetic-Drift Model
0. Applicaions of he Geneic-Drif Model 1) Deermining he probabiliy of forming any paricular combinaion of genoypes in he nex generaion: Example: If he parenal allele frequencies are p 0 = 0.35 and q 0
More informationThe complex Fourier series has an important limiting form when the period approaches infinity, i.e., T 0. 0 since it is proportional to 1/L, but
Fourier Transforms The complex Fourier series has an imporan limiing form when he period approaches infiniy, i.e., T or L. Suppose ha in his limi () k = nπ L remains large (ranging from o ) and (2) c n
More informationA NOTE ON S(t) AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION
Bull. London Mah. Soc. 39 2007 482 486 C 2007 London Mahemaical Sociey doi:10.1112/blms/bdm032 A NOTE ON S AND THE ZEROS OF THE RIEMANN ZETA-FUNCTION D. A. GOLDSTON and S. M. GONEK Absrac Le πs denoe he
More informationThe following report makes use of the process from Chapter 2 in Dr. Cumming s thesis.
Zaleski 1 Joseph Zaleski Mah 451H Final Repor Conformal Mapping Mehods and ZST Hele Shaw Flow Inroducion The Hele Shaw problem has been sudied using linear sabiliy analysis and numerical mehods, bu a novel
More information5.1 - Logarithms and Their Properties
Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We
More informationLecture 3: Fourier transforms and Poisson summation
Mah 726: L-funcions and modular forms Fall 2 Lecure 3: Fourier ransforms and Poisson summaion Insrucor: Henri Darmon Noes wrien by: Luca Candelori In he las lecure we showed how o derive he funcional equaion
More informationTHE SINE INTEGRAL. x dt t
THE SINE INTEGRAL As one learns in elemenary calculus, he limi of sin(/ as vanishes is uniy. Furhermore he funcion is even and has an infinie number of zeros locaed a ±n for n1,,3 Is plo looks like his-
More information= ( ) ) or a system of differential equations with continuous parametrization (T = R
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More information14 Autoregressive Moving Average Models
14 Auoregressive Moving Average Models In his chaper an imporan parameric family of saionary ime series is inroduced, he family of he auoregressive moving average, or ARMA, processes. For a large class
More informationReading from Young & Freedman: For this topic, read sections 25.4 & 25.5, the introduction to chapter 26 and sections 26.1 to 26.2 & 26.4.
PHY1 Elecriciy Topic 7 (Lecures 1 & 11) Elecric Circuis n his opic, we will cover: 1) Elecromoive Force (EMF) ) Series and parallel resisor combinaions 3) Kirchhoff s rules for circuis 4) Time dependence
More information( ) a system of differential equations with continuous parametrization ( T = R + These look like, respectively:
XIII. DIFFERENCE AND DIFFERENTIAL EQUATIONS Ofen funcions, or a sysem of funcion, are paramerized in erms of some variable, usually denoed as and inerpreed as ime. The variable is wrien as a funcion of
More informationPOSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION
Novi Sad J. Mah. Vol. 32, No. 2, 2002, 95-108 95 POSITIVE SOLUTIONS OF NEUTRAL DELAY DIFFERENTIAL EQUATION Hajnalka Péics 1, János Karsai 2 Absrac. We consider he scalar nonauonomous neural delay differenial
More informationHamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:
M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationarxiv:quant-ph/ v1 5 Jul 2004
Numerical Mehods for Sochasic Differenial Equaions Joshua Wilkie Deparmen of Chemisry, Simon Fraser Universiy, Burnaby, Briish Columbia V5A 1S6, Canada Sochasic differenial equaions (sdes) play an imporan
More informationBoundedness and Exponential Asymptotic Stability in Dynamical Systems with Applications to Nonlinear Differential Equations with Unbounded Terms
Advances in Dynamical Sysems and Applicaions. ISSN 0973-531 Volume Number 1 007, pp. 107 11 Research India Publicaions hp://www.ripublicaion.com/adsa.hm Boundedness and Exponenial Asympoic Sabiliy in Dynamical
More information