FRACTIONAL ORNSTEIN-UHLENBECK PROCESSES
|
|
- Richard Neal
- 5 years ago
- Views:
Transcription
1 FRACTIONAL ORNSTEIN-ULENBECK PROCESSES Prick Cheridio Deprmen of Mhemics, ET Zürich C-89 Zürich, Swizerlnd ideyuki Kwguchi Deprmen of Mhemics, Keio Universiy iyoshi, Yokohm 3-85, Jpn Mkoo Mejim Deprmen of Mhemics, Keio Universiy iyoshi, Yokohm 3-85, Jpn Absrc The clssicl sionry Ornsein-Uhlenbeck process cn be obined in wo differen wys. On he one hnd, i is sionry soluion of he Lngevin equion wih Brownin moion noise. On he oher hnd, i cn be obined from Brownin moion by he so clled Lmperi rnsformion. We show h he Lngevin equion wih frcionl Brownin moion noise lso hs sionry soluion nd h he decy of is uo-covrince funcion is like h of power funcion. Conrry o h, he sionry process obined from frcionl Brownin moion by he Lmperi rnsformion hs n uo-covrince funcion h decys eponenilly. Keywords Frcionl Brownin moion, Lngevin equion, long-rnge dependence, selfsimilr processes, Lmperi rnsformion AMS subjec clssificion primry: 6; secondry: 6G5, 6G8, 45F5 Inroducion Le Ω, F, P be probbiliy spce. Definiion. A frcionl Brownin moion wih urs prmeer,, is n lmos surely coninuous, cenered Gussin process B R wih Cov B, B s = + s s,, s R.. The firs uhor would like o hnk he Deprmen of Mhemics, Keio Universiy, for hving mde possible his sy in Yokohm. Curren ddress: Risk Mngemen Sysems Deprmen, Sumiomo Misui Bnking Corporion, -3-, Mrunouchi, Chiyod, Tokyo -5, Jpn; kwguchi hideyuki@dn.smbc.co.jp
2 For n in-deph inroducion o frcionl Brownin moions we refer he reder o Secion 7. of Smorodnisky nd Tqqu 994 or Chper 4 of Embrechs nd Mejim. I is cler h for ll,, B = lmos surely. Moreover, i cn be deduced from. h for ll,, B R hs sionry incremens nd is -selfsimilr, h is, for every c >, B c R d = c B R, where d = denoes equliy of ll finie-dimensionl disribuions. B R is wo-sided Brownin moion. In priculr, i hs independen incremens. For,,, B is no semimringle nd i cn be derived from. h for ll h R nd < < s, Cov Bh+ B h, B h+s+ h+s B = Cov B, Bs+ Bs = n= n n! n k s n, k= in priculr, for every N =,,..., for ll h R nd >, Cov Bh+ B h, B h+s+ B h+s N n n = k s n + Os N, s s.. n! n= k= This shows h for,, n= Cov B, Bn+ B n =, phenomenon referred o s long-rnge dependence or long memory of he incremens process Bn+ B n n=. The clssicl Ornsein-Uhlenbeck process wih prmeers λ > nd σ > sring R, is he unique srong soluion of he Lngevin equion wih Brownin moion noise X = ξ λ X s ds + σb,,.3 wih iniil condiion ξ =. I is given by he lmos surely coninuous Gussin Mrkov process Y, := e λ + σ e λu db u,. The unique srong soluion of.3 wih iniil condiion ξ = σ e λu db u, is given by he resricion o non-negive s of he sionry, lmos surely coninuous, cenered Gussin Mrkov process Y := σ e λ u db u, R.
3 I cn esily be checked h Cov Y, Y +s = σ λ e λ s,, s R. This implies h Y R is ergodic. Moreover, for ll R, Y Y, = e λ Y, s, lmos surely. From his i cn be derived h if Y is sionry process h solves.3 wih ny iniil condiion ξ L d Ω, hen Y = Y. Now le α >. Then, Z := e λ B, R, αe λ is lso sionry, lmos surely coninuous, cenered Gussin process, nd Cov Z, Z +s = αe λ s,, s R. ence, for α = σ λ, Y R = d Z R. I is shown in Lmperi 96 h for every >, sochsic process X is - selfsimilr if nd only if for ll λ, α >, he process X = e λ X α ep λ, R,.4 is sionry. We cll.4 he Lmperi rnsformion from selfsimilr processes o sionry processes nd X R he Lmperi rnsform of X. For =, frcionl Brownin moion cn be represened s follows: B = η, R, where η is sndrd norml rndom vrible. equion, X = ξ λ cn ph-wise be reduced o he ordinry differenil equions, For every iniil condiion ξ L Ω, he X s ds + σb,,.5 X ω = λx ω + σηω, ω Ω, wih iniil condiions which hve he unique soluions X ω = ξω, ω Ω, Y,ξ ω := e λ { ξω σ λ ηω } + σ λ ηω,, ω Ω. Equion.5 hs only sionry soluion for he iniil condiion ξ = σ λη. I is given by Y := σ λ η,, 3
4 which, for ll, equls he Lmperi rnsform Z := e λ B α epλ = αη, R, if α = σ λ. This leds us o he quesion wheher for,,, he Lngevin equion wih noise process σb hs sionry soluion, if is disribuion is unique nd if i is equl in some sense o he Lmperi rnsform Z := e λ B α ep λ, R, for n ppropriely chosen α >. The srucure of he pper is s follows. In Secion we show h for ll,, he Lngevin equion wih frcionl Brownin moion noise hs for ll iniil condiions ξ L Ω, unique srong soluion Y,ξ. Moreover, here eiss sionry, lmos surely coninuous, cenered Gussin process Y R such h Y solves he Lngevin equion wih frcionl Brownin moion noise, nd every oher sionry soluion is equl o Y in disribuion. The decy of he uo-covrince funcion of Y R is for ll,, similr o h of he incremens of B R see.. In priculr, Y R is ergodic, nd for,, i ehibis long-rnge dependence. In Secion 3 we show h for ll, he uo-covrince funcion of Z R decys eponenilly, which implies h for,,, Y R cnno hve he sme disribuion s Z R. Frcionl Ornsein-Uhlenbeck processes Le λ, σ > nd ξ L Ω. Since he Lngevin equion, X = ξ λ X s ds + N,, only involves n inegrl wih respec o, i cn be solved ph-wise for much more generl noise processes N hn Brownin moion. For emple, i follows from Proposiion A. h for ech, nd for every [,, e λu db u, >, eiss s ph-wise Riemnn-Sieljes inegrl, which is lmos surely coninuous in, nd Y,ξ := e ξ λ + σ e λu dbu,, is he unique lmos surely coninuous process h solves he equion, X = ξ λ X s ds + σb,.. 4
5 In priculr, he resricion o posiive s of he lmos surely coninuous process Y := σ solves. wih iniil condiion ξ = Y e λ u db u, R,. I is cler h Y R is Gussin process, nd i follows immediely from he sionriy of he incremens of frcionl Brownin moion h i is sionry. Furhermore, s in he Brownin moion cse, for every ξ L Ω, Y Y,ξ = e λ Y ξ, s, lmos surely, which implies h every sionry soluion of. hs he sme disribuion s Y cll Y,ξ sionry frcionl Ornsein-Uhlenbeck process. frcionl Ornsein-Uhlenbeck process wih iniil condiion ξ nd Y. We R In Pipirs nd Tqqu i is shown h for, nd wo rel-vlued mesurble funcions { } f, g f : fu fv u v dudv <, he wo inegrls fudb u, gudb u cn in consisen wy be defined s limis of inegrls of elemenry funcions, nd [ E fudbu gudbu = fugv u v dudv. For,, he kernel u v cnno be inegred over he digonl. owever, if f nd g re regulr enough nd he inersecion of heir suppors is of Lebesgue mesure zero, he sme holds rue. We will only need his resul for he cse where f nd g re given by fu = gu = e λu nd heir suppors re disjoin inervls. owever, he following lemm cn esily be generlized. Lemm. Le,,, λ > nd < b c < d <. Then [ b d b d E e λu dbu e λv dbv = e λu e λv v u dv du. c Proof. We firs ssume b = = c. By Proposiion A. we ge [ d E e λu dbu e λv dbv [ d = E e λ B λ e λu Bu du e λd B d λ e λv Bv dv = eλ e λd [ + d d + d λeλ e λv [ + v v dv λeλd e λu [ u + d d u du + d λ e λv e λu [ u + v v u du dv. 5 c
6 Afer pril inegrion wih respec o u, his becomes e λd e λu [ u d u du d +λ e λv e λu [ u v u du dv, which, by pril inegrion wih respec o v, is equl o e λu d e λv v u dv du. Now we ssume b = < c. I follows from bove h [ d [ d c E e λu dbu e λv dbv = E e λu dbu e λv dbv e λu dbu e λv dbv c [ d c = e λu e λv v u dv du e λu e λv v u dv du = e λu d c e λv v u dv du. For generl < b c < d <, he process B = B+b B b frcionl Brownin moion. Therefore, [ b E e λu dbu = = b b [ e λv dbv = E e λw+b d B w c b d b e λw+b e λ+b w d d c b d e λu e λv v u dv du, c d b c b dw, R, is gin e λ+b d B nd he proof is complee. Lemm. Le β <. Then for ech N =,,,..., N n e e y y β dy = β + β k β n + O β N, s, n= k= nd e e y y β dy = β + N n n β k β n + O β N, s, n= k= where n= mens. 6
7 Proof. We hve nd e e y y β dy = e e β + β e y y β dy =... = β + β β + ββ β ββ... β N + β N +e ββ... β N e y y β N dy, e e y y β N dy e e y β N dy = β N, which proves he firs sserion. On he oher hnd, nd e e y y β dy = e e β e β e y y β dy =... = β β β N ββ... β N + β N e e { β N ββ... β N + } e N ββ... β N e e y y β N dy e e y dy + e y y β N dy, e y β N dy e β N +. This proves he second pr of he lemm. Theorem.3 Le,, nd N =,,.... Then for fied R nd s, Cov Y, Y+s N n = λ n k s n + Os N. Proof. By Lemm., = E Cov Y [ σ = e λs E σ n=, Y+s = Cov Y [ σ e λu db u σ s e λu db u σ +σ e λs, Ys k= e λs v dbv λ e λv db v s e λu e λv v u dv du λ 7
8 by he chnge of vribles: w = λu, = λv λs = σ e λs e w e w d dw + Oe λs λ by he chnge of vribles: y = w, z = + w σ { λs y = e λs y e z dz dy λ y λs y } + y e z dz dy + Oe λs = = λs σ y e λs λ { λs e y y dy + λs σ λs {e λs λ e λs y y dy e y y dy + e λs The proof cn now be concluded by pplying Lemm.. Theorem.3 shows h for,,, he decy of is very similr o he decy of λs Cov Y, Y+s, for s, Cov B h+ B h, B h+s+ B h+s, for s } e y y dy + Oe λs } e y y dy + Oe λs. see.. In priculr, Y R is ergodic, nd for,, i ehibis long-rnge dependence. Remrk.4 Le s R. For ll,, he funcions f = { } e λ nd g = { s} e λ belong o he inner produc spce Λ defined on pge 89 of Pipirs nd Tqqu. ence, for ll, s R, Cov Y, Y+s is equl o σ e λs Γ + sinπ f, = σ g Λ π e is λ d.. + Therefore, he epression given in he he semen of Theorem.3 is n sympoic epnsion of he righ hnd side in. s s. The ne corollry shows h for he soluion Y, of. wih deerminisic iniil vlue Y, = R, Cov Y,, Y, +s, for s, decys like power funcion of he order s well. Corollry.5 Le,,, R nd N =,,.... Then for fied nd s, Cov Y,, Y, +s = N n { σ λ n k s n e λ + s n} + Os N. n= k= 8
9 Proof. Cov [ = E σ [ = E σ [ = E σ Y, e λs E, Y, +s e λ u db u σ e λ u db u σ +s +s e λ u dbu e λ [ σ e λ u dbu σ e λ+s v dbv e λ+s v dbv e λs e λ v dbv +s e λu dbu e λ+s v dbv e λ v dbv = Cov Y, Y+s e λ Cov Y, Y+s + Oe λs. Now, he corollry follows from Theorem.3. 3 The Lmperi rnsform of frcionl Brownin moion Le λ > nd α >. For ech,, we se Z := e λ B α ep λ, R. Theorem 3. Le, nd, s R. Then Cov { Z, Z+s α = e λ s + n n n= e λ n s }. 3. Proof. Wihou loss of generliy we cn ssume h s. Then, Cov Z, Z+s = e λ λ+s α e { = α eλs = α eλs = α { e λ+s + e λ e λ +s e λ } + e λs e λ s } { } + e λs e λ s n n n= { } e λs + e n λ n s, n n= which proves he heorem. I follows from Theorem 3. h for every N =,,..., for ech, nd ll R, Cov { Z, Z+s α = e λ s + N } e n λ n s n n= N+ + O e λ s, 9
10 s s. This shows h for ll,, he uo-covrince funcion of Z R decys eponenilly. I follows h for,,, Z R cnno hve he sme disribuion s Y R. For,, he leding erm in 3. for s, is α e λ s, wheres for,, i is α e λ s. Noe h for,, he leding erm of Cov Z, Z+s for s, is posiive, wheres he leding erm of Cov Y, Y+s for s, is negive see Theorem.3. Appendi: The Lngevin equion Lngevin 98 pioneered he following pproch o he movemen of free pricle immersed in liquid: e described he pricle s velociy v by he equion of moion dv d = f F v + m m A. where m > is he mss of he pricle, f > fricion coefficien nd F flucuing force resuling from impcs of he molecules of he surrounding medium. Uhlenbeck nd Ornsein 93 imposed probbiliy hypoheses on F nd hen derived h for v = R, v is e λ, for λ = f m nd σ = fkt m, normlly disribued wih men e λ nd vrince σ λ where k is he Bolzmnn consn nd T he emperure. Doob 94 noiced h if v is rndom vrible which is independen of F nd normlly disribued wih men zero nd vrince σ λ, hen he soluion v of A. is sionry nd {>} v λ ln,, is Brownin moion, from which he concluded h every soluion of A. hs lmos surely coninuous phs which re nowhere differenible. To void he embrrssing siuion h he equion A. involves he derivive of v bu leds o soluions v h do no hve derivive, he gve rigorous mening o sochsic differenil equions of he form dx = λx d + dn, A. for he cse h N is Lévy process nd showed h for ll R, he equion A. wih iniil condiion X = R, hs he unique soluion X = e λ + e λu dn u,. In he modern heory of sochsic differenil equions see e.g. Proer 99 he equion A. wih iniil condiion X = ξ L Ω is undersood s he inegrl equion X = ξ λ X s ds + N,, A.3
11 nd i cn be shown h he unique srong soluion of A.3 is given by X ξ := e λ ξ + e λu dn u,, whenever N is semimringle wih respec o he filrion genered by N nd ξ. Proposiion A. Le B R be frcionl Brownin moion wih urs prmeer, nd ξ L Ω. Le < nd λ, σ >. Then for lmos ll ω Ω, we hve he following: For ll >, e λu db u ω eiss s Riemnn-Sieljes inegrl nd is equl o b The funcion e λ B ω e λ B ω λ e λu db u ω, >, is coninuous in. c The unique coninuous funcion y h solves he equion, is given by y = ξω λ y = e λ {ξω + σ In priculr, he unique coninuous soluion of he equion, B u ωe λu du. ysds + σb ω,. A.4 y = σ e λu dbu ω λ } e λu dbu ω,. A.5 ysds + σb ω,, is given by y = σ e λ u dbu ω,. Proof. I cn esily be checked h B s := {s<} s B s + {s>} s B s, s R, is gin frcionl Brownin moion. I follows from he Kolmogorov-Čensov heorem see e.g. Theorem..8 of Krzs nd Shreve 99 h here eiss mesurble null se
12 N Ω, such h for every ω Ω \ N, Bs ω nd B s ω re coninuous in s, nd for ll β <, B s ω lim s s β =. This implies h for ll γ >, ence, for ll >, Bs ω lim s s γ =. B u ωe λu du eiss s Riemnn inegrl, which, by Theorem. of Wheeden nd Zygmund 977, implies h he Riemnn-Sieljes inegrl eiss oo nd is equl o e λu db u ω e λ B ω e λ B ω λ This proves. b follows from nd he fc h he funcion e λ B ω λ B u ωe λu du. B u ωe λu du, >, is coninuous in. A coninuous funcion y solves A.4 if nd only if he funcion z = solves he liner differenil equion: Since he unique soluion of A.6 is given by ysds,, z = λz + ξω + σb ω, z =. A.6 z = e λ e λu ξω + σbu ω du,, he unique coninuous funcion y h solves A.4 is given by λe λ e λu ξω + σbu ω du + ξω + σb ω,, which, by, is equl o A.5. This shows c.
13 Remrk A. Equion A.3 cn be solved ph-wise for ll sochsic processes N h hve lmos ll phs in { } L loc R + := h : R + R : h is mesurble nd, hs ds <, nd even when he consn λ is replced by sochsic process wih lmos ll phs in { } L loc R + := g : R + R : g is mesurble nd, sup gs <. s Indeed, if h L loc R + nd g L loc R +, hen i cn esily be checked h he funcion f := h + is in L loc R + nd solves he inegrl equion f = On he oher hnd, if f L loc R + is soluion of A.8, hen f f = gser s gudu hsds,, A.7 gsfsds + h,. A.8 nd i follows from vrin of Gronwll s lemm h [ gs f f ds,, f f =,. ence, A.7 is he only funcion in L loc R + h solves A.8. If he funcions g nd h re boh in L loc R + nd coninuous on R + \ C, where C is of Lebesgue mesure zero, hen i cn be deduced from Theorems 5.54 nd. of Wheeden nd Zygmund 977 h f cn be wrien s follows: R f = e gudu h + e R s gudu dhs,, where e R s gudu dhs is Riemnn-Sieljes inegrl. Noe h lmos ll phs of semimringle re righ-coninuous nd hve lef limis, in priculr, hey re in L loc R + nd hve mos counbly mny disconinuiies. References Doob, J.L. 94, The Brownin movemen nd sochsic equions, Ann. of Mh. 43, Embrechs, P. nd Mejim, M., Selfsimilr Processes, Princeon Series in Applied Mhemics, Princeon Universiy Press. 3
14 Krzs, I. nd Shreve, S.E. 99, Brownin Moion nd Sochsic Clculus, Grdue Tes in Mhemics, Springer-Verlg, New York. Lmperi, J.W. 96, Semi-sble sochsic processes, Trns. Amer. Mh. Soc. 4, Lngevin, P. 98, Sur l héorie du mouvemen brownien, C.R. Acd. Sci. Pris 46, Pipirs, V. nd Tqqu, M., Inegrion quesions reled o frcionl Brownin moion, Prob. Th. Rel. Fields 8, -9. Proer, P. 99, Sochsic Inegrion nd Differenil Equions, Springer-Verlg, Berlin. Smorodnisky, G. nd Tqqu, M.S. 994, Sble Non-Gussin Rndom Processes, Chpmn & ll, New York. Uhlenbeck, G.E. nd Ornsein, L.S. 93, On he heory of he Brownin moion, Physicl Review 36, Wheeden, R.L. nd Zygmund, A. 977, Mesure nd Inegrl, Mrcel Dekker, New York- Bsel. 4
Contraction Mapping Principle Approach to Differential Equations
epl Journl of Science echnology 0 (009) 49-53 Conrcion pping Principle pproch o Differenil Equions Bishnu P. Dhungn Deprmen of hemics, hendr Rn Cmpus ribhuvn Universiy, Khmu epl bsrc Using n eension of
More informationREAL ANALYSIS I HOMEWORK 3. Chapter 1
REAL ANALYSIS I HOMEWORK 3 CİHAN BAHRAN The quesions re from Sein nd Shkrchi s e. Chper 1 18. Prove he following sserion: Every mesurble funcion is he limi.e. of sequence of coninuous funcions. We firs
More informationENGR 1990 Engineering Mathematics The Integral of a Function as a Function
ENGR 1990 Engineering Mhemics The Inegrl of Funcion s Funcion Previously, we lerned how o esime he inegrl of funcion f( ) over some inervl y dding he res of finie se of rpezoids h represen he re under
More information3. Renewal Limit Theorems
Virul Lborories > 14. Renewl Processes > 1 2 3 3. Renewl Limi Theorems In he inroducion o renewl processes, we noed h he rrivl ime process nd he couning process re inverses, in sens The rrivl ime process
More informatione t dt e t dt = lim e t dt T (1 e T ) = 1
Improper Inegrls There re wo ypes of improper inegrls - hose wih infinie limis of inegrion, nd hose wih inegrnds h pproch some poin wihin he limis of inegrion. Firs we will consider inegrls wih infinie
More informationINTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q).
INTEGRALS JOHN QUIGG Eercise. Le f : [, b] R be bounded, nd le P nd Q be priions of [, b]. Prove h if P Q hen U(P ) U(Q) nd L(P ) L(Q). Soluion: Le P = {,..., n }. Since Q is obined from P by dding finiely
More information5.1-The Initial-Value Problems For Ordinary Differential Equations
5.-The Iniil-Vlue Problems For Ordinry Differenil Equions Consider solving iniil-vlue problems for ordinry differenil equions: (*) y f, y, b, y. If we know he generl soluion y of he ordinry differenil
More informationf t f a f x dx By Lin McMullin f x dx= f b f a. 2
Accumulion: Thoughs On () By Lin McMullin f f f d = + The gols of he AP* Clculus progrm include he semen, Sudens should undersnd he definie inegrl s he ne ccumulion of chnge. 1 The Topicl Ouline includes
More informationConvergence of Singular Integral Operators in Weighted Lebesgue Spaces
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 2, 2017, 335-347 ISSN 1307-5543 www.ejpm.com Published by New York Business Globl Convergence of Singulr Inegrl Operors in Weighed Lebesgue
More informationProbability, Estimators, and Stationarity
Chper Probbiliy, Esimors, nd Sionriy Consider signl genered by dynmicl process, R, R. Considering s funcion of ime, we re opering in he ime domin. A fundmenl wy o chrcerize he dynmics using he ime domin
More informationHermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals
Sud. Univ. Beş-Bolyi Mh. 6(5, No. 3, 355 366 Hermie-Hdmrd-Fejér ype inequliies for convex funcions vi frcionl inegrls İmd İşcn Asrc. In his pper, firsly we hve eslished Hermie Hdmrd-Fejér inequliy for
More informationGreen s Functions and Comparison Theorems for Differential Equations on Measure Chains
Green s Funcions nd Comprison Theorems for Differenil Equions on Mesure Chins Lynn Erbe nd Alln Peerson Deprmen of Mhemics nd Sisics, Universiy of Nebrsk-Lincoln Lincoln,NE 68588-0323 lerbe@@mh.unl.edu
More informationA LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR IAN KNOWLES
A LIMIT-POINT CRITERION FOR A SECOND-ORDER LINEAR DIFFERENTIAL OPERATOR j IAN KNOWLES 1. Inroducion Consider he forml differenil operor T defined by el, (1) where he funcion q{) is rel-vlued nd loclly
More informationSeptember 20 Homework Solutions
College of Engineering nd Compuer Science Mechnicl Engineering Deprmen Mechnicl Engineering A Seminr in Engineering Anlysis Fll 7 Number 66 Insrucor: Lrry Creo Sepember Homework Soluions Find he specrum
More informationGENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS
- TAMKANG JOURNAL OF MATHEMATICS Volume 5, Number, 7-5, June doi:5556/jkjm555 Avilble online hp://journlsmhkueduw/ - - - GENERALIZATION OF SOME INEQUALITIES VIA RIEMANN-LIOUVILLE FRACTIONAL CALCULUS MARCELA
More information4.8 Improper Integrals
4.8 Improper Inegrls Well you ve mde i hrough ll he inegrion echniques. Congrs! Unforunely for us, we sill need o cover one more inegrl. They re clled Improper Inegrls. A his poin, we ve only del wih inegrls
More informationON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX
Journl of Applied Mhemics, Sisics nd Informics JAMSI), 9 ), No. ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX MEHMET ZEKI SARIKAYA, ERHAN. SET
More informationMTH 146 Class 11 Notes
8.- Are of Surfce of Revoluion MTH 6 Clss Noes Suppose we wish o revolve curve C round n is nd find he surfce re of he resuling solid. Suppose f( ) is nonnegive funcion wih coninuous firs derivive on he
More informationIntegral Transform. Definitions. Function Space. Linear Mapping. Integral Transform
Inegrl Trnsform Definiions Funcion Spce funcion spce A funcion spce is liner spce of funcions defined on he sme domins & rnges. Liner Mpping liner mpping Le VF, WF e liner spces over he field F. A mpping
More informationA Kalman filtering simulation
A Klmn filering simulion The performnce of Klmn filering hs been esed on he bsis of wo differen dynmicl models, ssuming eiher moion wih consn elociy or wih consn ccelerion. The former is epeced o beer
More information0 for t < 0 1 for t > 0
8.0 Sep nd del funcions Auhor: Jeremy Orloff The uni Sep Funcion We define he uni sep funcion by u() = 0 for < 0 for > 0 I is clled he uni sep funcion becuse i kes uni sep = 0. I is someimes clled he Heviside
More informationEXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE BOUNDARY-VALUE PROBLEM
Elecronic Journl of Differenil Equions, Vol. 208 (208), No. 50, pp. 6. ISSN: 072-669. URL: hp://ejde.mh.xse.edu or hp://ejde.mh.un.edu EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR A SECOND-ORDER ITERATIVE
More informationMathematics 805 Final Examination Answers
. 5 poins Se he Weiersrss M-es. Mhemics 85 Finl Eminion Answers Answer: Suppose h A R, nd f n : A R. Suppose furher h f n M n for ll A, nd h Mn converges. Then f n converges uniformly on A.. 5 poins Se
More informationMotion. Part 2: Constant Acceleration. Acceleration. October Lab Physics. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.
Moion Accelerion Pr : Consn Accelerion Accelerion Accelerion Accelerion is he re of chnge of velociy. = v - vo = Δv Δ ccelerion = = v - vo chnge of velociy elpsed ime Accelerion is vecor, lhough in one-dimensionl
More informationHUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA
Communicions on Sochsic Anlysis Vol 6, No 4 2012 603-614 Serils Publicions wwwserilspublicionscom THE ITÔ FORMULA FOR A NEW STOCHASTIC INTEGRAL HUI-HSIUNG KUO, ANUWAT SAE-TANG, AND BENEDYKT SZOZDA Absrc
More informationSolutions to Problems from Chapter 2
Soluions o Problems rom Chper Problem. The signls u() :5sgn(), u () :5sgn(), nd u h () :5sgn() re ploed respecively in Figures.,b,c. Noe h u h () :5sgn() :5; 8 including, bu u () :5sgn() is undeined..5
More informationIX.2 THE FOURIER TRANSFORM
Chper IX The Inegrl Trnsform Mehods IX. The Fourier Trnsform November, 7 7 IX. THE FOURIER TRANSFORM IX.. The Fourier Trnsform Definiion 7 IX.. Properies 73 IX..3 Emples 74 IX..4 Soluion of ODE 76 IX..5
More informationSome Inequalities variations on a common theme Lecture I, UL 2007
Some Inequliies vriions on common heme Lecure I, UL 2007 Finbrr Hollnd, Deprmen of Mhemics, Universiy College Cork, fhollnd@uccie; July 2, 2007 Three Problems Problem Assume i, b i, c i, i =, 2, 3 re rel
More informationChapter 2: Evaluative Feedback
Chper 2: Evluive Feedbck Evluing cions vs. insrucing by giving correc cions Pure evluive feedbck depends olly on he cion ken. Pure insrucive feedbck depends no ll on he cion ken. Supervised lerning is
More informationFM Applications of Integration 1.Centroid of Area
FM Applicions of Inegrion.Cenroid of Are The cenroid of ody is is geomeric cenre. For n ojec mde of uniform meril, he cenroid coincides wih he poin which he ody cn e suppored in perfecly lnced se ie, is
More informationHow to Prove the Riemann Hypothesis Author: Fayez Fok Al Adeh.
How o Prove he Riemnn Hohesis Auhor: Fez Fok Al Adeh. Presiden of he Srin Cosmologicl Socie P.O.Bo,387,Dmscus,Sri Tels:963--77679,735 Emil:hf@scs-ne.org Commens: 3 ges Subj-Clss: Funcionl nlsis, comle
More informationSolutions for Nonlinear Partial Differential Equations By Tan-Cot Method
IOSR Journl of Mhemics (IOSR-JM) e-issn: 78-578. Volume 5, Issue 3 (Jn. - Feb. 13), PP 6-11 Soluions for Nonliner Pril Differenil Equions By Tn-Co Mehod Mhmood Jwd Abdul Rsool Abu Al-Sheer Al -Rfidin Universiy
More informationPositive and negative solutions of a boundary value problem for a
Invenion Journl of Reerch Technology in Engineering & Mngemen (IJRTEM) ISSN: 2455-3689 www.ijrem.com Volume 2 Iue 9 ǁ Sepemer 28 ǁ PP 73-83 Poiive nd negive oluion of oundry vlue prolem for frcionl, -difference
More informationHow to prove the Riemann Hypothesis
Scholrs Journl of Phsics, Mhemics nd Sisics Sch. J. Phs. Mh. S. 5; (B:5-6 Scholrs Acdemic nd Scienific Publishers (SAS Publishers (An Inernionl Publisher for Acdemic nd Scienific Resources *Corresonding
More information( ) ( ) ( ) ( ) ( ) ( y )
8. Lengh of Plne Curve The mos fmous heorem in ll of mhemics is he Pyhgoren Theorem. I s formulion s he disnce formul is used o find he lenghs of line segmens in he coordine plne. In his secion you ll
More informationCALDERON S REPRODUCING FORMULA FOR DUNKL CONVOLUTION
Avilble online hp://scik.org Eng. Mh. Le. 15, 15:4 ISSN: 49-9337 CALDERON S REPRODUCING FORMULA FOR DUNKL CONVOLUTION PANDEY, C. P. 1, RAKESH MOHAN AND BHAIRAW NATH TRIPATHI 3 1 Deprmen o Mhemics, Ajy
More informationAverage & instantaneous velocity and acceleration Motion with constant acceleration
Physics 7: Lecure Reminders Discussion nd Lb secions sr meeing ne week Fill ou Pink dd/drop form if you need o swich o differen secion h is FULL. Do i TODAY. Homework Ch. : 5, 7,, 3,, nd 6 Ch.: 6,, 3 Submission
More informationON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX.
ON NEW INEQUALITIES OF SIMPSON S TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE CONVEX. MEHMET ZEKI SARIKAYA?, ERHAN. SET, AND M. EMIN OZDEMIR Asrc. In his noe, we oin new some ineuliies
More informationFractional Calculus. Connor Wiegand. 6 th June 2017
Frcionl Clculus Connor Wiegnd 6 h June 217 Absrc This pper ims o give he reder comforble inroducion o Frcionl Clculus. Frcionl Derivives nd Inegrls re defined in muliple wys nd hen conneced o ech oher
More informationNumber variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric α-stable processes.
Number vrince from probbilisic perspecive: infinie sysems of independen Brownin moions nd symmeric α-sble processes. Ben Hmbly nd Liz Jones Mhemicl Insiue, Universiy of Oxford. July 26 Absrc Some probbilisic
More informationPHYSICS 1210 Exam 1 University of Wyoming 14 February points
PHYSICS 1210 Em 1 Uniersiy of Wyoming 14 Februry 2013 150 poins This es is open-noe nd closed-book. Clculors re permied bu compuers re no. No collborion, consulion, or communicion wih oher people (oher
More informationA Simple Method to Solve Quartic Equations. Key words: Polynomials, Quartics, Equations of the Fourth Degree INTRODUCTION
Ausrlin Journl of Bsic nd Applied Sciences, 6(6): -6, 0 ISSN 99-878 A Simple Mehod o Solve Quric Equions Amir Fhi, Poo Mobdersn, Rhim Fhi Deprmen of Elecricl Engineering, Urmi brnch, Islmic Ad Universi,
More informationLAPLACE TRANSFORMS. 1. Basic transforms
LAPLACE TRANSFORMS. Bic rnform In hi coure, Lplce Trnform will be inroduced nd heir properie exmined; ble of common rnform will be buil up; nd rnform will be ued o olve ome dierenil equion by rnforming
More informationNew Inequalities in Fractional Integrals
ISSN 1749-3889 (prin), 1749-3897 (online) Inernionl Journl of Nonliner Science Vol.9(21) No.4,pp.493-497 New Inequliies in Frcionl Inegrls Zoubir Dhmni Zoubir DAHMANI Lborory of Pure nd Applied Mhemics,
More informationChapter 2. Motion along a straight line. 9/9/2015 Physics 218
Chper Moion long srigh line 9/9/05 Physics 8 Gols for Chper How o describe srigh line moion in erms of displcemen nd erge elociy. The mening of insnneous elociy nd speed. Aerge elociy/insnneous elociy
More informationAnalytic solution of linear fractional differential equation with Jumarie derivative in term of Mittag-Leffler function
Anlyic soluion of liner frcionl differenil equion wih Jumrie derivive in erm of Mig-Leffler funcion Um Ghosh (), Srijn Sengup (2), Susmi Srkr (2b), Shnnu Ds (3) (): Deprmen of Mhemics, Nbdwip Vidysgr College,
More informationOn Hadamard and Fejér-Hadamard inequalities for Caputo k-fractional derivatives
In J Nonliner Anl Appl 9 8 No, 69-8 ISSN: 8-68 elecronic hp://dxdoiorg/75/ijn8745 On Hdmrd nd Fejér-Hdmrd inequliies for Cpuo -frcionl derivives Ghulm Frid, Anum Jved Deprmen of Mhemics, COMSATS Universiy
More information1. Introduction. 1 b b
Journl of Mhemicl Inequliies Volume, Number 3 (007), 45 436 SOME IMPROVEMENTS OF GRÜSS TYPE INEQUALITY N. ELEZOVIĆ, LJ. MARANGUNIĆ AND J. PEČARIĆ (communiced b A. Čižmešij) Absrc. In his pper some inequliies
More informationThe solution is often represented as a vector: 2xI + 4X2 + 2X3 + 4X4 + 2X5 = 4 2xI + 4X2 + 3X3 + 3X4 + 3X5 = 4. 3xI + 6X2 + 6X3 + 3X4 + 6X5 = 6.
[~ o o :- o o ill] i 1. Mrices, Vecors, nd Guss-Jordn Eliminion 1 x y = = - z= The soluion is ofen represened s vecor: n his exmple, he process of eliminion works very smoohly. We cn elimine ll enries
More informationFURTHER GENERALIZATIONS. QI Feng. The value of the integral of f(x) over [a; b] can be estimated in a variety ofways. b a. 2(M m)
Univ. Beogrd. Pul. Elekroehn. Fk. Ser. M. 8 (997), 79{83 FUTHE GENEALIZATIONS OF INEQUALITIES FO AN INTEGAL QI Feng Using he Tylor's formul we prove wo inegrl inequliies, h generlize K. S. K. Iyengr's
More informationChapter Direct Method of Interpolation
Chper 5. Direc Mehod of Inerpolion Afer reding his chper, you should be ble o:. pply he direc mehod of inerpolion,. sole problems using he direc mehod of inerpolion, nd. use he direc mehod inerpolns o
More information1.0 Electrical Systems
. Elecricl Sysems The ypes of dynmicl sysems we will e sudying cn e modeled in erms of lgeric equions, differenil equions, or inegrl equions. We will egin y looking fmilir mhemicl models of idel resisors,
More informationIX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704
Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 4, 7 699 IX. THE APACE TRANSFORM IX.. The plce Trnform Definiion 7 IX.. Properie 7 IX..3 Emple 7 IX..4 Soluion of IVP for ODE 74 IX..5 Soluion
More information..,..,.,
57.95. «..» 7, 9,,. 3 DOI:.459/mmph7..,..,., E-mil: yshr_ze@mil.ru -,,. -, -.. -. - - ( ). -., -. ( - ). - - -., - -., - -, -., -. -., - - -, -., -. : ; ; - ;., -,., - -, []., -, [].,, - [3, 4]. -. 3 (
More informationApplication on Inner Product Space with. Fixed Point Theorem in Probabilistic
Journl of Applied Mhemics & Bioinformics, vol.2, no.2, 2012, 1-10 ISSN: 1792-6602 prin, 1792-6939 online Scienpress Ld, 2012 Applicion on Inner Produc Spce wih Fixed Poin Theorem in Probbilisic Rjesh Shrivsv
More informationON THE OSTROWSKI-GRÜSS TYPE INEQUALITY FOR TWICE DIFFERENTIABLE FUNCTIONS
Hceepe Journl of Mhemics nd Sisics Volume 45) 0), 65 655 ON THE OSTROWSKI-GRÜSS TYPE INEQUALITY FOR TWICE DIFFERENTIABLE FUNCTIONS M Emin Özdemir, Ahme Ock Akdemir nd Erhn Se Received 6:06:0 : Acceped
More informationIX.1.1 The Laplace Transform Definition 700. IX.1.2 Properties 701. IX.1.3 Examples 702. IX.1.4 Solution of IVP for ODEs 704
Chper IX The Inegrl Trnform Mehod IX. The plce Trnform November 6, 8 699 IX. THE APACE TRANSFORM IX.. The plce Trnform Definiion 7 IX.. Properie 7 IX..3 Emple 7 IX..4 Soluion of IVP for ODE 74 IX..5 Soluion
More informationExample on p. 157
Example 2.5.3. Le where BV [, 1] = Example 2.5.3. on p. 157 { g : [, 1] C g() =, g() = g( + ) [, 1), var (g) = sup g( j+1 ) g( j ) he supremum is aken over all he pariions of [, 1] (1) : = < 1 < < n =
More informationJournal of Mathematical Analysis and Applications. Two normality criteria and the converse of the Bloch principle
J. Mh. Anl. Appl. 353 009) 43 48 Conens liss vilble ScienceDirec Journl of Mhemicl Anlysis nd Applicions www.elsevier.com/loce/jm Two normliy crieri nd he converse of he Bloch principle K.S. Chrk, J. Rieppo
More informationPhysics 2A HW #3 Solutions
Chper 3 Focus on Conceps: 3, 4, 6, 9 Problems: 9, 9, 3, 41, 66, 7, 75, 77 Phsics A HW #3 Soluions Focus On Conceps 3-3 (c) The ccelerion due o grvi is he sme for boh blls, despie he fc h he hve differen
More informationwhite strictly far ) fnf regular [ with f fcs)8( hs ) as function Preliminary question jointly speaking does not exist! Brownian : APA Lecture 1.
Am : APA Lecure 13 Brownin moion Preliminry quesion : Wh is he equivlen in coninuous ime of sequence of? iid Ncqe rndom vribles ( n nzn noise ( 4 e Re whie ( ie se every fm ( xh o + nd covrince E ( xrxs
More informationf(x) dx with An integral having either an infinite limit of integration or an unbounded integrand is called improper. Here are two examples dx x x 2
Impope Inegls To his poin we hve only consideed inegls f() wih he is of inegion nd b finie nd he inegnd f() bounded (nd in fc coninuous ecep possibly fo finiely mny jump disconinuiies) An inegl hving eihe
More informationECE Microwave Engineering. Fall Prof. David R. Jackson Dept. of ECE. Notes 10. Waveguides Part 7: Transverse Equivalent Network (TEN)
EE 537-635 Microwve Engineering Fll 7 Prof. Dvid R. Jcson Dep. of EE Noes Wveguides Pr 7: Trnsverse Equivlen Newor (N) Wveguide Trnsmission Line Model Our gol is o come up wih rnsmission line model for
More informationMAT 266 Calculus for Engineers II Notes on Chapter 6 Professor: John Quigg Semester: spring 2017
MAT 66 Clculus for Engineers II Noes on Chper 6 Professor: John Quigg Semeser: spring 7 Secion 6.: Inegrion by prs The Produc Rule is d d f()g() = f()g () + f ()g() Tking indefinie inegrls gives [f()g
More informationLAPLACE TRANSFORM OVERCOMING PRINCIPLE DRAWBACKS IN APPLICATION OF THE VARIATIONAL ITERATION METHOD TO FRACTIONAL HEAT EQUATIONS
Wu, G.-.: Lplce Trnsform Overcoming Principle Drwbcks in Applicion... THERMAL SIENE: Yer 22, Vol. 6, No. 4, pp. 257-26 257 Open forum LAPLAE TRANSFORM OVEROMING PRINIPLE DRAWBAKS IN APPLIATION OF THE VARIATIONAL
More informationRESPONSE UNDER A GENERAL PERIODIC FORCE. When the external force F(t) is periodic with periodτ = 2π
RESPONSE UNDER A GENERAL PERIODIC FORCE When he exernl force F() is periodic wih periodτ / ω,i cn be expnded in Fourier series F( ) o α ω α b ω () where τ F( ) ω d, τ,,,... () nd b τ F( ) ω d, τ,,... (3)
More informationA new model for limit order book dynamics
Anewmodelforlimiorderbookdynmics JeffreyR.Russell UniversiyofChicgo,GrdueSchoolofBusiness TejinKim UniversiyofChicgo,DeprmenofSisics Absrc:Thispperproposesnewmodelforlimiorderbookdynmics.Thelimiorderbookconsiss
More informationgraph of unit step function t
.5 Piecewie coninuou forcing funcion...e.g. urning he forcing on nd off. The following Lplce rnform meril i ueful in yem where we urn forcing funcion on nd off, nd when we hve righ hnd ide "forcing funcion"
More informationP441 Analytical Mechanics - I. Coupled Oscillators. c Alex R. Dzierba
Lecure 3 Mondy - Deceber 5, 005 Wrien or ls upded: Deceber 3, 005 P44 Anlyicl Mechnics - I oupled Oscillors c Alex R. Dzierb oupled oscillors - rix echnique In Figure we show n exple of wo coupled oscillors,
More informationAn integral having either an infinite limit of integration or an unbounded integrand is called improper. Here are two examples.
Improper Inegrls To his poin we hve only considered inegrls f(x) wih he is of inegrion nd b finie nd he inegrnd f(x) bounded (nd in fc coninuous excep possibly for finiely mny jump disconinuiies) An inegrl
More informationEXERCISE - 01 CHECK YOUR GRASP
UNIT # 09 PARABOLA, ELLIPSE & HYPERBOLA PARABOLA EXERCISE - 0 CHECK YOUR GRASP. Hin : Disnce beween direcri nd focus is 5. Given (, be one end of focl chord hen oher end be, lengh of focl chord 6. Focus
More informationThe order of reaction is defined as the number of atoms or molecules whose concentration change during the chemical reaction.
www.hechemisryguru.com Re Lw Expression Order of Recion The order of recion is defined s he number of oms or molecules whose concenrion chnge during he chemicl recion. Or The ol number of molecules or
More informationAnn. Funct. Anal. 2 (2011), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:
Ann. Func. Anal. 2 2011, no. 2, 34 41 A nnals of F uncional A nalysis ISSN: 2008-8752 elecronic URL: www.emis.de/journals/afa/ CLASSIFICAION OF POSIIVE SOLUIONS OF NONLINEAR SYSEMS OF VOLERRA INEGRAL EQUAIONS
More informationRepresentation of Stochastic Process by Means of Stochastic Integrals
Inernaional Journal of Mahemaics Research. ISSN 0976-5840 Volume 5, Number 4 (2013), pp. 385-397 Inernaional Research Publicaion House hp://www.irphouse.com Represenaion of Sochasic Process by Means of
More informationChapter 4. Lebesgue Integration
4.2. Lebesgue Integrtion 1 Chpter 4. Lebesgue Integrtion Section 4.2. Lebesgue Integrtion Note. Simple functions ply the sme role to Lebesgue integrls s step functions ply to Riemnn integrtion. Definition.
More informationHardy s inequality in L 2 ([0, 1]) and principal values of Brownian local times
Fields Insiue Communicions Volume, Hrdy s inequliy in [, ] nd principl vlues of Brownin locl imes Giovnni Pecci oroire de Proiliés e Modèles léoires Universié Pris VI & Universié Pris VII Pris, Frnce nd
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 informationCBSE 2014 ANNUAL EXAMINATION ALL INDIA
CBSE ANNUAL EXAMINATION ALL INDIA SET Wih Complee Eplnions M Mrks : SECTION A Q If R = {(, y) : + y = 8} is relion on N, wrie he rnge of R Sol Since + y = 8 h implies, y = (8 ) R = {(, ), (, ), (6, )}
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 informationContinuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationAvd. Matematisk statistik
Avd Maemaisk saisik TENTAMEN I SF294 SANNOLIKHETSTEORI/EXAM IN SF294 PROBABILITY THE- ORY WEDNESDAY THE 9 h OF JANUARY 23 2 pm 7 pm Examinaor : Timo Koski, el 79 7 34, email: jkoski@khse Tillåna hjälpmedel
More informationA Time Truncated Improved Group Sampling Plans for Rayleigh and Log - Logistic Distributions
ISSNOnline : 39-8753 ISSN Prin : 347-67 An ISO 397: 7 Cerified Orgnizion Vol. 5, Issue 5, My 6 A Time Trunced Improved Group Smpling Plns for Ryleigh nd og - ogisic Disribuions P.Kvipriy, A.R. Sudmni Rmswmy
More informationChapter 5 : Continuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll
More informationMATH 124 AND 125 FINAL EXAM REVIEW PACKET (Revised spring 2008)
MATH 14 AND 15 FINAL EXAM REVIEW PACKET (Revised spring 8) The following quesions cn be used s review for Mh 14/ 15 These quesions re no cul smples of quesions h will pper on he finl em, bu hey will provide
More informationAn Integral Two Space-Variables Condition for Parabolic Equations
Jornl of Mhemics nd Sisics 8 (): 85-9, ISSN 549-3644 Science Pblicions An Inegrl Two Spce-Vribles Condiion for Prbolic Eqions Mrhone, A.L. nd F. Lkhl Deprmen of Mhemics, Lborory Eqions Differenielles,
More informationTHREE IMPORTANT CONCEPTS IN TIME SERIES ANALYSIS: STATIONARITY, CROSSING RATES, AND THE WOLD REPRESENTATION THEOREM
THR IMPORTANT CONCPTS IN TIM SRIS ANALYSIS: STATIONARITY, CROSSING RATS, AND TH WOLD RPRSNTATION THORM Prof. Thoms B. Fomb Deprmen of conomics Souhern Mehodis Universi June 8 I. Definiion of Covrince Sionri
More informationOn the Pseudo-Spectral Method of Solving Linear Ordinary Differential Equations
Journl of Mhemics nd Sisics 5 ():136-14, 9 ISS 1549-3644 9 Science Publicions On he Pseudo-Specrl Mehod of Solving Liner Ordinry Differenil Equions B.S. Ogundre Deprmen of Pure nd Applied Mhemics, Universiy
More informationResearch Article New General Integral Inequalities for Lipschitzian Functions via Hadamard Fractional Integrals
Hindwi Pulishing orporion Inernionl Journl of Anlysis, Aricle ID 35394, 8 pges hp://d.doi.org/0.55/04/35394 Reserch Aricle New Generl Inegrl Inequliies for Lipschizin Funcions vi Hdmrd Frcionl Inegrls
More informationForms of Energy. Mass = Energy. Page 1. SPH4U: Introduction to Work. Work & Energy. Particle Physics:
SPH4U: Inroducion o ork ork & Energy ork & Energy Discussion Definiion Do Produc ork of consn force ork/kineic energy heore ork of uliple consn forces Coens One of he os iporn conceps in physics Alernive
More informationCopyright by Tianran Geng 2017
Copyrigh by Tinrn Geng 207 The Disserion Commiee for Tinrn Geng cerifies h his is he pproved version of he following disserion: Essys on forwrd porfolio heory nd finncil ime series modeling Commiee: Thlei
More informationarxiv: v1 [math.pr] 24 Sep 2015
RENEWAL STRUCTURE OF THE BROWNIAN TAUT STRING EMMANUEL SCHERTZER rxiv:59.7343v [mh.pr] 24 Sep 25 Absrc. In recen pper [LS5], M. Lifshis nd E. Seerqvis inroduced he u sring of Brownin moion w, defined s
More informationProcedia Computer Science
Procedi Compuer Science 00 (0) 000 000 Procedi Compuer Science www.elsevier.com/loce/procedi The Third Informion Sysems Inernionl Conference The Exisence of Polynomil Soluion of he Nonliner Dynmicl Sysems
More informationAN EIGENVALUE PROBLEM FOR LINEAR HAMILTONIAN DYNAMIC SYSTEMS
AN EIGENVALUE PROBLEM FOR LINEAR HAMILTONIAN DYNAMIC SYSTEMS Mrin Bohner Deprmen of Mhemics nd Sisics, Universiy of Missouri-Roll 115 Roll Building, Roll, MO 65409-0020, USA E-mil: ohner@umr.edu Romn Hilscher
More informationObservability of flow dependent structure functions and their use in data assimilation
Oserviliy of flow dependen srucure funcions nd heir use in d ssimilion Pierre Guhier nd Crisin Lupu Collorion wih Séphne Lroche, Mrk Buehner nd Ahmed Mhidji (Env. Cnd) 3rd meeing of he HORPEX DAOS-WG Monrél
More informationTutorial 4. b a. h(f) = a b a ln 1. b a dx = ln(b a) nats = log(b a) bits. = ln λ + 1 nats. = log e λ bits. = ln 1 2 ln λ + 1. nats. = ln 2e. bits.
Tutoril 4 Exercises on Differentil Entropy. Evlute the differentil entropy h(x) f ln f for the following: () The uniform distribution, f(x) b. (b) The exponentil density, f(x) λe λx, x 0. (c) The Lplce
More informationOn Gronwall s Type Integral Inequalities with Singular Kernels
Filoma 31:4 (217), 141 149 DOI 1.2298/FIL17441A Published by Faculy of Sciences and Mahemaics, Universiy of Niš, Serbia Available a: hp://www.pmf.ni.ac.rs/filoma On Gronwall s Type Inegral Inequaliies
More informationDISCRETE GRONWALL LEMMA AND APPLICATIONS
DISCRETE GRONWALL LEMMA AND APPLICATIONS JOHN M. HOLTE MAA NORTH CENTRAL SECTION MEETING AT UND 24 OCTOBER 29 Gronwall s lemma saes an inequaliy ha is useful in he heory of differenial equaions. Here is
More informationMinimum Squared Error
Minimum Squred Error LDF: Minimum Squred-Error Procedures Ide: conver o esier nd eer undersood prolem Percepron y i > 0 for ll smples y i solve sysem of liner inequliies MSE procedure y i i for ll smples
More informationAn random variable is a quantity that assumes different values with certain probabilities.
Probabiliy The probabiliy PrA) of an even A is a number in [, ] ha represens how likely A is o occur. The larger he value of PrA), he more likely he even is o occur. PrA) means he even mus occur. PrA)
More informationKEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
More information