Solution of Laplace s Differential Equation and Fractional Differential Equation of That Type
|
|
- Terence Weaver
- 6 years ago
- Views:
Transcription
1 Applie Mheics Pulishe Olie oveer 3 (hp://wwwscirporg/jourl/) hp://oiorg/436/34a5 Soluio of Lplce s iffereil Equio Frciol iffereil Equio of Th Type Tohru Mori Ke-ichi So Tohou Uiversiy Sei Jp College of Egieerig iho Uiversiy Koriy Jp Eil: se@jcohoeejp eceive Augus 9 3; revise Sepeer 9 3; ccepe Sepeer 6 3 Copyrigh 3 Tohru Mori Ke-ichi So This is ope ccess ricle isriue uer he Creive Coos Ariuio Licese which peris uresrice use isriuio reproucio i y eiu provie he origil wor is properly cie ABSTACT I preceig pper we iscusse he soluio of Lplce s iffereil equio y usig operiol clculus i he frewor of isriuio heory We here suie he soluio of h iffereil equio wih ihoogeeous er lso frciol iffereil equio of he ype of Lplce s iffereil equio We here cosiere erivives of fucio u o whe u is loclly iegrle o he iegrl u coverges We ow iscr he ls coiio h u shoul coverge iscuss he se prole I Appeices polyoil for of priculr soluios re give for he iffereil equios suie Herie s iffereil equio wih specil ihoogeeous ers Keywors: Lplce s iffereil Equio; Kuer s iffereil Equio; Frciol iffereil Equio; isriuio Theory; Operiol Clculus; Ihoogeeous Equio; Polyoil Soluio Iroucio Yosi [] iscusse he soluio of Lplce s iffereil equio (E) which is lier E wih coefficies which re lier fucios of he vrile The E which he es up is y y y () l l for l re coss His iscussio is se o Miusińsi s operiol clculus [3] Yosi [] gve here oly oe of he soluios of he E () I he preceig pper [4] we iscusse he soluio of frciol iffereil equio (fe) of he ype of E () h is give y u u u f u () for Here for > is he ie-liouville frciol erivive (f) efie i Secio We use o eoe he se of ll rel uers : u u Whe is equl o ieger > Whe () is he ihoogeeous E for () We use o eoe he se of ll iegers : : : for sisfyig < We use for o eoe he les ieger h is o less h I [4] we op operiol clculus i he frewor of isriuio heory evelope for he soluio of he fe wih cos coefficies i [56] I [4] we give he recipe of oiig he soluio of he ihoogeeous equio s well s he hoogeeous oe we show how he se of wo soluios of he hoogeeous equio is ie I [4] we op he usul efiiio of he ie- -Liouville f which efies f oly for such loclly iegrle fucio f o > h f is fiie Prciclly we op Coiio B i [4] which is Ope Access
2 T MOITA K SATO 7 u H Coiio IB f H re epresse s lier coiio of g for > Here H is Hevisie s sep fucio whe f is efie o > f H is ssue o e equl o f whe > o whe g is efie y g (3) for > is he g fucio I [4] we e up Kuer s E s eple which is u c u u (4) c re coss If c oe of he soluios give i [78] is c F c ; ; : (5)! for > The oher soluio is c F c c ; ; (6) I [4] if c < we oi oh of he soluios Bu whe c (6) oes o sisfy Coiio IB we coul o ge i I rece review [9] we iscusse he lyic coiuios of f lyic coiuio of ie- -Liouville f f is such h he f eiss eve for such loclly iegrle fucio f o > h f iverges I he prese pper we op his lyic coiuio of f I plce of he ove Coiio IB we ow op he followig coiio Coiio A u H f H re epresse s lier coiio of g H for S S is se of > M \ < for soe M > As cosequece we c ow chieve oriry soluios for () of For (4) we oi oh soluios (5) (6) if c I is he purpose his pper o show how he preseio i [4] shoul e revise wih he chge of efiiio of f he replcee of Coiio IB wih Coiio A I Secio we prepre he efiiio of ie- Liouville f he epli how he fucio u is f i () re covere io he correspoig isriuio u is f i isriuio heory lso how u is covere c io u Afer hese preprio recipe is give o e use i solvig he fe () wih he i of operiol culculus i Secio 3 I his recipe he soluio is oie oly whe Whe is lso re- quire A eplio of his fc is give i Appeices C of [4] I Secio 4 we pply he recipe o () of which specil oe is Kuer s E This is eple which Yosi [] es up I Secio 5 we pply he recipe o he fe wih ssuig For he Herie E wih ihoogeeous er Levie Mle [] showe h here eis priculr soluios i he for of polyoil I Appeices A C we show h such soluio eiss for he E fe suie i Secios 4 5 respecively I Appei B we show how he resuls presee i [] re erive fro hose i Appei A Foruls We ow op Coiio A We he epress u H s follows; u H u g H () S u re coss Le For \ < g g () Proof By (3) for < we hve g g ie-liouville Frciol Iegrl erivive e loclly iegrle o We he efie he ie-liouville frciol iegrl f of orer > y f f (3) Le f H We he efie he ie-liouville f of orer y u u u > (4) if i eiss u u for > For > we hve g \ g (5) Ope Access
3 8 T MOITA K SATO If we ssue h es cople vlue g y efiiio (3) is lyic fucio of i he oi e < g efie y (4) is is lyic coiuio o he whole cople ple If we ssue h lso es cople vlue g efie y (4) is lyic fucio of i he oi e > The lyic coiuio s fucio of ws lso suie The rgue is urlly coclue h (5) shoul pply for he lyic coiuio eve i e ecep he pois < ; see [9] We ow op his lyic coiuio of g o represe g hece we ccep he followig le Le (5) hols for every \ < By () (5) we hve For u is loclly iegrle o S u u g efie y () we oe h M u H Frciol Iegrl erivive of isriuio (6) We cosier isriuios elogig o Whe fucio h is loclly iegrle o hs suppor oue o he lef i elogs o is clle regulr isriuio The isriuios i re clle righ-sie isriuios A copc forl efiiio of isriuio i is frciol iegrl erivive is give i Appei A of [4] Le f H e regulr isriuio The f H for is lso regulr isriu- io isriuio f H is efie y f H f H (7) Le le h e such regulr isri- > uio h h is coiuous iffereile o for every The h is efie y h h (8) Le > le f H e coiuous iffereile o The for every f H f H Whe h is regulr isriuio fie for ll h he ie lw: Le 3 For h h is vli for every irc s el fucio H fie y Le g g for e efie y Le 4 If > h (9) is e- () is he isriuio e- () g g H () Proof By puig f i (7) usig () (5) we oi H H g H By operig o his usig (9) (5) we oi () Correspoig o u epresse y () we efie u y u u g (3) The S u f re epresse s u uˆ f fˆ (4) uˆ u (5) S Becuse of () we hve g g \ g (6) Le 5 Le The g g (7) (8) The ls erivive wih respec o is e regrig s vrile A proof of (7) for > is give i Appei B of [4] Proof Whe > y Les 4 g g H g H g Ope Access
4 T MOITA K SATO 9 The firs equliy i (8) is oie fro (7) vice vers y usig () The followig le is cosequece of his le Le 6 Le u e epresse s lier coiio of g for The u uˆ uˆ (9) 3 Fro u o u Vice Vers Le 7 Le \ < > sisfy > The () g g H () g g H Proof Forul () is erive y pplyig (3) () (6) o he righh Forul () follows fro () y replcig g g y g g g g respecively y usig () (7) By usig Le 7 o (6) we oi S u g () u H S u H u g (3) Le 8 Le \ < > sisfy > The g H g (4) This follows fro () Coiio B u is epresse s lier coiio of g for S S is se of > M \ < for soe M > Whe his coiio is sisfie u is epresse s (3) wih S replce y S ~ Le 9 Le u sisfy Coiio B The he correspoig u is oie fro u y u H u (5) is epresse y () wih S replce y S Le Le u u e give y (3) () respecively The u u H re rele y u u H u S (6) S u u H if > u sisfies M Proof By (3) (6) we hve u u g S S u g (7) (8) Usig () i he firs er o he righh sie we oi (6) Muliplyig (8) y oig h he firs er o he righh sie is he equl o (3) we oi (7) 3 ecipe of Solvig Lplce s E fe of Th Type We ow epress he E/fE () o e solve s follows: l l l u f (3) l or 5 we suy his E for respecively I Secios 4 his fe for 3 efor o E/fE for isriuio Usig Le we epress (3) s l l l u f v l (3) l v l l ul (33) l l 3 Soluio Vi Operiol Clculus By usig (4) (9) we epress (3) s l l ˆ ˆ l u l u l l A uˆ Buˆ fˆ vˆ (34) Ope Access
5 3 T MOITA K SATO l l A l l l l l l B l vˆ u u l l l l l l l I orer o solve he Equio (34) for ˆ u u we solve he followig equio for fucio rel vrile : (35) (36) û of A u ˆ Bu ˆ (37) fˆ vˆ Le The copleery soluio (C-soluio) of equio (37) is give y û C ˆ C is rirry cos ˆ ep B C A (38) he iegrl is he iefiie iegrl C is y cos Le Le ˆ e he C-soluio of (37) û e he priculr soluio (P-soluio) of (37) whe he ihoogeeous er is for The uˆ ˆ C3 ˆ A ˆ (39) C 3 is y cos Sice f H sisfies Coiio A ˆv is give y (36) he P-soluio û of (37) is epresse s lier coiio of u ˆ for > of uˆ M for > respecively Fro he soluio û of (37) û is oie y susiuig y The we cofir h (34) is sisfie y h û opere o 33 eu Series Epsio Filly he oie epressio of û is epe io eu series [] Prciclly we ep i io he su of ers of egive powers of he we oi he soluio û of (34) If he oie û is lier coiio of for > M wih soe M > he uˆ is he soluio u of (3) If i sisfies Coiio B i is covere o soluio u of (3) for > wih he i of Le 9 34 ecipe of Oiig he Soluio of (3) ˆf ) We prepre he : y (4) A B ˆv y (35) (36) ) We oi ˆ y (38) The C-soluio of (3) is give y u C ˆ If v he C-soluio of (3) is oie fro his wih he i of Le 9 3) If ˆ f or vˆ we oi û give y (39) vˆ c f he so- 4) If luio of (3) is give y u C ˆ c ˆ u (3) c re coss The C-soluio of (3) is he oie fro his wih he i of Le 9 5) If ˆ f he P-soluio of (3) is give y u uˆ > M re coss The P-soluio of (3) wih ihoogeeous er c f g is oie fro his wih he i of Le 9 35 Coes o he ecipe I he ove recipe we firs oi he C-soluio of (37) h is û C ˆ I gives he C-soluio û of (34) hece he C-soluios u of (3) A C-soluio u of (3) is he oie wih he i of Le 9 We e oi he P-soluio û of (37) whe he ihoogeeous pr is for As oe ove he P-soluios û of (37) for f ˆ for ˆv re epresse s lier coiio of û for > M of uˆ for > respecively The su of he P-soluios û of (37) for ˆf for ˆv gives he P-soluio û of (34) hece he P-soluio u of (3) The C-soluio u of (3) coes fro he C-soluio of (37) he P-soluio of (37) for ˆv 36 ers Whe we oi û he e of Secio 3 we us eie wheher i is copile wih Coiio B We will fi h if l for l > he oû is o cceple Hece we hve o solve ie Ope Access
6 T MOITA K SATO 3 he prole ssuig h l for ll l > Whe we pu Whe we pu = iscussio of his prole is give i Appeices C of [4] I he prese cse he iscussio us e re ig Coiio B here o represe he prese Coiio B 4 Lplce s Kuer s E We ow cosier he cse of σ = = The (3) reuces o By (35) (36) u u u f B A B vˆ u A ˆv re (4) (4) (43) 4 Copleery Soluio of (37) (3) (4) ˆ I orer o oi he C-soluio of (37) y usig (38) we epress B A s follows: B (44) A B() is ow epresse s (45) B By usig (38) we oi ˆ! he ioil coefficies The C-soluio of (3) is give y (46) for > re ˆ ˆ u u C If > C C (47) we oi C-soluio of (4) y usig Le 9: u H C C H C! C H F ; ; H (48) er I Iroucio Kuer s E is give y (4) I is equl o (4) for c I his cse c c (49) We he cofir h he epressio (48) for c > grees wih (6) which is oe of he C-soluios of Kuer s E give i [78] 4 Priculr Soluio of (37) We ow oi he P-soluio of (37) whe he ihoogeeous er is equl o for Whe he C-soluio of (37) is ˆ he P-soluio of (37) is give y (39) By usig (4) (46) he followig resul is oie i [4]: ˆ u C ˆ 3 C (4) pp C p p p p (4) Le 3 Whe p p < C p efie y p (4) is epresse s Ope Access
7 3 T MOITA K SATO? Cp p Cp p p p C p p p p p This le is prove i [4] 43 Priculr Soluios of (3) (4) (4) Equio (4) shows h if he ihoogeeous er is for he P-soluio of (3) is give y u C (43) Theore Le \ < < f g The we hve P-soluio u of (4) give y u u H u H (44) H H Proof Applyig Le 9 o (43) we oi u H C H (45) (46) By usig (4) i (46) we oi (44) wih (45) We oe h u H is epresse s u H! H F ; ; H 44 Copleery Soluio of (4) (47) (48) By (43) (45) vˆ u Whe fˆ vˆ he P-soluio of (47) is give y uˆ u uˆ (49) By usig (44) for if < we oi C-soluio of (4): u H u u H u! H u F ; ; H (4) I Secio 4 we hve (48) h is oher C-soluio of (4) If we copre (48) wih (45) whe > i c e epresse s (4) C H C u H Proposiio Whe he copleery soluio of (4) uliplie y H is give y he su of he righh sies of (48) of (4) which C u H u u H re equl o respecively er As se i er for Kuer s E re give i (49) c (4) We he cofir h if c he se of (48) (4) grees wih he se of (45) (46) 45 ers I [] i ws show h here eis P-soluios epresse y polyoil for ihoogeeous Herie s E e l We c oi he correspoig resul for Lplce s E We iscuss his prole i Appei A he iscuss he P-soluio of ihoogeeous Herie s E i he prese forulio i Appei B 5 Soluio of fe (3) for I his secio we cosier he cse of The he Equio (3) o e solve is u u u f > ow (35) (36) re epresse s A B M vˆ u (5) (5) (53) Ope Access
8 T MOITA K SATO 33 5 Copleery Soluio of (37) By usig (5) B A is epresse s B A (54) (55) By (38) he C-soluio ˆ of (37) is give y ˆ 5 Copleery Soluio of (3) (5) The C-soluio of (3) is give y ˆ ˆ u u C If > C (56) (57) y pplyig Le 9 o his we oi he C-soluio of (5): u H C H C H (58) 53 Priculr Soluio of (3) (5) ˆ give y By usig he epressios of A (5) (56) i (39) we oi he P-soluio of (37) whe he ihoogeeous er is for : uˆ C 3 C ˆ (59) of (5) give y u H u H u H (5) H (5) I Appei C iscussio is give o show h here eis P-soluios i he for of polyoil for (5) 54 Copleery Soluio of (5) We oi he soluio u oly for < Eve hough we hve P-soluios of (3) for v vˆ whe ˆv is give y (53) wih ozero vlues of u i oes o sisfy Coiio B oes o give soluio of (5) Hece u give y (58) is he oly C-soluio of (5) If we copre (58) wih (5) we oi he followig proposiio Proposiio Le > The he C-soluio of (5) is give y (5) C H C u H EFEECES [] K Yosi The Algeric erivive Lplce s iffereil Equio Proceeigs of he Jp Acey Vol 59 Ser A 983 pp -4 [] K Yosi Operiol Clculus Spriger-Verlg ew Yor 98 Chper VII [3] J Miusińsi Operiol Clculus Pergo Press Loo 959 [4] T Mori K So ers o he Soluio of Lplce s iffereil Equio Frciol iffereil Equio of Th Type Applie Mheics Vol 4 o A 3 pp 3- [5] T Mori K So Soluio of Frciol iffereil Equio i Ters of isriuio Theory Ieris- C cipliry Iforio Scieces Vol o 6 pp is efie y (4) is give y 7-83 (4) if < [6] T Mori K So eu-series Soluio of By usig (4) i (59) we c show h if he i- Frciol iffereil Equio Ieriscipliry Iforio Scieces Vol 6 o pp 7-37 hoogeeous er is for he P-solu- io of (3) is u û By pplyig Le [7] M Arowiz I A Segu Hoo of Mhe- 9 o his we oi he followig heore icl Fucios wih Foruls Grphs Mhe- Theore Le < < icl Tles over Pul Ic ew Yor 97 Chper 3 f The we hve P-soluio u [8] M Mgus F Oerheiger Foruls Theo- res for he Fucios of Mheicl Physics Chelse Ope Access
9 34 T MOITA K SATO Pul Co ew Yor 949 Chper VI [9] T Mori K So Liouville ie-liouville Frciol erivives vi Coour Iegrls Frciol Clculus Applie Alysis Vol 6 o 3 3 pp [] L Levie Mleh Polyoil Soluios of he Clssicl Equios of Herie Legere Cheyshev Ieriol Jourl of Mheicl Eucio i Sciece Techology Vol 34 3 pp 95-3 [] F iesz B Sz-gy Fuciol Alysis over Pul Ic ew Yor 99 p 46 Ope Access
10 T MOITA K SATO 35 Appei A: Polyoil For of P-Soluio of (4) Le \ < > The (45) gives u H C H (A) C C u C H (A) C (A3) We oi he followig heores fro (A) wih he of P-soluio of (4): i of Proposiio Theore 3 Le > < < f g The we hve he polyoil for u H H Theore 4 Le > > > polyoil for of P-soluio of (4): f g (A4) for The we hve he u H H (A5) Appei B: Polyoil For of P-Soluio of Herie E We ow cosier he ihoogeeous Herie E give y y ycy (B) for > We pu u y The he equio for u is give y 4 4 u u c u (B) This is Lplce s E (4) wih preers c (B3) c c g he ihoogeeous er f Theore 5 Le c > The we hve he polyoil for of P- soluio of (B): u H H (B4) 4 3 Proof I his cse By Theore 3 we oi his resul Theore 6 Le c < > The we hve he polyoil for of P- soluio of (B): Ope Access
11 36 T MOITA K SATO 3 Proof I his cse By Theore 4 we oi his resul u H H 4 (B5) 3 4 Theore 7 Le c > The we hve he polyoil for of P- soluio of (B): u H H (B6) 3 Proof I his cse By Theore 4 we oi his resul Theore 8 Le c < > The we hve he polyoil for of P- soluio of (B): u H H (B7) 4 Proof I his cse By Theore 3 we oi his resul er 3 We cofir h Theores 7 5 res- pecively gree wih Theores i [] Appei C: Polyoil For of P-Soluio of (5) Le > The (5) gives u H C H C (C) u C H (C) C C (C3) We oi he followig heore fro (C) wih he i of Proposiio Theore 9 Le > > > f g for The we hve he polyoil for of P-soluio of (5): u H H (C4) () Ope Access
Existence Of Solutions For Nonlinear Fractional Differential Equation With Integral Boundary Conditions
Reserch Ivey: Ieriol Jourl Of Egieerig Ad Sciece Vol., Issue (April 3), Pp 8- Iss(e): 78-47, Iss(p):39-6483, Www.Reserchivey.Com Exisece Of Soluios For Nolier Frciol Differeil Equio Wih Iegrl Boudry Codiios,
More informationWeek 8 Lecture 3: Problems 49, 50 Fourier analysis Courseware pp (don t look at French very confusing look in the Courseware instead)
Week 8 Lecure 3: Problems 49, 5 Fourier lysis Coursewre pp 6-7 (do look Frech very cofusig look i he Coursewre ised) Fourier lysis ivolves ddig wves d heir hrmoics, so i would hve urlly followed fer he
More informationFractional Fourier Series with Applications
Aeric Jourl o Couiol d Alied Mheics 4, 4(6): 87-9 DOI: 593/jjc446 Frciol Fourier Series wih Alicios Abu Hd I, Khlil R * Uiversiy o Jord, Jord Absrc I his er, we iroduce coorble rciol Fourier series We
More informationOn Absolute Indexed Riesz Summability of Orthogonal Series
Ieriol Jourl of Couiol d Alied Mheics. ISSN 89-4966 Volue 3 Nuer (8). 55-6 eserch Idi Pulicios h:www.riulicio.co O Asolue Ideed iesz Suiliy of Orhogol Series L. D. Je S. K. Piry *. K. Ji 3 d. Sl 4 eserch
More informationSpecial Functions. Leon M. Hall. Professor of Mathematics University of Missouri-Rolla. Copyright c 1995 by Leon M. Hall. All rights reserved.
Specil Fucios Leo M. Hll Professor of Mhemics Uiversiy of Missouri-Roll Copyrigh c 995 y Leo M. Hll. All righs reserved. Chper 5. Orhogol Fucios 5.. Geerig Fucios Cosider fucio f of wo vriles, ( x,), d
More informationFirst-Passage Time moment Approximation For The Birth Death Diffusion Process To A General moving Barrier
Ieriol Jourl of emics Sisics Iveio IJSI E-ISSN: 3 4767 P-ISSN: 3-4759 Volume 5 Issue 9 Jury 8 PP-4-8 Firs-Pssge Time mome pproimio For Te Bir De Diffusio Process To Geerl movig Brrier Bsel. l-eie KuwiUiversiy
More informationSupplement: Gauss-Jordan Reduction
Suppleme: Guss-Jord Reducio. Coefficie mri d ugmeed mri: The coefficie mri derived from sysem of lier equios m m m m is m m m A O d he ugmeed mri derived from he ove sysem of lier equios is [ ] m m m m
More informationLOCUS 1. Definite Integration CONCEPT NOTES. 01. Basic Properties. 02. More Properties. 03. Integration as Limit of a Sum
LOCUS Defiie egrio CONCEPT NOTES. Bsic Properies. More Properies. egrio s Limi of Sum LOCUS Defiie egrio As eplied i he chper iled egrio Bsics, he fudmel heorem of clculus ells us h o evlue he re uder
More informationBoundary Value Problems of Conformable. Fractional Differential Equation with Impulses
Applied Meicl Scieces Vol 2 28 o 8 377-397 HIKARI Ld www-irico ps://doiorg/2988/s28823 Boudry Vlue Probles of Coforble Frciol Differeil Equio wi Ipulses Arisr Tgvree Ci Tipryoo d Apisi Ppogpu Depre of
More informationExtension of Hardy Inequality on Weighted Sequence Spaces
Jourl of Scieces Islic Reublic of Ir 20(2): 59-66 (2009) Uiversiy of ehr ISS 06-04 h://sciecesucir Eesio of Hrdy Iequliy o Weighed Sequece Sces R Lshriour d D Foroui 2 Dere of Mheics Fculy of Mheics Uiversiy
More informationA new approach to Kudryashov s method for solving some nonlinear physical models
Ieriol Jourl of Physicl Scieces Vol. 7() pp. 860-866 0 My 0 Avilble olie hp://www.cdeicourls.org/ijps DOI: 0.897/IJPS.07 ISS 99-90 0 Acdeic Jourls Full Legh Reserch Pper A ew pproch o Kudryshov s ehod
More informationON PRODUCT SUMMABILITY OF FOURIER SERIES USING MATRIX EULER METHOD
Ieriol Jourl o Advces i Egieerig & Techology Mrch IJAET ISSN: 3-963 N PRDUCT SUMMABILITY F FURIER SERIES USING MATRIX EULER METHD BPPdhy Bii Mlli 3 UMisr d 4 Mhedr Misr Depre o Mheics Rold Isiue o Techology
More informationAn Extension of Hermite Polynomials
I J Coemp Mh Scieces, Vol 9, 014, o 10, 455-459 HIKARI Ld, wwwm-hikricom hp://dxdoiorg/101988/ijcms0144663 A Exesio of Hermie Polyomils Ghulm Frid Globl Isiue Lhore New Grde Tow, Lhore, Pkis G M Hbibullh
More informationERROR ESTIMATES FOR APPROXIMATING THE FOURIER TRANSFORM OF FUNCTIONS OF BOUNDED VARIATION
ERROR ESTIMATES FOR APPROXIMATING THE FOURIER TRANSFORM OF FUNCTIONS OF BOUNDED VARIATION N.S. BARNETT, S.S. DRAGOMIR, AND G. HANNA Absrc. I his pper we poi ou pproximio for he Fourier rsform for fucios
More informationIntegration and Differentiation
ome Clculus bckgroud ou should be fmilir wih, or review, for Mh 404 I will be, for he mos pr, ssumed ou hve our figerips he bsics of (mulivrible) fucios, clculus, d elemer differeil equios If here hs bee
More informationF.Y. Diploma : Sem. II [CE/CR/CS] Applied Mathematics
F.Y. Diplom : Sem. II [CE/CR/CS] Applied Mhemics Prelim Quesio Pper Soluio Q. Aemp y FIVE of he followig : [0] Q. () Defie Eve d odd fucios. [] As.: A fucio f() is sid o e eve fucio if f() f() A fucio
More informationSLOW INCREASING FUNCTIONS AND THEIR APPLICATIONS TO SOME PROBLEMS IN NUMBER THEORY
VOL. 8, NO. 7, JULY 03 ISSN 89-6608 ARPN Jourl of Egieerig d Applied Sciece 006-03 Ai Reerch Publihig Nework (ARPN). All righ reerved. www.rpjourl.com SLOW INCREASING FUNCTIONS AND THEIR APPLICATIONS TO
More informationDETERMINATION OF PARTICULAR SOLUTIONS OF NONHOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS BY DISCRETE DECONVOLUTION
U.P.B. ci. Bull. eries A Vol. 69 No. 7 IN 3-77 DETERMINATION OF PARTIULAR OLUTION OF NONHOMOGENEOU LINEAR DIFFERENTIAL EQUATION BY DIRETE DEONVOLUTION M. I. ÎRNU e preziă o ouă meoă e eermiare a soluţiilor
More information* power rule: * fraction raised to negative exponent: * expanded power rule:
Mth 15 Iteredite Alger Stud Guide for E 3 (Chpters 7, 8, d 9) You use 3 5 ote crd (oth sides) d scietific clcultor. You re epected to kow (or hve writte o our ote crd) foruls ou eed. Thik out rules d procedures
More informationSuggested Solution for Pure Mathematics 2011 By Y.K. Ng (last update: 8/4/2011) Paper I. (b) (c)
per I. Le α 7 d β 7. The α d β re he roos o he equio, such h α α, β β, --- α β d αβ. For, α β For, α β α β αβ 66 The seme is rue or,. ssume Cosider, α β d α β y deiiio α α α α β or some posiive ieer.
More informationLocal Fractional Kernel Transform in Fractal Space and Its Applications
From he SelecedWorks of Xio-J Yg 22 Locl Frciol Kerel Trsform i Frcl Spce d Is Applicios Yg Xioj Aville : hps://works.epress.com/yg_ioj/3/ Advces i Compuiol Mhemics d is Applicios 86 Vol. No. 2 22 Copyrigh
More informationHypergeometric Functions and Lucas Numbers
IOSR Jourl of Mthetis (IOSR-JM) ISSN: 78-78. Volue Issue (Sep-Ot. ) PP - Hypergeoetri utios d us Nuers P. Rjhow At Kur Bor Deprtet of Mthetis Guhti Uiversity Guwhti-78Idi Astrt: The i purpose of this pper
More informationxp (X = x) = P (X = 1) = θ. Hence, the method of moments estimator of θ is
Exercise 7 / page 356 Noe ha X i are ii from Beroulli(θ where 0 θ a Meho of momes: Sice here is oly oe parameer o be esimae we ee oly oe equaio where we equae he rs sample mome wih he rs populaio mome,
More informationBINOMIAL THEOREM OBJECTIVE PROBLEMS in the expansion of ( 3 +kx ) are equal. Then k =
wwwskshieduciocom BINOMIAL HEOREM OBJEIVE PROBLEMS he coefficies of, i e esio of k e equl he k /7 If e coefficie of, d ems i e i AP, e e vlue of is he coefficies i e,, 7 ems i e esio of e i AP he 7 7 em
More informationHOMEWORK 6 - INTEGRATION. READING: Read the following parts from the Calculus Biographies that I have given (online supplement of our textbook):
MAT 3 CALCULUS I 5.. Dokuz Eylül Uiversiy Fculy of Sciece Deprme of Mhemics Isrucors: Egi Mermu d Cell Cem Srıoğlu HOMEWORK 6 - INTEGRATION web: hp://kisi.deu.edu.r/egi.mermu/ Tebook: Uiversiy Clculus,
More informationUnit 1 Chapter-3 Partial Fractions, Algebraic Relationships, Surds, Indices, Logarithms
Uit Chpter- Prtil Frctios, Algeric Reltioships, Surds, Idices, Logriths. Prtil Frctios: A frctio of the for 7 where the degree of the uertor is less th the degree of the deoitor is referred to s proper
More informationForced Oscillation of Nonlinear Impulsive Hyperbolic Partial Differential Equation with Several Delays
Jourl of Applied Mhemics d Physics, 5, 3, 49-55 Published Olie November 5 i SciRes hp://wwwscirporg/ourl/mp hp://dxdoiorg/436/mp5375 Forced Oscillio of Nolier Impulsive Hyperbolic Pril Differeil Equio
More informationReinforcement Learning
Reiforceme Corol lerig Corol polices h choose opiml cios Q lerig Covergece Chper 13 Reiforceme 1 Corol Cosider lerig o choose cios, e.g., Robo lerig o dock o bery chrger o choose cios o opimize fcory oupu
More informationFourier Series and Applications
9/7/9 Fourier Series d Applictios Fuctios epsio is doe to uderstd the better i powers o etc. My iportt probles ivolvig prtil dieretil equtios c be solved provided give uctio c be epressed s iiite su o
More informationA GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD
Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method
More informationON SOME FRACTIONAL PARABOLIC EQUATIONS DRIVEN BY FRACTIONAL GAUSSIAN NOISE
IJRRAS 6 3) Februry www.rppress.com/volumes/vol6issue3/ijrras_6_3_.pdf ON SOME FRACIONAL ARABOLIC EQUAIONS RIVEN BY FRACIONAL GAUSSIAN NOISE Mhmoud M. El-Bori & hiri El-Sid El-Ndi Fculy of Sciece Alexdri
More informationDepartment of Mathematical and Statistical Sciences University of Alberta
MATH 4 (R) Wier 008 Iermediae Calculus I Soluios o Problem Se # Due: Friday Jauary 8, 008 Deparme of Mahemaical ad Saisical Scieces Uiversiy of Albera Quesio. [Sec.., #] Fid a formula for he geeral erm
More informationMath 153: Lecture Notes For Chapter 1
Mth : Lecture Notes For Chpter Sectio.: Rel Nubers Additio d subtrctios : Se Sigs: Add Eples: = - - = - Diff. Sigs: Subtrct d put the sig of the uber with lrger bsolute vlue Eples: - = - = - Multiplictio
More informationEE757 Numerical Techniques in Electromagnetics Lecture 9
EE757 uericl Techiques i Elecroeics Lecure 9 EE757 06 Dr. Mohed Bkr Diereil Equios Vs. Ierl Equios Ierl equios ke severl ors e.. b K d b K d Mos diereil equios c be epressed s ierl equios e.. b F d d /
More informationNOTES ON BERNOULLI NUMBERS AND EULER S SUMMATION FORMULA. B r = [m = 0] r
NOTES ON BERNOULLI NUMBERS AND EULER S SUMMATION FORMULA MARK WILDON. Beroulli umbers.. Defiiio. We defie he Beroulli umbers B m for m by m ( m + ( B r [m ] r r Beroulli umbers re med fer Joh Beroulli
More informationCoefficient Inequalities for Certain Subclasses. of Analytic Functions
I. Jourl o Mh. Alysis, Vol., 00, o. 6, 77-78 Coeiie Iequliies or Ceri Sulsses o Alyi Fuios T. Rm Reddy d * R.. Shrm Deprme o Mhemis, Kkiy Uiversiy Wrgl 506009, Adhr Prdesh, Idi reddyr@yhoo.om, *rshrm_005@yhoo.o.i
More informationInternational Journal of Mathematical Archive-5(1), 2014, Available online through ISSN
Itertiol Jourl of Mthemticl Archive-5( 4 93-99 Avilble olie through www.ijm.ifo ISSN 9 546 GENERALIZED FOURIER TRANSFORM FOR THE GENERATION OF COMPLE FRACTIONAL MOMENTS M. Gji F. Ghrri* Deprtmet of Sttistics
More informationTaylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best
Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =
More informationGraphing Review Part 3: Polynomials
Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)
More information[ m] x = 0.25cos 20 t sin 20 t m
. x.si ( 5 s [ ] CHAPER OSCILLAIONS x ax (.( ( 5 6. s s ( ( ( xax. 5.7 s s. x.si [] x. cos s Whe, x a x.5. s 5s.6 s x. x( x cos + si a f ( ( [ ] x.5cos +.59si. ( ( cos α β cosαcos β + siαsi β x Acos φ
More informationName: Period: Date: 2.1 Rules of Exponents
SM NOTES Ne: Period: Dte:.1 Rules of Epoets The followig properties re true for ll rel ubers d b d ll itegers d, provided tht o deoitors re 0 d tht 0 0 is ot cosidered. 1 s epoet: 1 1 1 = e.g.) 7 = 7,
More informationON BILATERAL GENERATING FUNCTIONS INVOLVING MODIFIED JACOBI POLYNOMIALS
Jourl of Sciece d Ars Yer 4 No 227-6 24 ORIINAL AER ON BILATERAL ENERATIN FUNCTIONS INVOLVIN MODIFIED JACOBI OLYNOMIALS CHANDRA SEKHAR BERA Muscri received: 424; Acceed er: 3524; ublished olie: 3624 Absrc
More informationModified Ratio and Product Estimators for Estimating Population Mean in Two-Phase Sampling
America Joural of Operaioal esearch 06, 6(3): 6-68 DOI: 0.593/j.ajor.060603.0 Moifie aio a Prouc Esimaors for Esimaig Populaio Mea i Two-Phase Samplig Subhash Kumar Yaav, Sa Gupa, S. S. Mishra 3,, Alok
More informationf(t)dt 2δ f(x) f(t)dt 0 and b f(t)dt = 0 gives F (b) = 0. Since F is increasing, this means that
Uiversity of Illiois t Ur-Chmpig Fll 6 Mth 444 Group E3 Itegrtio : correctio of the exercises.. ( Assume tht f : [, ] R is cotiuous fuctio such tht f(x for ll x (,, d f(tdt =. Show tht f(x = for ll x [,
More informationk=1 s k (x) (3) and that the corresponding infinite series may also converge; moreover, if it converges, then it defines a function S through its sum
0. L Hôpital s rule You alreay kow from Lecture 0 that ay sequece {s k } iuces a sequece of fiite sums {S } through S = s k, a that if s k 0 as k the {S } may coverge to the it k= S = s s s 3 s 4 = s k.
More informationONE RANDOM VARIABLE F ( ) [ ] x P X x x x 3
The Cumulive Disribuio Fucio (cd) ONE RANDOM VARIABLE cd is deied s he probbiliy o he eve { x}: F ( ) [ ] x P x x - Applies o discree s well s coiuous RV. Exmple: hree osses o coi x 8 3 x 8 8 F 3 3 7 x
More informationCalculus Limits. Limit of a function.. 1. One-Sided Limits...1. Infinite limits 2. Vertical Asymptotes...3. Calculating Limits Using the Limit Laws.
Limi of a fucio.. Oe-Sided..... Ifiie limis Verical Asympoes... Calculaig Usig he Limi Laws.5 The Squeeze Theorem.6 The Precise Defiiio of a Limi......7 Coiuiy.8 Iermediae Value Theorem..9 Refereces..
More informationResearch Article Sums of Products of Cauchy Numbers, Including Poly-Cauchy Numbers
Hiawi Publishig Corporatio Joural of Discrete Matheatics Volue 2013, Article ID 373927, 10 pages http://.oi.org/10.1155/2013/373927 Research Article Sus of Proucts of Cauchy Nubers, Icluig Poly-Cauchy
More informationDavid Randall. ( )e ikx. k = u x,t. u( x,t)e ikx dx L. x L /2. Recall that the proof of (1) and (2) involves use of the orthogonality condition.
! Revised April 21, 2010 1:27 P! 1 Fourier Series David Radall Assume ha u( x,) is real ad iegrable If he domai is periodic, wih period L, we ca express u( x,) exacly by a Fourier series expasio: ( ) =
More informationLIPSCHITZ ESTIMATES FOR MULTILINEAR COMMUTATOR OF MARCINKIEWICZ OPERATOR
Reseh d ouiios i heis d hei Siees Vo. Issue Pges -46 ISSN 9-699 Puished Oie o Deee 7 Joi Adei Pess h://oideiess.e IPSHITZ ESTIATES FOR UTIINEAR OUTATOR OF ARINKIEWIZ OPERATOR DAZHAO HEN Dee o Siee d Ioio
More informationEXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D. S. Palimkar
Ieraioal Joural of Scieific ad Research Publicaios, Volue 2, Issue 7, July 22 ISSN 225-353 EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D S Palikar Depare of Maheaics, Vasarao Naik College, Naded
More informationConvergence rates of approximate sums of Riemann integrals
Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsuku Tsuku Irki 5-857 Jp tski@mth.tsuku.c.jp Keywords : covergece rte; Riem sum; Riem
More information«A first lesson on Mathematical Induction»
Bcgou ifotio: «A fist lesso o Mtheticl Iuctio» Mtheticl iuctio is topic i H level Mthetics It is useful i Mtheticl copetitios t ll levels It hs bee coo sight tht stuets c out the poof b theticl iuctio,
More informationAlgebra 2 Important Things to Know Chapters bx c can be factored into... y x 5x. 2 8x. x = a then the solutions to the equation are given by
Alger Iportt Thigs to Kow Chpters 8. Chpter - Qudrtic fuctios: The stdrd for of qudrtic fuctio is f ( ) c, where 0. c This c lso e writte s (if did equl zero, we would e left with The grph of qudrtic fuctio
More informationLaplace Transform. Definition of Laplace Transform: f(t) that satisfies The Laplace transform of f(t) is defined as.
Lplce Trfor The Lplce Trfor oe of he hecl ool for olvg ordry ler dfferel equo. - The hoogeeou equo d he prculr Iegrl re olved oe opero. - The Lplce rfor cover he ODE o lgerc eq. σ j ple do. I he pole o
More informationDERIVING THE DEMAND CURVE ASSUMING THAT THE MARGINAL UTILITY FUNCTIONS ARE LINEAR
Bllei UASVM, Horilre 65(/008 pissn 1843-554; eissn 1843-5394 DERIVING THE DEMAND CURVE ASSUMING THAT THE MARGINAL UTILITY FUNCTIONS ARE LINEAR Crii C. MERCE Uiveriy of Agrilrl iee d Veeriry Mediie Clj-Npo,
More informationComparison between Fourier and Corrected Fourier Series Methods
Malaysia Joural of Mahemaical Scieces 7(): 73-8 (13) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Joural homepage: hp://eispem.upm.edu.my/oural Compariso bewee Fourier ad Correced Fourier Series Mehods 1
More informationMATRIX ALGEBRA, Systems Linear Equations
MATRIX ALGEBRA, Systes Lier Equtios Now we chge to the LINEAR ALGEBRA perspective o vectors d trices to reforulte systes of lier equtios. If you fid the discussio i ters of geerl d gets lost i geerlity,
More informationSome Properties of Semi-E-Convex Function and Semi-E-Convex Programming*
The Eighh Ieraioal Symposium o Operaios esearch ad Is Applicaios (ISOA 9) Zhagjiajie Chia Sepember 2 22 29 Copyrigh 29 OSC & APOC pp 33 39 Some Properies of Semi-E-Covex Fucio ad Semi-E-Covex Programmig*
More information1 Notes on Little s Law (l = λw)
Copyrigh c 26 by Karl Sigma Noes o Lile s Law (l λw) We cosider here a famous ad very useful law i queueig heory called Lile s Law, also kow as l λw, which assers ha he ime average umber of cusomers i
More informationConvergence rates of approximate sums of Riemann integrals
Jourl of Approximtio Theory 6 (9 477 49 www.elsevier.com/locte/jt Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsukub, Tsukub Ibrki
More informationIntegral Equations and their Relationship to Differential Equations with Initial Conditions
Scece Refleco SR Vol 6 wwwscecereflecocom Geerl Leers Mhemcs GLM 6 3-3 Geerl Leers Mhemcs GLM Wese: hp://wwwscecereflecocom/geerl-leers--mhemcs/ Geerl Leers Mhemcs Scece Refleco Iegrl Equos d her Reloshp
More informationChapter 2: Time-Domain Representations of Linear Time-Invariant Systems. Chih-Wei Liu
Caper : Time-Domai Represeaios of Liear Time-Ivaria Sysems Ci-Wei Liu Oulie Iroucio Te Covoluio Sum Covoluio Sum Evaluaio Proceure Te Covoluio Iegral Covoluio Iegral Evaluaio Proceure Iercoecios of LTI
More informationOn computing two special cases of Gauss hypergeometric function
O comuig wo secil cses of Guss hyergeomeric fucio Mohmmd Msjed-Jmei Wolfrm Koef b * Derme of Mhemics K.N.Toosi Uiversiy of Techology P.O.Bo 65-68 Tehr Ir E-mil: mmjmei@u.c.ir mmjmei@yhoo.com b* Isiue of
More informationTransient Solution of the M/M/C 1 Queue with Additional C 2 Servers for Longer Queues and Balking
Jourl of Mhemics d Sisics 4 (): 2-25, 28 ISSN 549-3644 28 Sciece ublicios Trsie Soluio of he M/M/C Queue wih Addiiol C 2 Servers for Loger Queues d Blkig R. O. Al-Seedy, A. A. El-Sherbiy,,2 S. A. EL-Shehwy
More informationAPPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES
Scietific Reserch of the Istitute of Mthetics d Coputer Sciece 3() 0, 5-0 APPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES Jolt Borows, Le Łcińs, Jowit Rychlews Istitute of Mthetics,
More informationOrthogonal Function Solution of Differential Equations
Royal Holloway Uiversity of Loo Departet of Physics Orthogoal Fuctio Solutio of Differetial Equatios trouctio A give oriary ifferetial equatio will have solutios i ters of its ow fuctios Thus, for eaple,
More informationSampling Example. ( ) δ ( f 1) (1/2)cos(12πt), T 0 = 1
Samplig Example Le x = cos( 4π)cos( π). The fudameal frequecy of cos 4π fudameal frequecy of cos π is Hz. The ( f ) = ( / ) δ ( f 7) + δ ( f + 7) / δ ( f ) + δ ( f + ). ( f ) = ( / 4) δ ( f 8) + δ ( f
More informationCape Cod Community College
Cpe Cod Couity College Deprtetl Syllus Prepred y the Deprtet of Mthetics Dte of Deprtetl Approvl: Noveer, 006 Dte pproved y Curriculu d Progrs: Jury 9, 007 Effective: Fll 007 1. Course Nuer: MAT110 Course
More informationIdeal Amplifier/Attenuator. Memoryless. where k is some real constant. Integrator. System with memory
Liear Time-Ivaria Sysems (LTI Sysems) Oulie Basic Sysem Properies Memoryless ad sysems wih memory (saic or dyamic) Causal ad o-causal sysems (Causaliy) Liear ad o-liear sysems (Lieariy) Sable ad o-sable
More informationS.E. Sem. III [EXTC] Applied Mathematics - III
S.E. Sem. III [EXTC] Applied Mhemic - III Time : 3 Hr.] Prelim Pper Soluio [Mrk : 8 Q.() Fid Lplce rform of e 3 co. [5] A.: L{co }, L{ co } d ( ) d () L{ co } y F.S.T. 3 ( 3) Le co 3 Q.() Prove h : f (
More informationdegree non-homogeneous Diophantine equation in six unknowns represented by x y 2z
Scholrs Jourl of Egieerig d Techology (SJET Sch. J. Eg. Tech., ; (A:97- Scholrs Acdeic d Scietific Pulisher (A Itertiol Pulisher for Acdeic d Scietific Resources www.sspulisher.co ISSN -X (Olie ISSN 7-9
More informationFormulae For. Standard Formulae Of Integrals: x dx k, n 1. log. a dx a k. cosec x.cot xdx cosec. e dx e k. sec. ax dx ax k. 1 1 a x.
Forule For Stndrd Forule Of Integrls: u Integrl Clculus By OP Gupt [Indir Awrd Winner, +9-965 35 48] A B C D n n k, n n log k k log e e k k E sin cos k F cos sin G tn log sec k OR log cos k H cot log sin
More informationSUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11
SUTCLIFFE S NOTES: CALCULUS SWOKOWSKI S CHAPTER Ifiite Series.5 Altertig Series d Absolute Covergece Next, let us cosider series with both positive d egtive terms. The simplest d most useful is ltertig
More informationInternational Journal of Mathematical Archive-3(1), 2012, Page: Available online through
eril Jrl f Mhemicl rchive-3 Pge: 33-39 vilble lie hrgh wwwijmif NTE N UNFRM MTRX SUMMBLTY Shym Ll Mrdl Veer Sigh d Srbh Prwl 3* Deprme f Mhemics Fcly f Sciece Brs Hid Uiversiy Vrsi UP - ND E-mil: shym_ll@rediffmilcm
More information1. Solve by the method of undetermined coefficients and by the method of variation of parameters. (4)
7 Differeial equaios Review Solve by he mehod of udeermied coefficies ad by he mehod of variaio of parameers (4) y y = si Soluio; we firs solve he homogeeous equaio (4) y y = 4 The correspodig characerisic
More informationOn New Prajapati-Shukla Functions And Polynomials
Mhei Toy Vol.7De- 4-39 ISSN 976-38 O New Prji-Shul Fuio Polyoil J. C. Prji. K. Shul Dere of Mheil Siee Fuly of Tehology gieerig Chror Uiveriy of Siee Tehology Chg -388 4 Ii. Dere of lie Mhei Huiie S.V.Niol
More informationFractional Order EOQ Model with Linear Trend of Time-Dependent Demand
I.J. Iellige Sysems d Applicios, 5,, -5 Published Olie Februry 5 i MES (hp://.mecs-press.org/ OI:.585/ijis.5..6 Frciol Order EOQ Model ih Lier red o ime-epede emd Asim Kumr s, p Kumr Roy eprme o Applied
More informationOH BOY! Story. N a r r a t iv e a n d o bj e c t s th ea t e r Fo r a l l a g e s, fr o m th e a ge of 9
OH BOY! O h Boy!, was or igin a lly cr eat ed in F r en ch an d was a m a jor s u cc ess on t h e Fr en ch st a ge f or young au di enc es. It h a s b een s een by ap pr ox i ma t ely 175,000 sp ect at
More informationTopic 4 Fourier Series. Today
Topic 4 Fourier Series Toy Wves with repetig uctios Sigl geertor Clssicl guitr Pio Ech istrumet is plyig sigle ote mile C 6Hz) st hrmoic hrmoic 3 r hrmoic 4 th hrmoic 6Hz 5Hz 783Hz 44Hz A sigle ote will
More informationMean Square Convergent Finite Difference Scheme for Stochastic Parabolic PDEs
America Joural of Compuaioal Mahemaics, 04, 4, 80-88 Published Olie Sepember 04 i SciRes. hp://www.scirp.org/joural/ajcm hp://dx.doi.org/0.436/ajcm.04.4404 Mea Square Coverge Fiie Differece Scheme for
More informationLecture 6 Testing Nonlinear Restrictions 1. The previous lectures prepare us for the tests of nonlinear restrictions of the form:
Eco 75 Lecture 6 Testig Noliear Restrictios The previous lectures prepare us for the tests of oliear restrictios of the form: H 0 : h( 0 ) = 0 versus H : h( 0 ) 6= 0: () I this lecture, we cosier Wal,
More informationMathematical Induction (selected questions)
Mtheticl Iductio (selected questios). () Let P() e the propositio : For P(), L.H.S. R.H.S., P() is true. Assue P() is true for soe turl uer, tht is, () For P( ),, y () By the Priciple of Mtheticl Iductio,
More informationReview for the Midterm Exam.
Review for he iderm Exm Rememer! Gross re e re Vriles suh s,, /, p / p, r, d R re gross res 2 You should kow he disiio ewee he fesile se d he udge se, d kow how o derive hem The Fesile Se Wihou goverme
More information[Nachlass] of the Theory of the Arithmetic-Geometric Mean and the Modulus Function.
[Nchlss] of he Theory of he Arihmeic-Geomeric Me d he Modulus Fucio Defiiio d Covergece of he Algorihm [III 6] Le d e wo posiive rel mgiudes d le From hem we form he wo sequeces: i such wy h y wo correspodig
More informationSTK4080/9080 Survival and event history analysis
STK48/98 Survival ad eve hisory aalysis Marigales i discree ime Cosider a sochasic process The process M is a marigale if Lecure 3: Marigales ad oher sochasic processes i discree ime (recap) where (formally
More informationDegree of Approximation of Conjugate of Signals (Functions) by Lower Triangular Matrix Operator
Alied Mhemics 2 2 448-452 doi:.4236/m.2.2226 Pulished Olie Decemer 2 (h://www.scirp.org/jourl/m) Degree of Aroimio of Cojuge of Sigls (Fucios) y Lower Trigulr Mri Oeror Asrc Vishu Nry Mishr Huzoor H. Kh
More informationFIXED FUZZY POINT THEOREMS IN FUZZY METRIC SPACE
Mohia & Samaa, Vol. 1, No. II, December, 016, pp 34-49. ORIGINAL RESEARCH ARTICLE OPEN ACCESS FIED FUZZY POINT THEOREMS IN FUZZY METRIC SPACE 1 Mohia S. *, Samaa T. K. 1 Deparme of Mahemaics, Sudhir Memorial
More informationFunctions, Limit, And Continuity
Fucios, Limi d coiuiy of fucio Fucios, Limi, Ad Coiuiy. Defiiio of fucio A fucio is rule of correspodece h ssocies wih ech ojec i oe se clled f from secod se. The se of ll vlues so oied is he domi, sigle
More informationVARIATIONAL ITERATION METHOD (VIM) FOR SOLVING PARTIAL INTEGRO-DIFFERENTIAL EQUATIONS
Jourl of Theoreicl d Applied Iformio Techology h Jue 16. Vol.88. No. 5-16 JATIT & LLS. All righs reserved. ISSN: 199-8645 www.ji.org E-ISSN: 1817-195 VARIATIONAL ITERATION ETHOD VI FOR SOLVING PARTIAL
More informationWestchester Community College Elementary Algebra Study Guide for the ACCUPLACER
Westchester Commuity College Elemetry Alger Study Guide for the ACCUPLACER Courtesy of Aims Commuity College The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetry
More informationSolutions to selected problems from the midterm exam Math 222 Winter 2015
Soluios o seleced problems from he miderm eam Mah Wier 5. Derive he Maclauri series for he followig fucios. (cf. Pracice Problem 4 log( + (a L( d. Soluio: We have he Maclauri series log( + + 3 3 4 4 +...,
More informationAssessment Center Elementary Algebra Study Guide for the ACCUPLACER (CPT)
Assessmet Ceter Elemetr Alger Stud Guide for the ACCUPLACER (CPT) The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetr Alger test. Reviewig these smples will give
More informationmywbut.com Lesson 13 Representation of Sinusoidal Signal by a Phasor and Solution of Current in R-L-C Series Circuits
wut.co Lesson 3 Representtion of Sinusoil Signl Phsor n Solution of Current in R-L-C Series Circuits wut.co In the lst lesson, two points were escrie:. How sinusoil voltge wvefor (c) is generte?. How the
More information(A) 1 (B) 1 + (sin 1) (C) 1 (sin 1) (D) (sin 1) 1 (C) and g be the inverse of f. Then the value of g'(0) is. (C) a. dx (a > 0) is
[STRAIGHT OBJECTIVE TYPE] l Q. Th vlu of h dfii igrl, cos d is + (si ) (si ) (si ) Q. Th vlu of h dfii igrl si d whr [, ] cos cos Q. Vlu of h dfii igrl ( si Q. L f () = d ( ) cos 7 ( ) )d d g b h ivrs
More informationResearch Article Refinements of Aczél-Type Inequality and Their Applications
Hidwi Pulishig Corportio Jourl of Applied Mthetics Volue 04, Article ID 58354, 7 pges http://dxdoiorg/055/04/58354 Reserch Article Refieets of Aczél-Type Iequlity d Their Applictios Jigfeg Ti d We-Li Wg
More informationIn an algebraic expression of the form (1), like terms are terms with the same power of the variables (in this case
Chpter : Algebr: A. Bckgroud lgebr: A. Like ters: I lgebric expressio of the for: () x b y c z x y o z d x... p x.. we cosider x, y, z to be vribles d, b, c, d,,, o,.. to be costts. I lgebric expressio
More informationFOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),
FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To -periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the
More informationDerivation of the Metal-Semiconductor Junction Current
.4.4. Derivio of e Mel-Seiouor uio Curre.4.4.1.Derivio of e iffuio urre We r fro e epreio for e ol urre e iegre i over e wi of e epleio regio: q( µ + D (.4.11 wi be rewrie b uig -/ uliplig bo ie of e equio
More informationAn arithmetic interpretation of generalized Li s criterion
A riheic ierpreio o geerlized Li crierio Sergey K. Sekkii Lboroire de Phyique de l Mière Vive IPSB Ecole Polyechique Fédérle de Lue BSP H 5 Lue Swizerld E-il : Serguei.Sekki@epl.ch Recely we hve eblihed
More information