Systems Variables and Structural Controllability: An Inverted Pendulum Case
|
|
- Derek Little
- 5 years ago
- Views:
Transcription
1 Reserch Journl of Applied Sciences, Engineering nd echnology 6(: 46-4, 3 ISSN: ; e-issn: Mxwell Scienific Orgniion, 3 Submied: Jnury 5, 3 Acceped: Mrch 7, 3 Published: November, 3 Sysems Vribles nd Srucurl Conrollbiliy: An Invered Pendulum Cse Qing M nd Xio-Qun Li School of Mechnicl nd Auomoive Engineering, Hubei Universiy of Ars nd Science, XingYng, Chin Deprmen of Auomobile Engineering, ChengDe Peroleum College, Cheng De, Chin Absrc: In order o explore he essence of srucurl conrollbiliy, srucurl conrollbiliy of invered pendulum is nlyed in his presened sudy nd sufficien conrollbiliy condiion of clss of perurbed liner sysem is obined, which is essenil o prove he srucurl conrollbiliy for he perurbed invered pendulum. wo differen srucured models of invered pendulum re consruced. Srucurl conrollbiliy of boh cses re discussed nd compred, which shows h he usul model used in conroller design for invered pendulum is jus specil cse of norml model for invered pendulum. Keywords: Perurbed liner sysem, Rionl Funcions Mrix wih muli-prmeer (RFM, srucurl conrollbiliy, sysem vrible INRODUCION Here consider liner ime-invrin sysem denoed by differenil equion: x Ax Bu ( where, we denoe by x R n, u R m, A R n n, B R n m he sysem ses, sysem inpus, coefficien mrix nd inpu mrix, respecively. Define is conrollbiliy mrix by Q c = [B, AB,, A n- B] nd composie mrix by Q (s = [si - A, B]. I is clerly h sysem ( is compleely conrollble if nd only if rnk (Q c = n, which is equivlen o h for ll s C (C is he complex number field, rnk (Q (s = n. Generlly, conrollbiliy is generic propery (Lee nd Mrkus, 967, h is, if we denoe ll he conrollble sysem se by Y, hen Y is open nd dense se. In rdiionl sysemic heory, he mhemic model ( is described excly, h is, ll he elemens of coefficien mrix nd inpu mrix re ccurely numericl numbers. However, from he view poin of engineering, i is difficul o ge such ccure model, which is o sy h here my exis unceriny in elemens of A nd B. mos of hese uncerinies re resuled from he uncerinies of sysem vribles. If we include hese sysem vribles in physicl model, hen i urns ou o be sysem model wih vribles, which we cn sy srucurl model. he invered pendulum is muli-prmeer sysem nd mos reserch on i is conrol sregy. Vrible srucure mehod is used o conrol he invered pendulum, in which he sliding mold conrol lw ws obined by Lypunov mehod (M nd Lu, 9. An nher Lypunov bsed conrol mehod ws proposed o conrol he invered pendulum (Ibne e l., 5. Oher mehod such h he pssive ful olern mehod ws used o conrol double invered pendulum (Niemnn nd Sousrup, 5. On he oher hnd, he rod opiml selecion problem in muli-link invered pendulum ws reserched (Bei-Chen nd Shung, 6. By using he conrollbiliy mrix Q c which is obined by differen rods combinion, he smlles Eigen vlues of differen conrollbiliy mrices Q c re clculed. hen he sysem which is corresponding o he smlles Eigen vlue is considered o be he opiml choice. In h sudy, he problem h rod is considered o be sysem vrible is concerned, neverheless, he clculion is sill process over he rel number field. From he view poin of srucurl model, i is worh reserching wheher he sysem vribles ffec he sysem propery (such s conrollbiliy. In his sudy srucurl conrollbiliy of invered pendulum is nlyed. wo srucurl models of invered pendulum re consruced. For clss of perurbed liner sysem sufficien condiion of conrollbiliy is presened, which is bsemen for nlying srucurl conrollbiliy of he second ype of srucurl model. Also his sudy gives reserch h wheher sysem vribles ffec he conrollbiliy of invered pendulum from he view of srucurl model poin, h is, srucurl conrollbiliy. In order o explore he essence of srucurl conrollbiliy, in his sudy we jus consider he lineried invered pendulum model. Corresponding Auhor: Qing M, School of Mechnicl nd Auomoive Engineering, Hubei Universiy of Ars nd Science, Xingyng, 4453, Chin, el.:
2 Res. J. Appl. Sci. Eng. echnol., 6(: 46-4, 3 SOME DEFINIIONS AND WO INERMEDIAE CONCLUSION For liner ime-invrin sysem (, n imporn concepion is conrollbiliy, h is, wheher we cn find some suible inpus u such h for ny given iniil se x i cn be seered o ny desired desinion se. Wih he liner ime-invrin sysems, here exis some definiions bou conrollbiliy s follow: Definiion : For liner ime-invrin sysem ( if here exis some dmissible conrol u ( (wih he consrin: u B x A ( e W C(, ( P where, (x - P is he iniil se ime. hen he soluion x ( of liner ime-invrin sysem ( ime cn be denoed o: A A( x( e ( x P e Bu( A A( A WC (, ( e ( x P e BB e x P A A A A e W (, C e ( x P e e BB ( x P u( A A e ( x P e W (, W (, ( x P C C during he ime inervl (, nd rnsfer he iniil se x ( o x ( =, hen se x ( is sid o be conrollble over he inervl (,, or conrollble ime. Definiion : If ny se of liner ime-invrin sysem ( is conrollble over he ime inervl (,, hen he sysem ( is sid o be compleely conrollble. Here we consider clss of liner ime-invrin sysem ( wih perurbion denoed by: x Ax Bu P ( where, we denoe by x R n, u R m, A R n n, B R n m nd P R n he sysem ses, sysem inpus, coefficien mrix, inpu mrix nd consn perurbion respecively. Now define symmeric mrix: A e ( x P e ( x P A By he Definiion, liner ime-invrin sysem ( is conrollble. Lemm : For liner ime-invrin sysem (, he non-singulriy of conrollbiliy Grmin mrix W C (, is equivlen o h rnk (Q c = n. hen for sysem (, we define mrix denoed by c, which is lso sid o be he conrollbiliy mrix. Anoher conclusion bou conrollbiliy of sysem ( is immedie. heorem : Liner ime-invrin sysem ( is conrollble if is conrollbiliy mrix c is of full rnk, h is, rnk ( c = n. Proof: By he heorem nd Lemm, he proof is immedie. A ( A ( (3 W (, e B( B ( e d C SOME NOAIONS OF SRUCURAL CONROLLABILIY For sysem ( symmeric mrix W C (, is sid o be he conrollbiliy Grmin mrix. Similrly, for sysem ( we cn lso define is symmeric conrollbiliy Grmin mrix denoed by C (, which differ from h of sysem (. For sysem (, here gives conclusion bou is conrollbiliy. heorem : Liner ime-invrin sysem ( is conrollble if is conrollbiliy Grmin mrix C (, is nonsingulr. Proof: Becuse C (, is nonsingulr, hen (, exiss. Suppose he sysem inpu is wih he form s follow: 47 In clssicl conrol heory, he elemens of mrices A nd B re considered o be ccure excly, bu for some physicl resons some of hese elemens re pproxime numericl numbers. So someimes he vribles re preserved in sysem mhemic model. he model wih vribles is considered o be more resonble nd only he known eros re decided o be ccure. he concepion of srucurl conrollbiliy is presened firs (Lin, 974. During he nlysis of srucurl conrollbiliy only he srucure of mrices A nd B is concerned, which mens h he objecive reserched is no he numericlly known mrices A nd B, bu he corresponding sme dimension srucured mrices denoed by nd, respecively.
3 Res. J. Appl. Sci. Eng. echnol., 6(: 46-4, 3 Definiion 3: he elemen of srucured mrix is eher ero or undeermined vlue differing from ny oher undeermined elemen. If ll he elemens of srucured mrix re concree numericl numbers, hen i is considered o be dmissible numericl reliion. If he numericl mrices M nd N re boh he dmissible numericl reliion of srucured mrix, hen hey re srucurlly equivlen. Oher srucured mrices re proposed by reserchers (Anderson nd Clemens, 98; Ymd nd Luenberger, 985; Muro, 999 fer Lin s srucurl mrix. However, here exiss problem h he inverse mrix of hese srucured mrices is no longer o be heir corresponding srucured mrix. Clculing inverse mrix is frequen during exploring he properies of sysem ( nd (, so ll hese srucured mrices re no fied. hen clss of more generl srucured mrix ws proposed, which is cll rionl funcions mrix wih Muli-Prmeer (RFM (Lu nd Wei, 99, 994. Le,, be he prmeer vecor in sysem ( or (. R q is clled he definiion domin of r or prmeer spce. Le F (r denoe he field of ll rionl funcions wih rel coefficiens in q prmeers r,, r q. F (r [λ] denoe he polynomil ring in λ wih coefficiens in members over F (r. Definiion 4: If ech member of mrix M is member over F (r, hen he mrix M is sid o be rionl mrix or mrix over F (r. If ll he mrices of sysems re consider o be RFM, hen he sysems is sid o be rionl funcion sysems wih muliprmeers r or sysems over F (r. he mos dvnge of RFM over oher srucured mrices is h he inverse mrix of RFM is sill RFM, which is more convenien in nlying sysem properies. he srucured mrix used in his presened sudy is RFM defined wih Definiion 4. Now we cn define he mhemic model of liner ime-invrin sysem over F (r: x Arx ( Bru ( (4 or liner ime-invrin sysem wih consn perurbion over F (r: x Ar ( xbru ( P (5 Fig. : One sge invered pendulum sysem Proof: By heorem nd Definiion 4, he proof is immedie. SRUCURAL CONROLLABILIY ANALYSES FOR INVERED PENDULUM he invered pendulum now is considered o be norml sysem, so here we jus give he simple illusrion figure shown by Fig., if ny doub my refer o Shen e l. (3. In order o exploring he srucurl properies here he one sge lineried invered pendulum is concerned. In his secion wo differen lineried cses re considered s follow: Lineriion up-vericl equilibrium poin: his is he usul cse sudied. he dynmic behvior of one sge invered pendulum is described by 4 ses, h is, posiion of cr, velociy of cr, ngulr of rod nd velociy of rod ngulr, denoed by,, nd respecively. In order o deermine he mhemic model governing his sysem, we firs concern he horionl posiion of he cr, menwhile he posiion of he pendulum is +lsin. hen by Newon s second lw, h is, mss ccelerion = force, in he horionl direcion yields: d M d ( l sin m u (6 Agin pplying he Newon s second lw o pendulum mss, in he up-vericl direcion o he rod, we cn ge: m cos ml mg sin (7 where, r denoes he prmeer vecor in sysems (4 nd (5. heorem 3: Liner ime-invrin sysem wih consn perurbion (5 is srucurl conrollble if is conrollbiliy mrix c (r is of full rnk, h is, rnk c (r = n. 48 where, noion g is he ccelerion of grviy in he downwrd direcion. he differenil Eq. (6 nd (7 re nonliner, we need furher simplificion. Now le, where is he equilibrium poin, which we cll i he sic opering poin. is consn, is he deviion h
4 Res. J. Appl. Sci. Eng. echnol., 6(: 46-4, 3 deprs from. When = nd is no lrge, we cn ge is lineried mhemic model described by se spce form: where, X A( r X B( r U (8 mg M Ar ( M Br ( ( M m g Ml Ml X Le prmeer r = (M, m, l, g R 4, hen he conrollbiliy mrix Q C (r is: mg M M l mg M M l ( M mg Ml M l ( M m g Ml M l 3 Qc ( r B AB A B A B Since deerminn of Q C (r is: g M l M m l m nd when (M + m l = m, Q c (r hs full rnk nd he sysem is compleely conrollble. I is known h relion (M + m l = m is n lgebric vriey. Le se S be S = {r Q c (r = }, hen he Lebesgue mesuremen of invered pendulum (8 being nonsrucurl conrollble is ero, which mens h invered pendulum (8 is srucurl conrollble. sin cos (9 cos cos( cos cos sin sin cos sin cos ( he effeciveness of Eq. ( depends on he ssumpion h 6. becuse of sin. According o Eq. (9, we hve: d sin cos d sin cos ( ( Subsiuing he Equliies ( nd ( ino Eq. (6, we cn hve n pproxime differenil equion for he invered pendulum sysem: ( M m ml cos u (3 Agin pplying he sme mehod o Eq. (7, we hve: cos l g g (4 Becuse of: d cos sin ( Equion (4 kes he form: cos l g g (5 cos sin According o Eq. (3 nd (4, we hve: Lineriion ny up opering poin: Wh we hve discussed in he bove subsecion is he usul lineried cse, under which condiion h he sic opering poin is =. Now we will discuss lrge rnge h he ngulr kes vlue from he inervl [-6, 6 ]. If we do his, hen we re sure h dynmic model described by differenil Eq. (6 nd (7 is conrollble he opering poin = is excly relible when 6. Becuse he equilibrium poin, now we pply, where is smll perurbion, which is close o ero. According o he fc h cos nd sin, hen pplying hese o he rigonomeric expnsions, we hve: sin sin( sin cos cos sin 49 ( M m g cos cos ( M m mcos l ( M m mcos ( M m g sin ( M m m cos l u l M m ( g ( M m mcos l cos cos g sin u (6 M m mgcos mgcos sin u M m mcos M m mcos M m mcos mgcos mgcos sin u M m mcos M m mcos M m mcos (7
5 Res. J. Appl. Sci. Eng. echnol., 6(: 46-4, 3 Now we denoe he se vecor o being wih he form: X hen we my wrie Eq. (4 nd (7 in se spce form: X mg cos M m m cos ( M m g cos ( M m m cos l mg cossin + M mmcos M mmcos (8 u cos ( M m gsin ( M mmcos l ( M mmcos l Here we consider he cse in subsecion A h nd subsiue i ino Eq. (8. hen i urns ou Eq. (6, which mens h Eq. (8 is relible. Supposing h,, i.e., his invered pendulum runs he sic poin. hen by Eq. (6 we cn ge h: u ( M m g sin gg (9 ( M m cos Subsiuing Eq. (9 ino (7 yields: ( M m g sin mgcos sin M m mcos cos gg ( I is clerly h (M + m = (M + m gg, which is in fc he Newon s second lw. Le u = u + Δu where u = (M + m gg, hen by subsiuing hem ino Eq. (7 we hve: hen by subsiuing u = u + Δu nd u = (M + m gg ino Eq. (6 we ge: ( M m g cos ( M m m cos l cos u M m m l ( cos ( According o Eq. ( nd ( we cn ge noher se spce form wih he form s follow: X mg cos M m m cos ( M m g cos ( M m m cos l M m m cos cos (3 u ( M m m cos l I is clerly h bove sysem (3 is he sme form of sysem (5. Now le us es he rnk of is conrollbiliy mrix c (r. Firs le prmeer vecor r be (M, m, l, g,. hen for clculing esily, we le N denoe (M + m - m cos, hen we ge he conrollbiliy mrix c (r wih he form: mg cos cos N N l mg cos cos N N l Qc ( r cos ( M m gcos cos Nl N l cos ( M m gcos cos Nl N l I is cler h rnk c (r = 4, which mens h by heorem 3 sysem (3 is srucurl conrollble. CONCLUSION mg cos M m m cos ( M m gg u mg cos sin M m m cos = mg cos + u ( M m mcos M m m cos where gg. 4 From he srucurl model of view poin, he srucurl conrollbiliy of lineried one-sge invered pendulum is nlyed in his presened sudy. For he wo cses h lineriion up-vericl equilibrium poin nd lineriion ny up opering poin, srucurl conrollbiliy of he wo cses re discussed, which proves h he former cse is he specil form of ler cse. he liner srucurl model obined in he second cse is no he sndrd liner srucurl model (4, bu i is sndrd liner srucurl model wih consn perurbion. So he conrollbiliy
6 Res. J. Appl. Sci. Eng. echnol., 6(: 46-4, 3 condiion of srucurl model (5 is explored. However, he condiion obined in his presened sudy is only sufficien condiion, so exploring he necessry condiion is he fuure work. ACKNOWLEDGMEN he firs uhor wishes o hnk professor Lu Kisheng, whose some useful suggesion helps him improving his sudy. Professor Lu s pioneering work in reserching liner sysems nd elecricl neworks over F (r mkes him ineresed in srucurl properies reserch. he uhors lso wish o give heir hnks o nonymous reviewers. his sudy of he firs uhor ws suppored in pr by Nionl Nure Science Foundion of Chin under grn: No REFERENCES Anderson, B.D.O. nd D.J. Clemens, 98. Algebric chrceriion of fixed modes in decenrlied conrol sysems. Auomic, 7(5: Bei-Chen, J. nd C. Shung, 6. Opiml link group selecion for riple invered pendulum. Compu. Simul., 3: 58-6, (In Chinese. Ibne, C.A., O.G. Fris nd M.S. Csnon, 5. Lypunov bsed conroller for he invered pendulum cr sysem. Nonliner Dynm., 4: Lee, E.B. nd L. Mrkus, 967. Foundions of Opiml Conrol heory, J. Willey, New York. Lin, C.., 974. Srucurl conrollbiliy [J]. IEEE. Auom. Conr., 9(3: -8. Lu, K.S. nd J.N. Wei, 99. Rionl funcion mrices nd srucurl conrollbiliy nd observbiliy. IEE Proceedings-D Con. heory Appl., 38: Lu, K.S. nd J.N. Wei, 994. Reducibiliy condiion of clss of rionl funcion mrices. SIAM J. Mrix. Anl. Appl., 5: 5-6. M, Q. nd K.S. Lu, 9. Design nd reliion of invered pendulum conrol bsed on vrible srucure. Proceeding of Inernionl Conference on Informion Engineering nd Compuer Science (ICIECS, pp: -3. Muro, K., 999. On he degree of mixed polynomil mrices. SIAM J. Mrix Anl. Appl., : Niemnn, H. nd J. Sousrup, 5. Pssive ful olern conrol of double inver ed pendulum: A cse sudy. Con. Eng. Prc., 3: Shen,., S. Mei, H. Wng e l., 3. he reference design issue of robus conrol: Conrol of n invered pendulum. Con. heory Appl., (6: , (In Chinese. Ymd,. nd D.G. Luenberger, 985. Generic properies of column-srucured mrices. Liner Algebr. Appl., 65:
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationTax Audit and Vertical Externalities
T Audi nd Vericl Eernliies Hidey Ko Misuyoshi Yngihr Ngoy Keizi Universiy Ngoy Universiy 1. Inroducion The vericl fiscl eernliies rise when he differen levels of governmens, such s he federl nd se governmens,
More information3 Motion with constant acceleration: Linear and projectile motion
3 Moion wih consn ccelerion: Liner nd projecile moion cons, In he precedin Lecure we he considered moion wih consn ccelerion lon he is: Noe h,, cn be posiie nd neie h leds o rie of behiors. Clerl similr
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 information2k 1. . And when n is odd number, ) The conclusion is when n is even number, an. ( 1) ( 2 1) ( k 0,1,2 L )
Scholrs Journl of Engineering d Technology SJET) Sch. J. Eng. Tech., ; A):8-6 Scholrs Acdemic d Scienific Publisher An Inernionl Publisher for Acdemic d Scienific Resources) www.sspublisher.com ISSN -X
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 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 information1. Find a basis for the row space of each of the following matrices. Your basis should consist of rows of the original matrix.
Mh 7 Exm - Prcice Prolem Solions. Find sis for he row spce of ech of he following mrices. Yor sis shold consis of rows of he originl mrix. 4 () 7 7 8 () Since we wn sis for he row spce consising of rows
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 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 informationMinimum Squared Error
Minimum Squred Error LDF: Minimum Squred-Error Procedures Ide: conver o esier nd eer undersood prolem Percepron y i > for ll smples y i solve sysem of liner inequliies MSE procedure y i = i for ll smples
More information22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 9: The High Beta Tokamak
.65, MHD Theory of Fusion Sysems Prof. Freidberg Lecure 9: The High e Tokmk Summry of he Properies of n Ohmic Tokmk. Advnges:. good euilibrium (smll shif) b. good sbiliy ( ) c. good confinemen ( τ nr )
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 informationM r. d 2. R t a M. Structural Mechanics Section. Exam CT5141 Theory of Elasticity Friday 31 October 2003, 9:00 12:00 hours. Problem 1 (3 points)
Delf Universiy of Technology Fculy of Civil Engineering nd Geosciences Srucurl echnics Secion Wrie your nme nd sudy numer he op righ-hnd of your work. Exm CT5 Theory of Elsiciy Fridy Ocoer 00, 9:00 :00
More informationAsymptotic relationship between trajectories of nominal and uncertain nonlinear systems on time scales
Asympoic relionship beween rjecories of nominl nd uncerin nonliner sysems on ime scles Fim Zohr Tousser 1,2, Michel Defoor 1, Boudekhil Chfi 2 nd Mohmed Djemï 1 Absrc This pper sudies he relionship beween
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 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 informationNeural assembly binding in linguistic representation
Neurl ssembly binding in linguisic represenion Frnk vn der Velde & Mrc de Kmps Cogniive Psychology Uni, Universiy of Leiden, Wssenrseweg 52, 2333 AK Leiden, The Neherlnds, vdvelde@fsw.leidenuniv.nl Absrc.
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 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 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 informationFault-Tolerant Guaranteed Cost Control of Uncertain Networked Control Systems with Time-varying Delay
IJCSNS Inernionl Journl of Compuer Science nd Nework Securiy VOL.9 No.6 June 9 3 Ful-olern Gurneed Cos Conrol of Uncerin Neworked Conrol Sysems wih ime-vrying Dely Guo Yi-nn Zhng Qin-ying Chin Universiy
More informationA 1.3 m 2.5 m 2.8 m. x = m m = 8400 m. y = 4900 m 3200 m = 1700 m
PHYS : Soluions o Chper 3 Home Work. SSM REASONING The displcemen is ecor drwn from he iniil posiion o he finl posiion. The mgniude of he displcemen is he shores disnce beween he posiions. Noe h i is onl
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 informationA new model for solving fuzzy linear fractional programming problem with ranking function
J. ppl. Res. Ind. Eng. Vol. 4 No. 07 89 96 Journl of pplied Reserch on Indusril Engineering www.journl-prie.com new model for solving fuzzy liner frcionl progrmming prolem wih rning funcion Spn Kumr Ds
More informationThink of the Relationship Between Time and Space Again
Repor nd Opinion, 1(3),009 hp://wwwsciencepubne sciencepub@gmilcom Think of he Relionship Beween Time nd Spce Agin Yng F-cheng Compny of Ruid Cenre in Xinjing 15 Hongxing Sree, Klmyi, Xingjing 834000,
More informationProperties of Logarithms. Solving Exponential and Logarithmic Equations. Properties of Logarithms. Properties of Logarithms. ( x)
Properies of Logrihms Solving Eponenil nd Logrihmic Equions Properies of Logrihms Produc Rule ( ) log mn = log m + log n ( ) log = log + log Properies of Logrihms Quoien Rule log m = logm logn n log7 =
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 informationFRACTIONAL-order differential equations (FDEs) are
Proceedings of he Inernionl MuliConference of Engineers nd Compuer Scieniss 218 Vol I IMECS 218 Mrch 14-16 218 Hong Kong Comprison of Anlyicl nd Numericl Soluions of Frcionl-Order Bloch Equions using Relible
More informationA new model for limit order book dynamics
Anewmodelforlimiorderbookdynmics JeffreyR.Russell UniversiyofChicgo,GrdueSchoolofBusiness TejinKim UniversiyofChicgo,DeprmenofSisics Absrc:Thispperproposesnewmodelforlimiorderbookdynmics.Thelimiorderbookconsiss
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 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 informationLecture 20: Riccati Equations and Least Squares Feedback Control
34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he
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 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 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 informationON THE STABILITY OF DELAY POPULATION DYNAMICS RELATED WITH ALLEE EFFECTS. O. A. Gumus and H. Kose
Mhemicl nd Compuionl Applicions Vol. 7 o. pp. 56-67 O THE STABILITY O DELAY POPULATIO DYAMICS RELATED WITH ALLEE EECTS O. A. Gumus nd H. Kose Deprmen o Mhemics Selcu Universiy 47 Kony Turey ozlem@selcu.edu.r
More informationDevelopment of a New Scheme for the Solution of Initial Value Problems in Ordinary Differential Equations
IOS Journl o Memics IOSJM ISSN: 78-78 Volume Issue July-Aug PP -9 www.iosrjournls.org Developmen o New Sceme or e Soluion o Iniil Vlue Problems in Ordinry Dierenil Equions Ogunrinde. B. dugb S. E. Deprmen
More information3D Transformations. Computer Graphics COMP 770 (236) Spring Instructor: Brandon Lloyd 1/26/07 1
D Trnsformions Compuer Grphics COMP 770 (6) Spring 007 Insrucor: Brndon Lloyd /6/07 Geomery Geomeric eniies, such s poins in spce, exis wihou numers. Coordines re nming scheme. The sme poin cn e descried
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 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 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 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 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 information1 jordan.mcd Eigenvalue-eigenvector approach to solving first order ODEs. -- Jordan normal (canonical) form. Instructor: Nam Sun Wang
jordnmcd Eigenvlue-eigenvecor pproch o solving firs order ODEs -- ordn norml (cnonicl) form Insrucor: Nm Sun Wng Consider he following se of coupled firs order ODEs d d x x 5 x x d d x d d x x x 5 x x
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 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 informationLECTURES ON RECONSTRUCTION ALGEBRAS I
LETURES ON REONSTRUTION ALGEBRAS I MIHAEL WEMYSS. Inroducion Noncommuive lgebr (=quivers) cn be used o solve boh explici nd non-explici problems in lgebric geomery, nd hese lecures will ry o explin some
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 informationPhysics 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 informationPhysic 231 Lecture 4. Mi it ftd l t. Main points of today s lecture: Example: addition of velocities Trajectories of objects in 2 = =
Mi i fd l Phsic 3 Lecure 4 Min poins of od s lecure: Emple: ddiion of elociies Trjecories of objecs in dimensions: dimensions: g 9.8m/s downwrds ( ) g o g g Emple: A foobll pler runs he pern gien in he
More informationYan Sun * 1 Introduction
Sun Boundry Vlue Problems 22, 22:86 hp://www.boundryvlueproblems.com/conen/22//86 R E S E A R C H Open Access Posiive soluions of Surm-Liouville boundry vlue problems for singulr nonliner second-order
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 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 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 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 information[5] Solving Multiple Linear Equations A system of m linear equations and n unknown variables:
[5] Solving Muliple Liner Equions A syse of liner equions nd n unknown vribles: + + + nn = b + + + = b n n : + + + nn = b n n A= b, where A =, : : : n : : : : n = : n A = = = ( ) where, n j = ( ); = :
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 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 information..,..,.,
57.95. «..» 7, 9,,. 3 DOI:.459/mmph7..,..,., E-mil: yshr_ze@mil.ru -,,. -, -.. -. - - ( ). -., -. ( - ). - - -., - -., - -, -., -. -., - - -, -., -. : ; ; - ;., -,., - -, []., -, [].,, - [3, 4]. -. 3 (
More informationON THE DEGREES OF RATIONAL KNOTS
ON THE DEGREES OF RATIONAL KNOTS DONOVAN MCFERON, ALEXANDRA ZUSER Absrac. In his paper, we explore he issue of minimizing he degrees on raional knos. We se a bound on hese degrees using Bézou s heorem,
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 informationTemperature Rise of the Earth
Avilble online www.sciencedirec.com ScienceDirec Procedi - Socil nd Behviorl Scien ce s 88 ( 2013 ) 220 224 Socil nd Behviorl Sciences Symposium, 4 h Inernionl Science, Socil Science, Engineering nd Energy
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 informationNonlinear System Modelling: How to Estimate the. Highest Significant Order
IEEE Insrumenion nd Mesuremen Technology Conference nchorge,, US, - My Nonliner Sysem Modelling: ow o Esime he ighes Significn Order Neophyos Chirs, Ceri Evns nd Dvid Rees, Michel Solomou School of Elecronics,
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 information(b) 10 yr. (b) 13 m. 1.6 m s, m s m s (c) 13.1 s. 32. (a) 20.0 s (b) No, the minimum distance to stop = 1.00 km. 1.
Answers o Een Numbered Problems Chper. () 7 m s, 6 m s (b) 8 5 yr 4.. m ih 6. () 5. m s (b).5 m s (c).5 m s (d) 3.33 m s (e) 8. ().3 min (b) 64 mi..3 h. ().3 s (b) 3 m 4..8 mi wes of he flgpole 6. (b)
More information6. Gas dynamics. Ideal gases Speed of infinitesimal disturbances in still gas
6. Gs dynmics Dr. Gergely Krisóf De. of Fluid echnics, BE Februry, 009. Seed of infiniesiml disurbnces in sill gs dv d, dv d, Coninuiy: ( dv)( ) dv omenum r r heorem: ( ( dv) ) d 3443 4 q m dv d dv llievi
More informationThermal neutron self-shielding factor in foils: a universal curve
Proceedings of he Inernionl Conference on Reserch Recor Uilizion, Sfey, Decommissioning, Fuel nd Wse Mngemen (Snigo, Chile, -4 Nov.3) Pper IAEA-CN-/, IAEA Proceedings Series, Vienn, 5 Therml neuron self-shielding
More informationOrdinary Differential Equations
Ordinary Differenial Equaions 5. Examples of linear differenial equaions and heir applicaions We consider some examples of sysems of linear differenial equaions wih consan coefficiens y = a y +... + a
More informationThe motions of the celt on a horizontal plane with viscous friction
The h Join Inernaional Conference on Mulibody Sysem Dynamics June 8, 18, Lisboa, Porugal The moions of he cel on a horizonal plane wih viscous fricion Maria A. Munisyna 1 1 Moscow Insiue of Physics and
More informationAvailable online at Pelagia Research Library. Advances in Applied Science Research, 2011, 2 (3):
Avilble online www.pelgireserchlibrry.com Pelgi Reserch Librry Advnces in Applied Science Reserch 0 (): 5-65 ISSN: 0976-860 CODEN (USA): AASRFC A Mhemicl Model of For Species Syn-Ecosymbiosis Comprising
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 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 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 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 informationCalculation method of flux measurements by static chambers
lculion mehod of flux mesuremens by sic chmbers P.S. Kroon Presened he NiroEurope Workshop, 15h - 17h December 28, openhgen, Denmrk EN-L--9-11 December 28 lculion mehod of flux mesuremens by sic chmbers
More information