Gradient Descent for General Reinforcement Learning
|
|
- Basil Austin
- 5 years ago
- Views:
Transcription
1 To appar n M. S. Karns, S. A. Solla, and D. A. Cohn, dors, Advancs n Nral Informaon Procssng Sysms, MIT Prss, Cambrdg, MA, 999. Gradn Dscn for Gnral Rnforcmn Larnng Lmon Bard Andrw Moor lmon@cs.cm.d awm@cs.cm.d Compr Scnc Dparmn Compr Scnc Dparmn 5000 Forbs Avn 5000 Forbs Avn Carng Mllon Unvrsy Carng Mllon Unvrsy Psbrgh, PA Psbrgh, PA Absrac A smpl larnng rl s drvd, h VAPS algorhm, whch can b nsanad o gnra a wd rang of nw rnforcmnlarnng algorhms. Ths algorhms solv a nmbr of opn problms, dfn svral nw approachs o rnforcmn larnng, and nfy dffrn approachs o rnforcmn larnng ndr a sngl hory. Ths algorhms all hav garand convrgnc, and ncld modfcaons of svral xsng algorhms ha wr known o fal o convrg on smpl MDPs. Ths ncld Q- larnng, SARSA, and advanag larnng. In addon o hs val-basd algorhms also gnras pr polcy-sarch rnforcmn-larnng algorhms, whch larn opmal polcs who larnng a val fncon. In addon, allows polcysarch and val-basd algorhms o b combnd, hs nfyng wo vry dffrn approachs o rnforcmn larnng no a sngl Val and Polcy Sarch (VAPS algorhm. And hs algorhms convrg for POMDPs who rqrng a propr blf sa. Smlaons rsls ar gvn, and svral aras for fr rsarch ar dscssd. CONVERGENCE OF GREEDY EXPLORATION Many rnforcmn-larnng algorhms ar known ha s a paramrzd fncon approxmaor o rprsn a val fncon, and adjs h wghs ncrmnally drng larnng. Exampls ncld Q-larnng, SARSA, and advanag larnng. Thr ar smpl MDPs whr h orgnal form of hs algorhms fals o convrg, as smmarzd n Tabl. For h cass wh, h algorhms ar garand o convrg ndr rasonabl assmpons sch as
2 Tabl. Crrn convrgnc rsls for ncrmnal, val-basd RL algorhms. Rsdal algorhms changd vry X n h frs wo colmns o. Th nw algorhms proposd n hs papr chang vry X o a. Fxd dsrbon (on-polcy Fxd dsrbon Usallygrdy dsrbon Lookp abl Markov Avragr chan Lnar X Nonlnar X X Lookp abl MDP Avragr X Lnar X X X Nonlnar X X X Lookp abl X POMDP Avragr X Lnar X X X Nonlnar X X X convrgnc garand Xconrxampl s known ha hr dvrgs or oscllas bwn h bs and wors possbl polcs. dcayng larnng ras. For h cass wh X, hr ar known conrxampls whr wll hr dvrg or osclla bwn h bs and wors possbl polcs, whch hav vry-dffrn vals. Ths can happn vn wh nfn ranng m and slowly-dcrasng larnng ras (Bard, 95, Gordon, 96. Each X n h frs wo colmns can b changd o a and mad o convrg by sng a modfd form of h algorhm, h rsdal form (Bard 95. B hs s only possbl whn larnng wh a fxd ranng dsrbon, and ha s rarly praccal. For mos larg problms, s sfl o xplor wh a polcy ha s sally-grdy wh rspc o h crrn val fncon, and ha changs as h val fncon changs. In ha cas (h rghmos colmn of h char, h crrn convrgnc garans ar no vry good. On way o garan convrgnc n all hr colmns s o modfy h algorhm so ha s prformng sochasc gradn dscn on som avrag rror fncon, whr h avrag s wghd by sa-vsaon frqncs for h crrn sally-grdy polcy. Thn h wghng changs as h polcy changs. I mgh appar ha hs gradn s dffcl o comp. Consdr Q- larnng xplorng wh a Bolzman dsrbon ha s sally grdy wh rspc o h larnd Q fncon. I sms dffcl o calcla gradns, snc changng a sngl wgh wll chang many Q vals, changng a sngl Q val wll chang many acon-choc probabls n ha sa, and changng a sngl acon-choc probably may affc h frqncy wh whch vry sa n h MDP s vsd. Alhogh hs mgh sm dffcl, s no. Srprsngly, nbasd smas of h gradns of vsaon dsrbons wh rspc o h wghs can b calclad qckly, and h rslng algorhms can p a n vry cas n Tabl. DERIVATION OF THE VAPS EQUATION Consdr a sqnc of ransons obsrvd whl followng a parclar sochasc polcy on an MDP. L s {x 0, 0,R 0, x,,r, x -, -,R -, x,,r } b h sqnc of sas, acons, and rnforcmns p o m, whr prformng acon n sa x ylds rnforcmn R and a ranson o sa x +. Th
3 sochasc polcy may b a fncon of a vcor of wghs w. Assm h MDP has a sngl sar sa namd x 0. If h MDP has rmnal sas, and x s a rmnal sa, hn x + x 0. L S b h s of all possbl sqncs from m 0 o. L (s b a gvn rror fncon ha calclas an rror on ach m sp, sch as h sqard Bllman rsdal a m, or som ohr rror occrrng a m. If s a fncon of h wghs, hn ms b a smooh fncon of h wghs. Consdr a prod of m sarng a m 0 and ndng wh probably P(nd s afr h sqnc s occrs. Th probabls ms b sch ha h xpcd sqard prod lngh s fn. L B b h xpcd oal rror drng ha prod, whr h xpcaon s wghd accordng o h sa-vsaon frqncs gnrad by h gvn polcy: B P(prod nds a mt afr rajcory s T T 0 st ST 0 0 s S ( s P( s T ( s ( ( whr: P( s P( s P( R s 0 P( s P( R s P( s s [ P( nd s ] + No ha on h frs ln, for a parclar s, h rror (s wll b addd n o B onc for vry sqnc ha sars wh s. Each of hs rms wll b wghd by h probably of a compl rajcory ha sars wh s. Th sm of h probabls of all rajcors ha sar wh s s smply h probably of s bng obsrvd, snc h prod s assmd o nd vnally wh probably on. So h scond ln qals h frs. Th hrd ln s h probably of h sqnc, of whch only h P( x facor mgh b a fncon of w. If so, hs probably ms b a smooh fncon of h wghs and nonzro vrywhr. Th paral drvav of B wh rspc o w, a parclar lmn of h wgh vcor w, s: B w [ P( s ] w j j + ( s P( s ( s P( s 0 s j P( j s j S w P( s ( s + ( s w w 0 s S j ln ( P( s j j Spac hr s lmd, and may no b clar from h shor skch of hs drvaon, b smmng (5 ovr an nr prod dos gv an nbasd sma of B, h xpcd oal rror drng a prod. An ncrmnal algorhm o prform sochasc gradn dscn on B s h wgh pda gvn on h lf sd of Tabl, whr h smmaon ovr prvos m sps s rplacd wh a rac T for ach wgh. Ths algorhm s mor gnral han prvosly-pblshd algorhms of hs form, n ha can b a fncon of all prvos sas, acons, and rnforcmns, rahr han js h crrn rnforcmn. Ths s wha allows VAPS o do boh val and polcy sarch. Evry algorhm proposd n hs papr s a spcal cas of h VAPS qaon on h lf sd of Tabl. No ha no modl s ndd for hs algorhm. Th only probably ndd n h algorhm s h polcy, no h ranson probably from h MDP. Ths s sochasc gradn dscn on B, and h pda rl s only corrc f h obsrvd ransons ar sampld from rajcors fond by followng (3 (4 (5
4 Tabl. Th gnral VAPS algorhm (lf, and svral nsanaons of (rgh. Ths sngl algorhm nclds boh val-basd and polcy-sarch approachs and hr combnaon, and gvs garand convrgnc n vry cas. [ ( s + ( s T ] w α w T ln( P( s w [ R + γq( x, Q( x ], SARSA ( s E s E [ R + γ max Q( x, Q( x ], Q larnng ( advanag ( s R E + γ max A( x, + K A( x, ( max A( x, K ( max [ ( ] ( val raon s E R + γv x V x SARSA polcy ( s ( β SARSA ( s + β ( b γ R h crrn, sochasc polcy. Boh and P shold b smooh fncons of w, and for any gvn w vcor, shold b bondd. Th algorhm s smpl, b acally gnras a larg class of dffrn algorhms dpndng on h choc of and whn h rac s rs o zro. For a sngl sqnc, sampld by followng h crrn polcy, h sm of w along h sqnc wll gv an nbasd sma of h r gradn, wh fn varanc. Thrfor, drng larnng, f wgh pdas ar mad a h nd of ach ral, and f h wghs say whn a bondd rgon, and h larnng ra approachs zro, hn B wll convrg wh probably on. Addng a wgh-dcay rm (a consan ms h -norm of h wgh vcor ono B wll prvn wgh dvrgnc for small nal larnng ras. Thr s no garan ha a global mnmm wll b fond whn sng gnral fncon approxmaors, b a las wll convrg. Ths s r for backprop as wll. 3 INSTANTIATING THE VAPS ALGORITHM Many rnforcmn-larnng algorhms ar val-basd; hy ry o larn a val fncon ha sasfs h Bllman qaon. Exampls ar Q-larnng, whch larns a val fncon, acor-crc algorhms, whch larn a val fncon and h polcy whch s grdy wh rspc o, and TD(, whch larns a val fncon basd on fr rwards. Ohr algorhms ar pr polcy-sarch algorhms; hy drcly larn a polcy ha rrns hgh rwards. Ths ncld REINFORCE (Wllams, 988, backprop hrogh m, larnng aomaa, and gnc algorhms. Th algorhms proposd hr combn h wo approachs: hy prform Val And Polcy Sarch (VAPS. Th gnral VAPS qaon s nsanad by choosng an xprsson for. Ths can b a Bllman rsdal (yldng val-basd, h rnforcmn (yldng polcy-sarch, or a lnar combnaon of h wo (yldng Val And Polcy Sarch. Th sngl VAPS pda rl on h lf sd of Tabl gnras a vary of dffrn yps of algorhms, som of whch ar dscrbd n h followng scons. 3. REDUCING MEAN SQUARED RESIDUAL PER TRIAL If h MDP has rmnal sas, and a ral s h m from h sar nl a rmnal sa s rachd, hn s possbl o mnmz h xpcd oal rror pr ral by rsng h rac o zro a h sar of ach ral. Thn, a convrgn form of SARSA, Q-larnng, ncrmnal val raon, or advanag larnng can b gnrad by choosng o b h sqard Bllman rsdal, as shown on h rgh sd of Tabl. In ach cas, h xpcd val s akn ovr all possbl (x,,r
5 rpls, gvn s -. Th polcy ms b a smooh, nonzro fncon of h wghs. So cold no b an ε-grdy polcy ha chooss h grdy acon wh probably (-ε and chooss nformly ohrws. Tha wold cas a dsconny n h gradn whn wo Q vals n a sa wr qal. B h polcy cold b somhng ha approachs ε-grdy as a posv mprar c approachs zro: ε P( x + n ( ε + + ' Q( x, / c Q( x, ' / c ( whr n s h nmbr of possbl acons n ach sa. For ach nsanc n Tabl ohr han val raon, h gradn of can b smad sng wo, ndpndn, nbasd smas of h xpcd val. For xampl: w s SARSA ( s γφ Q( x', ' Q( x, (7 w w SARSA ( Whn φ, hs s an sma of h r gradn. Whn φ<, hs s a rsdal algorhm, as dscrbd n (Bard, 96, and rans garand convrgnc, b may larn mor qckly han pr gradn dscn for som vals of φ. No ha h gradn of Q(x, a m ss prmd varabls. Tha mans a nw sa and acon a m wr gnrad ndpndnly from h sa and acon a m -. Of cors, f h MDP s drmnsc, hn h prmd varabls ar h sam as h nprmd. If h MDP s nondrmnsc b h modl s known, hn h modl ms b valad on addonal m o g h ohr sa. If h modl s no known, hn hr ar hr chocs. Frs, a modl cold b larnd from pas daa, and hn valad o gv hs ndpndn sampl. Scond, h ss cold b gnord, smply rsng h nprmd varabls n plac of h prmd varabls. Ths may affc h qaly of h larnd fncon (dpndng on how random h MDP s, b dosn sop convrgnc, and b an accpabl approxmaon n pracc. Thrd, all pas ransons cold b rcordd, and h prmd varabls cold b fond by sarchng for all h ms (x -, - has bn sn bfor, and randomly choosng on of hos ransons and sng s sccssor sa and acon as h prmd varabls. Ths s qvaln o larnng h crany qvalnc modl, and samplng from, and so s a spcal cas of h frs choc. For xrmly larg sa-acon spacs wh many sarng sas, hs s lkly o gv h sam rsl n pracc as smply rsng h nprmd varabls as h prmd varabls. No, ha whn wghs do no ffc h polcy a all, hs algorhms rdc o sandard rsdal algorhms (Bard, 95. I s also possbl o rdc h man sqard rsdal pr sp, rahr han pr ral. Ths s don by makng prod lnghs ndpndn of h polcy, so mnmzng rror pr prod wll also mnmz h rror pr sp. For xampl, a prod mgh b dfnd o b h frs 00 sps, afr whch h racs ar rs, and h sa s rrnd o h sar sa. No ha f vry sa-acon par has a posv chanc of bng sn n h frs 00 sps, hn hs wll no js b solvng a fn-horzon problm. I wll b acally b solvng h dscond, nfn-horzon problm, by rdcng h Bllman rsdal n vry sa. B h wghng of h rsdals wll b drmnd only by wha happns drng h frs 00 sps. Many dffrn problms can b solvd by h VAPS algorhm by nsanang h dfnon of "prod" n dffrn ways. 3. POLICY-SEARCH AND VALUE-BASED LEARNING I s also possbl o add a rm ha rs o maxmz rnforcmn drcly. For xampl, cold b dfnd o b SARSA-polcy rahr han SARSA. from Tabl, and (6
6 A 0000 sar B Trals 000 nd Ba Fgr. A POMDP and h nmbr of rals ndd o larn vs. β. A combnaon of polcy-sarch and val-basd RL oprforms hr alon. h rac rs o zro afr ach rmnal sa s rachd. Th consan b dos no affc h xpcd gradn, b dos affc h nos dsrbon, as dscssd n (Wllams, 88. Whn β0, h algorhm wll ry o larn a Q fncon ha sasfs h Bllman qaon, js as bfor. Whn β, drcly larns a polcy ha wll mnmz h xpcd oal dscond rnforcmn. Th rslng Q fncon may no vn b clos o conanng r Q vals or o sasfyng h Bllman qaon, wll js gv a good polcy. Whn β s n bwn, hs algorhm rs o boh sasfy h Bllman qaon and gv good grdy polcs. A smlar modfcaon can b mad o any of h algorhms n Tabl. In h spcal cas whr β, hs algorhm rdcs o h REINFORCE algorhm (Wllams, 988. REINFORCE has bn rdrvd for h spcal cas of gassan acon dsrbons (Trsp & Hofman, 995, and xnsons of appar n (Marbach, 998. Ths cas of pr polcy sarch s parclarly nrsng, bcas for β, hr s no nd for any knd of modl or of gnrang wo ndpndn sccssors. Ohr algorhms hav bn proposd for fndng polcs drcly, sch as hos gvn n (Gllapall, 9 and h varos algorhms from larnng aomaa hory smmarzd n (Narndra & Thahachar, 89. Th VAPS algorhms proposd hr appars o b h frs on nfyng hs wo approachs o rnforcmn larnng, fndng a val fncon ha boh approxmas a Bllman-qaon solon and drcly opmzs h grdy polcy. Fgr shows smlaon rsls for h combnd algorhm. A rn s sad o hav larnd whn h grdy polcy s opmal for 000 conscv rals. Th graph shows h avrag plo of 00 rns, wh dffrn nal random wghs bwn ±0-6. Th larnng ra was opmzd sparaly for ach β val. R whn lavng sa A, R whn lavng sa B or nrng nd, and R0 ohrws. γ0.9. Th algorhm sd was h modfd Q-larnng from Tabl, wh xploraon as n qaon 6, and ϕc, b0, ε0.. Sas A and B shar h sam paramrs, so ordnary SARSA or grdy Q-larnng cold nvr convrg, as shown n (Gordon, 96. Whn β0 (pr val-basd, h nw algorhm convrgs, b of cors canno larn h opmal polcy n h sar sa, snc hos wo Q vals larn o b qal. Whn β (pr polcy-sarch, larnng convrgs o opmaly, b slowly, snc hr s no val fncon cachng h rsls n h long sqnc of sas nar h nd. By combnng h wo approachs, h nw algorhm larns mch mor qckly han hr alon. I s nrsng ha h VAPS algorhms dscrbd n h las hr scons can b appld drcly o a Parally Obsrvabl Markov Dcson Procss (POMDP, whr h r sa s hddn, and all ha s avalabl on ach m sp s an
7 ambgos obsrvaon, whch s a fncon of h r sa. Normally, an algorhm sch as SARSA only has garand convrgnc whn appld o an MDP. Th VAPS algorhms wll convrg n sch cass. 4 CONCLUSION A nw algorhm has bn prsnd. Spcal cass of gv nw algorhms smlar o Q-larnng, SARSA, and advanag larnng, b wh garand convrgnc for a wdr rang of problms han was prvosly possbl, ncldng POMDPs. For h frs m, hs can b garand o convrg, vn whn h xploraon polcy changs drng larnng. Ohr spcal cass allow nw approachs o rnforcmn larnng, whr hr s a radoff bwn sasfyng h Bllman qaon and mprovng h grdy polcy. For on MDP, smlaon showd ha hs combnd algorhm larnd mor qckly han hr approach alon. Ths nfd hory, nfyng for h frs m boh val-basd and polcysarch rnforcmn larnng, s of horcal nrs, and also was of praccal val for h smlaons prformd. Fr rsarch wh hs nfd framwork may b abl o mprcally or analycally addrss h old qson of whn s br o larn val fncons and whn s br o larn h polcy drcly. I may also shd lgh on h nw qson, of whn s bs o do boh a onc. Acknowldgmns Ths rsarch was sponsord n par by h U.S. Ar Forc. Rfrncs Bard, L. C. (995. Rsdal Algorhms: Rnforcmn Larnng wh Fncon Approxmaon. In Armand Prds & Sar Rssll, ds. Machn Larnng: Procdngs of h Twlfh Inrnaonal Confrnc, 9- Jly, Morgan Kafman Pblshrs, San Francsco, CA. Gordon, G. (996. Sabl fd rnforcmn larnng. In G. Tsaro, M. Mozr, and M. Hasslmo (ds., Advancs n Nral Informaon Procssng Sysms 8, pp MIT Prss, Cambrdg, MA. Gllapall, V. (99. Rnforcmn Larnng and Is Applcaon o Conrol. Dssraon and COINS Tchncal Rpor 9-0, Unvrsy of Massachss, Amhrs, MA. Kalblng, L. P., Lman, M. L. & Cassandra, A., Plannng and Acng n Parally Obsrvabl Sochasc Domans. Arfcal Inllgnc, o appar. Avalabl now a hp:// Marbach, P. (998. Smlaon-Basd Opmzaon of Markov Dcson Procsss. Thss LIDS-TH 49, Massachss Ins of Tchnology. McCallm (995, A. Rnforcmn larnng wh slcv prcpon and hddn sa. Dssraon, Dparmn of Compr Scnc, Unvrsy of Rochsr, Rochsr, NY. Narndra, K., & Thahachar, M.A.L. (989. Larnng aomaa: An nrodcon. Prnc Hall, Englwood Clffs, NJ. Trsp, V., & R. Hofman (995. "Mssng and nosy daa n nonlnar m-srs prdcon". In Procdngs of Nral Nworks for Sgnal Procssng 5, F. Gros, J. Makhol, E. Manolakos and E. Wlson, ds., IEEE Sgnal Procssng Socy, Nw York, Nw York, 995, pp. -0. Wllams, R. J. (988. Toward a hory of rnforcmn-larnng conncons sysms. Tchncal rpor NU-CCS-88-3, Norhasrn Unvrsy, Boson, MA.
Consider a system of 2 simultaneous first order linear equations
Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm
More informationThe Variance-Covariance Matrix
Th Varanc-Covaranc Marx Our bggs a so-ar has bn ng a lnar uncon o a s o daa by mnmzng h las squars drncs rom h o h daa wh mnsarch. Whn analyzng non-lnar daa you hav o us a program l Malab as many yps o
More informationSummary: Solving a Homogeneous System of Two Linear First Order Equations in Two Unknowns
Summary: Solvng a Homognous Sysm of Two Lnar Frs Ordr Equaons n Two Unknowns Gvn: A Frs fnd h wo gnvalus, r, and hr rspcv corrspondng gnvcors, k, of h coffcn mar A Dpndng on h gnvalus and gnvcors, h gnral
More information9. Simple Rules for Monetary Policy
9. Smpl Ruls for Monar Polc John B. Talor, Ma 0, 03 Woodford, AR 00 ovrvw papr Purpos s o consdr o wha xn hs prscrpon rsmbls h sor of polc ha conomc hor would rcommnd Bu frs, l s rvw how hs sor of polc
More informationBoosting and Ensemble Methods
Boosng and Ensmbl Mhods PAC Larnng modl Som dsrbuon D ovr doman X Eampls: c* s h arg funcon Goal: Wh hgh probably -d fnd h n H such ha rrorh,c* < d and ar arbrarly small. Inro o ML 2 Wak Larnng
More informationImplementation of the Extended Conjugate Gradient Method for the Two- Dimensional Energized Wave Equation
Lonardo Elcronc Jornal of raccs and Tchnolos ISSN 58-078 Iss 9 Jl-Dcmbr 006 p. -4 Implmnaon of h Endd Cona Gradn Mhod for h Two- Dmnsonal Enrd Wav Eqaon Vcor Onoma WAZIRI * Snda Ass REJU Mahmacs/Compr
More informationAdvanced Queueing Theory. M/G/1 Queueing Systems
Advand Quung Thory Ths slds ar rad by Dr. Yh Huang of Gorg Mason Unvrsy. Sudns rgsrd n Dr. Huang's ourss a GMU an ma a sngl mahn-radabl opy and prn a sngl opy of ah sld for hr own rfrn, so long as ah sld
More informationinnovations shocks white noise
Innovaons Tm-srs modls ar consrucd as lnar funcons of fundamnal forcasng rrors, also calld nnovaons or shocks Ths basc buldng blocks sasf var σ Srall uncorrlad Ths rrors ar calld wh nos In gnral, f ou
More informationSupplementary Figure 1. Experiment and simulation with finite qudit. anharmonicity. (a), Experimental data taken after a 60 ns three-tone pulse.
Supplmnar Fgur. Eprmn and smulaon wh fn qud anharmonc. a, Eprmnal daa akn afr a 6 ns hr-on puls. b, Smulaon usng h amlonan. Supplmnar Fgur. Phagoran dnamcs n h m doman. a, Eprmnal daa. Th hr-on puls s
More information(heat loss divided by total enthalpy flux) is of the order of 8-16 times
16.51, Rok Prolson Prof. Manl Marnz-Sanhz r 8: Convv Ha ransfr: Ohr Effs Ovrall Ha oss and Prforman Effs of Ha oss (1) Ovrall Ha oss h loal ha loss r n ara s q = ρ ( ) ngrad ha loss s a S, and sng m =
More informationt=0 t>0: + vr - i dvc Continuation
hapr Ga Dlay and rcus onnuaon s rcu Equaon >: S S Ths dffrnal quaon, oghr wh h nal condon, fully spcfs bhaor of crcu afr swch closs Our n challng: larn how o sol such quaons TUE/EE 57 nwrk analys 4/5 NdM
More informationOUTLINE FOR Chapter 2-2. Basic Laws
0//8 OUTLINE FOR Chapr - AERODYNAMIC W-- Basc Laws Analss of an problm n fld mchancs ncssarl nclds samn of h basc laws gornng h fld moon. Th basc laws, whch applcabl o an fld, ar: Consraon of mass Conn
More informationState Observer Design
Sa Obsrvr Dsgn A. Khak Sdgh Conrol Sysms Group Faculy of Elcrcal and Compur Engnrng K. N. Toos Unvrsy of Tchnology Fbruary 2009 1 Problm Formulaon A ky assumpon n gnvalu assgnmn and sablzng sysms usng
More informationHomework: Introduction to Motion
Homwork: Inroducon o Moon Dsanc vs. Tm Graphs Nam Prod Drcons: Answr h foowng qusons n h spacs provdd. 1. Wha do you do o cra a horzona n on a dsancm graph? 2. How do you wak o cra a sragh n ha sops up?
More informationLecture 4 : Backpropagation Algorithm. Prof. Seul Jung ( Intelligent Systems and Emotional Engineering Laboratory) Chungnam National University
Lcur 4 : Bacpropagaon Algorhm Pro. Sul Jung Inllgn Sm and moonal ngnrng Laboraor Chungnam Naonal Unvr Inroducon o Bacpropagaon algorhm 969 Mn and Papr aac. 980 Parr and Wrbo dcovrd bac propagaon algorhm.
More informationWave Superposition Principle
Physcs 36: Was Lcur 5 /7/8 Wa Suroson Prncl I s qu a common suaon for wo or mor was o arr a h sam on n sac or o xs oghr along h sam drcon. W wll consdr oday sral moran cass of h combnd ffcs of wo or mor
More informationMathematical Statistics. Chapter VIII Sampling Distributions and the Central Limit Theorem
Mahmacal ascs 8 Chapr VIII amplg Dsrbos ad h Cral Lm Thorm Fcos of radom arabls ar sall of rs sascal applcao Cosdr a s of obsrabl radom arabls L For ampl sppos h arabls ar a radom sampl of s from a poplao
More informationTheoretical Seismology
Thorcal Ssmology Lcur 9 Sgnal Procssng Fourr analyss Fourr sudd a h Écol Normal n Pars, augh by Lagrang, who Fourr dscrbd as h frs among Europan mn of scnc, Laplac, who Fourr rad lss hghly, and by Mong.
More informationVertical Sound Waves
Vral Sond Wavs On an drv h formla for hs avs by onsdrn drly h vral omonn of momnm qaon hrmodynam qaon and h onny qaon from 5 and hn follon h rrbaon mhod and assmn h snsodal solons. Effvly h frs ro and
More informationANALYTICITY THEOREM FOR FRACTIONAL LAPLACE TRANSFORM
Sc. Rs. hm. ommn.: (3, 0, 77-8 ISSN 77-669 ANALYTIITY THEOREM FOR FRATIONAL LAPLAE TRANSFORM P. R. DESHMUH * and A. S. GUDADHE a Prof. Ram Mgh Insttt of Tchnology & Rsarch, Badnra, AMRAVATI (M.S. INDIA
More informationMicroscopic Flow Characteristics Time Headway - Distribution
CE57: Traffic Flow Thory Spring 20 Wk 2 Modling Hadway Disribuion Microscopic Flow Characrisics Tim Hadway - Disribuion Tim Hadway Dfiniion Tim Hadway vrsus Gap Ahmd Abdl-Rahim Civil Enginring Dparmn,
More informationLecture 1: Numerical Integration The Trapezoidal and Simpson s Rule
Lcur : Numrical ngraion Th Trapzoidal and Simpson s Rul A problm Th probabiliy of a normally disribud (man µ and sandard dviaion σ ) vn occurring bwn h valus a and b is B A P( a x b) d () π whr a µ b -
More informationFAULT TOLERANT SYSTEMS
FAULT TOLERANT SYSTEMS hp://www.cs.umass.du/c/orn/faultolransysms ar 4 Analyss Mhods Chapr HW Faul Tolranc ar.4.1 Duplx Sysms Boh procssors xcu h sam as If oupus ar n agrmn - rsul s assumd o b corrc If
More information10/7/14. Mixture Models. Comp 135 Introduction to Machine Learning and Data Mining. Maximum likelihood estimation. Mixture of Normals in 1D
Comp 35 Introducton to Machn Larnng and Data Mnng Fall 204 rofssor: Ron Khardon Mxtur Modls Motvatd by soft k-mans w dvlopd a gnratv modl for clustrng. Assum thr ar k clustrs Clustrs ar not rqurd to hav
More informationSIMEON BALL AND AART BLOKHUIS
A BOUND FOR THE MAXIMUM WEIGHT OF A LINEAR CODE SIMEON BALL AND AART BLOKHUIS Absrac. I s shown ha h paramrs of a lnar cod ovr F q of lngh n, dmnson k, mnmum wgh d and maxmum wgh m sasfy a cran congrunc
More informationFUZZY NEURAL NETWORK CONTROL FOR GRAVURE PRINTING
Conrol Unvrsy of Bah UK Spmbr D-9 FUZZY NEURAL NETWRK CNTRL FR GRAVURE RNTNG L. Dng.E. Bamforh M.R. ackson R. M. arkn Wolfson School of Mchancal & Manfacrng Engnrng Loghborogh Unvrsy Ashby Road Loghborogh
More information"Science Stays True Here" Journal of Mathematics and Statistical Science, Volume 2016, Science Signpost Publishing
"Scnc Says r Hr" Jornal of Mahmacs and Sascal Scnc Volm 6 343-356 Scnc Sgnpos Pblshng Mhod for a Solon o Som Class of Qas-Sac Problms n Lnar Vscolascy hory as Appld o Problms of Lnar orson of a Prsmac
More informationELEN E4830 Digital Image Processing
ELEN E48 Dgal Imag Procssng Mrm Eamnaon Sprng Soluon Problm Quanzaon and Human Encodng r k u P u P u r r 6 6 6 6 5 6 4 8 8 4 P r 6 6 P r 4 8 8 6 8 4 r 8 4 8 4 7 8 r 6 6 6 6 P r 8 4 8 P r 6 6 8 5 P r /
More informationFrequency Response. Response of an LTI System to Eigenfunction
Frquncy Rsons Las m w Rvsd formal dfnons of lnary and m-nvaranc Found an gnfuncon for lnar m-nvaran sysms Found h frquncy rsons of a lnar sysm o gnfuncon nu Found h frquncy rsons for cascad, fdbac, dffrnc
More information4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b
4. Th Uniform Disribuion Df n: A c.r.v. has a coninuous uniform disribuion on [a, b] whn is pdf is f x a x b b a Also, b + a b a µ E and V Ex4. Suppos, h lvl of unblivabiliy a any poin in a Transformrs
More informationLucas Test is based on Euler s theorem which states that if n is any integer and a is coprime to n, then a φ(n) 1modn.
Modul 10 Addtonal Topcs 10.1 Lctur 1 Prambl: Dtrmnng whthr a gvn ntgr s prm or compost s known as prmalty tstng. Thr ar prmalty tsts whch mrly tll us whthr a gvn ntgr s prm or not, wthout gvng us th factors
More informationGauge Theories. Elementary Particle Physics Strong Interaction Fenomenology. Diego Bettoni Academic year
Gau Thors Elmary Parcl Physcs Sro Iraco Fomoloy o Bo cadmc yar - Gau Ivarac Gau Ivarac Whr do Laraas or Hamloas com from? How do w kow ha a cra raco should dscrb a acual hyscal sysm? Why s h lcromac raco
More informationA Note on Estimability in Linear Models
Intrnatonal Journal of Statstcs and Applcatons 2014, 4(4): 212-216 DOI: 10.5923/j.statstcs.20140404.06 A Not on Estmablty n Lnar Modls S. O. Adymo 1,*, F. N. Nwob 2 1 Dpartmnt of Mathmatcs and Statstcs,
More informationNeutron electric dipole moment on the lattice
ron lcrc dol on on h lac go Shnan Unv. of Tkba 3/6/006 ron lcrc dol on fro lac QCD Inrodcon arar Boh h ha of CKM arx and QCD vac ffc conrb o CP volaon P and T volaon arar. CP odd QCD 4 L arg d CKM f f
More informationAR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 )
AR() Procss Th firs-ordr auorgrssiv procss, AR() is whr is WN(0, σ ) Condiional Man and Varianc of AR() Condiional man: Condiional varianc: ) ( ) ( Ω Ω E E ) var( ) ) ( var( ) var( σ Ω Ω Ω Ω E Auocovarianc
More informationThe Penalty Cost Functional for the Two-Dimensional Energized Wave Equation
Lonardo Jornal of Scncs ISSN 583-033 Iss 9, Jly-Dcmbr 006 p. 45-5 Th Pnalty Cost Fnctonal for th Two-Dmnsonal Enrgd Wav Eqaton Vctor Onoma WAZIRI, Snday Agsts REJU Mathmatcs/Comptr Scnc dpartmnt, Fdral
More informationSoft k-means Clustering. Comp 135 Machine Learning Computer Science Tufts University. Mixture Models. Mixture of Normals in 1D
Comp 35 Machn Larnng Computr Scnc Tufts Unvrsty Fall 207 Ron Khardon Th EM Algorthm Mxtur Modls Sm-Suprvsd Larnng Soft k-mans Clustrng ck k clustr cntrs : Assocat xampls wth cntrs p,j ~~ smlarty b/w cntr
More informationUNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Answr Ky Nam: Da: UNIT # EXPONENTIAL AND LOGARITHMIC FUNCTIONS Par I Qusions. Th prssion is quivaln o () () 6 6 6. Th ponnial funcion y 6 could rwrin as y () y y 6 () y y (). Th prssion a is quivaln o
More informationSafety and Reliability of Embedded Systems. (Sicherheit und Zuverlässigkeit eingebetteter Systeme) Stochastic Reliability Analysis
(Schrh und Zuvrlässgk ngbr Sysm) Sochasc Rlably Analyss Conn Dfnon of Rlably Hardwar- vs. Sofwar Rlably Tool Asssd Rlably Modlng Dscrpons of Falurs ovr Tm Rlably Modlng Exampls of Dsrbuon Funcons Th xponnal
More informationSafety and Reliability of Embedded Systems. (Sicherheit und Zuverlässigkeit eingebetteter Systeme) Stochastic Reliability Analysis
Safy and Rlably of Embddd Sysms (Schrh und Zuvrlässgk ngbr Sysm) Sochasc Rlably Analyss Safy and Rlably of Embddd Sysms Conn Dfnon of Rlably Hardwar- vs. Sofwar Rlably Tool Asssd Rlably Modlng Dscrpons
More informationCONTINUOUS TIME DYNAMIC PROGRAMMING
Eon. 511b Sprng 1993 C. Sms I. Th Opmaon Problm CONTINUOUS TIME DYNAMIC PROGRAMMING W onsdr h problm of maxmng subj o and EU(C, ) d (1) j ^ d = (C, ) d + σ (C, ) dw () h(c, ), (3) whr () and (3) hold for
More information, t 1. Transitions - this one was easy, but in general the hardest part is choosing the which variables are state and control variables
Opmal Conrol Why Use I - verss calcls of varaons, opmal conrol More generaly More convenen wh consrans (e.g., can p consrans on he dervaves More nsghs no problem (a leas more apparen han hrogh calcls of
More informationThe Fourier Transform
/9/ Th ourr Transform Jan Baptst Josph ourr 768-83 Effcnt Data Rprsntaton Data can b rprsntd n many ways. Advantag usng an approprat rprsntaton. Eampls: osy ponts along a ln Color spac rd/grn/blu v.s.
More informationMidterm exam 2, April 7, 2009 (solutions)
Univrsiy of Pnnsylvania Dparmn of Mahmaics Mah 26 Honors Calculus II Spring Smsr 29 Prof Grassi, TA Ashr Aul Midrm xam 2, April 7, 29 (soluions) 1 Wri a basis for h spac of pairs (u, v) of smooh funcions
More informationOn the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument
Inrnaional Rsarch Journal of Applid Basic Scincs 03 Aailabl onlin a wwwirjabscom ISSN 5-838X / Vol 4 (): 47-433 Scinc Eplorr Publicaions On h Driais of Bssl Modifid Bssl Funcions wih Rspc o h Ordr h Argumn
More informationby Lauren DeDieu Advisor: George Chen
b Laren DeDe Advsor: George Chen Are one of he mos powerfl mehods o nmercall solve me dependen paral dfferenal eqaons PDE wh some knd of snglar shock waves & blow-p problems. Fed nmber of mesh pons Moves
More informationEE243 Advanced Electromagnetic Theory Lec # 10: Poynting s Theorem, Time- Harmonic EM Fields
Appl M Fall 6 Nuruhr Lcur # r 9/6/6 4 Avanc lcromagnc Thory Lc # : Poynng s Thorm Tm- armonc M Fls Poynng s Thorm Consrvaon o nrgy an momnum Poynng s Thorm or Lnar sprsv Ma Poynng s Thorm or Tm-armonc
More informationChapter 9 Transient Response
har 9 Transn sons har 9: Ouln N F n F Frs-Ordr Transns Frs-Ordr rcus Frs ordr crcus: rcus conan onl on nducor or on caacor gornd b frs-ordr dffrnal quaons. Zro-nu rsons: h crcu has no ald sourc afr a cran
More information10.5 Linear Viscoelasticity and the Laplace Transform
Scn.5.5 Lnar Vclacy and h Lalac ranfrm h Lalac ranfrm vry uful n cnrucng and analyng lnar vclac mdl..5. h Lalac ranfrm h frmula fr h Lalac ranfrm f h drvav f a funcn : L f f L f f f f f c..5. whr h ranfrm
More informationEconomics 600: August, 2007 Dynamic Part: Problem Set 5. Problems on Differential Equations and Continuous Time Optimization
THE UNIVERSITY OF MARYLAND COLLEGE PARK, MARYLAND Economcs 600: August, 007 Dynamc Part: Problm St 5 Problms on Dffrntal Equatons and Contnuous Tm Optmzaton Quston Solv th followng two dffrntal quatons.
More informationGrand Canonical Ensemble
Th nsmbl of systms mmrsd n a partcl-hat rsrvor at constant tmpratur T, prssur P, and chmcal potntal. Consdr an nsmbl of M dntcal systms (M =,, 3,...M).. Thy ar mutually sharng th total numbr of partcls
More informationTwo-Dimensional Quantum Harmonic Oscillator
D Qa Haroc Oscllaor Two-Dsoal Qa Haroc Oscllaor 6 Qa Mchacs Prof. Y. F. Ch D Qa Haroc Oscllaor D Qa Haroc Oscllaor ch5 Schrödgr cosrcd h cohr sa of h D H.O. o dscrb a classcal arcl wh a wav ack whos cr
More informationDynamic Power Allocation in MIMO Fading Systems Without Channel Distribution Information
PROC. IEEE INFOCOM 06 Dynamc Powr Allocaon n MIMO Fadng Sysms Whou Channl Dsrbuon Informaon Hao Yu and Mchal J. Nly Unvrsy of Souhrn Calforna Absrac Ths papr consdrs dynamc powr allocaon n MIMO fadng sysms
More informationSpring 2006 Process Dynamics, Operations, and Control Lesson 2: Mathematics Review
Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw. conx and dircion Imagin a sysm ha varis in im; w migh plo is oupu vs. im. A plo migh imply an quaion, and h quaion is usually an
More informationCPSC 211 Data Structures & Implementations (c) Texas A&M University [ 259] B-Trees
CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 259] B-Trs Th AVL r and rd-black r allowd som variaion in h lnghs of h diffrn roo-o-laf pahs. An alrnaiv ida is o mak sur ha all roo-o-laf pahs
More informationOutlier-tolerant parameter estimation
Outlr-tolrant paramtr stmaton Baysan thods n physcs statstcs machn larnng and sgnal procssng (SS 003 Frdrch Fraundorfr fraunfr@cg.tu-graz.ac.at Computr Graphcs and Vson Graz Unvrsty of Tchnology Outln
More informationSeptember 27, Introduction to Ordinary Differential Equations. ME 501A Seminar in Engineering Analysis Page 1. Outline
Introucton to Ornar Dffrntal Equatons Sptmbr 7, 7 Introucton to Ornar Dffrntal Equatons Larr artto Mchancal Engnrng AB Smnar n Engnrng Analss Sptmbr 7, 7 Outln Rvw numrcal solutons Bascs of ffrntal quatons
More informationa 1and x is any real number.
Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars
More informationProblem 1: Consider the following stationary data generation process for a random variable y t. e t ~ N(0,1) i.i.d.
A/CN C m Sr Anal Profor Òcar Jordà Wnr conomc.c. Dav POBLM S SOLIONS Par I Analcal Quon Problm : Condr h followng aonar daa gnraon proc for a random varabl - N..d. wh < and N -. a Oban h populaon man varanc
More informationBlack-Scholes Partial Differential Equation In The Mellin Transform Domain
INTRNATIONAL JOURNAL OF SCINTIFIC & TCHNOLOGY RSARCH VOLUM 3, ISSU, Dcmbr 4 ISSN 77-866 Blac-Schols Paral Dffrnal qaon In Th Mlln Transform Doman Fadgba Snday mmanl, Ognrnd Rosln Bosd Absrac: Ths ar rsns
More informationApplying Software Reliability Techniques to Low Retail Demand Estimation
Applyng Sofwar Rlably Tchnqus o Low Ral Dmand Esmaon Ma Lndsy Unvrsy of Norh Txas ITDS Dp P.O. Box 30549 Dnon, TX 7603-549 940 565 3174 lndsym@un.du Robr Pavur Unvrsy of Norh Txas ITDS Dp P.O. Box 30549
More informationOscillations of Hyperbolic Systems with Functional Arguments *
Avll ://vmd/gs/9/s Vol Iss Dcmr 6 95 Prvosly Vol No Alcons nd Ald mcs AA: An Inrnonl Jornl Asrc Oscllons of Hyrolc Sysms w Fnconl Argmns * Y So Fcly of Engnrng nzw Unvrsy Isw 9-9 Jn E-ml: so@nzw-c Noro
More informationGuaranteed Cost Control for a Class of Uncertain Delay Systems with Actuator Failures Based on Switching Method
49 Inrnaonal Journal of Conrol, Ru Wang Auomaon, and Jun and Zhao Sysms, vol. 5, no. 5, pp. 49-5, Ocobr 7 Guarand Cos Conrol for a Class of Uncran Dlay Sysms wh Acuaor Falurs Basd on Swchng Mhod Ru Wang
More informationChapter 8 Theories of Systems
~~ 7 Char Thor of Sm - Lala Tranform Solon of Lnar Sm Lnar Sm F : Conr n a n- n- a n- n- a a f L n n- ' ' ' n n n a a a a f Eg - an b ranform no ' ' b an b Lala ranform Sol Lf ]F-f 7 C 7 C C C ] a L a
More informationEngineering Circuit Analysis 8th Edition Chapter Nine Exercise Solutions
Engnrng rcu naly 8h Eon hapr Nn Exrc Soluon. = KΩ, = µf, an uch ha h crcu rpon oramp. a For Sourc-fr paralll crcu: For oramp or b H 9V, V / hoo = H.7.8 ra / 5..7..9 9V 9..9..9 5.75,.5 5.75.5..9 . = nh,
More informationOn the Speed of Heat Wave. Mihály Makai
On h Spd of Ha Wa Mihály Maai maai@ra.bm.hu Conns Formulaion of h problm: infini spd? Local hrmal qulibrium (LTE hypohsis Balanc quaion Phnomnological balanc Spd of ha wa Applicaion in plasma ranspor 1.
More informationCHAPTER: 3 INVERSE EXPONENTIAL DISTRIBUTION: DIFFERENT METHOD OF ESTIMATIONS
CHAPTER: 3 INVERSE EXPONENTIAL DISTRIBUTION: DIFFERENT METHOD OF ESTIMATIONS 3. INTRODUCTION Th Ivrs Expoal dsrbuo was roducd by Kllr ad Kamah (98) ad has b sudd ad dscussd as a lfm modl. If a radom varabl
More information10. If p and q are the lengths of the perpendiculars from the origin on the tangent and the normal to the curve
0. If p and q ar h lnghs of h prpndiculars from h origin on h angn and h normal o h curv + Mahmaics y = a, hn 4p + q = a a (C) a (D) 5a 6. Wha is h diffrnial quaion of h family of circls having hir cnrs
More informationChapter 7 Stead St y- ate Errors
Char 7 Say-Sa rror Inroucon Conrol ym analy an gn cfcaon a. rann ron b. Sably c. Say-a rror fnon of ay-a rror : u c a whr u : nu, c: ouu Val only for abl ym chck ym ably fr! nu for ay-a a nu analy U o
More information1. Inverse Matrix 4[(3 7) (02)] 1[(0 7) (3 2)] Recall that the inverse of A is equal to:
Rfrncs Brnank, B. and I. Mihov (1998). Masuring monary policy, Quarrly Journal of Economics CXIII, 315-34. Blanchard, O. R. Proi (00). An mpirical characrizaion of h dynamic ffcs of changs in govrnmn spnding
More informationCOMPLEX NUMBER PAIRWISE COMPARISON AND COMPLEX NUMBER AHP
ISAHP 00, Bal, Indonsa, August -9, 00 COMPLEX NUMBER PAIRWISE COMPARISON AND COMPLEX NUMBER AHP Chkako MIYAKE, Kkch OHSAWA, Masahro KITO, and Masaak SHINOHARA Dpartmnt of Mathmatcal Informaton Engnrng
More informationChapter 13 Laplace Transform Analysis
Chapr aplac Tranorm naly Chapr : Ouln aplac ranorm aplac Tranorm -doman phaor analy: x X σ m co ω φ x X X m φ x aplac ranorm: [ o ] d o d < aplac Tranorm Thr condon Unlaral on-dd aplac ranorm: aplac ranorm
More informationCONSISTENT EARTHQUAKE ACCELERATION AND DISPLACEMENT RECORDS
APPENDX J CONSSTENT EARTHQUAKE ACCEERATON AND DSPACEMENT RECORDS Earhqake Acceleraons can be Measred. However, Srcres are Sbjeced o Earhqake Dsplacemens J. NTRODUCTON { XE "Acceleraon Records" }A he presen
More informationLecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields
Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration
More informationFinal Exam : Solutions
Comp : Algorihm and Daa Srucur Final Exam : Soluion. Rcuriv Algorihm. (a) To bgin ind h mdian o {x, x,... x n }. Sinc vry numbr xcp on in h inrval [0, n] appar xacly onc in h li, w hav ha h mdian mu b
More information2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35
MATH 5 PS # Summr 00.. Diffrnial Equaions and Soluions PS.# Show ha ()C #, 4, 7, 0, 4, 5 ( / ) is a gnral soluion of h diffrnial quaion. Us a compur or calculaor o skch h soluions for h givn valus of h
More informationJournal of Theoretical and Applied Information Technology 10 th January Vol. 47 No JATIT & LLS. All rights reserved.
Journal o Thortcal and Appld Inormaton Tchnology th January 3. Vol. 47 No. 5-3 JATIT & LLS. All rghts rsrvd. ISSN: 99-8645 www.att.org E-ISSN: 87-395 RESEARCH ON PROPERTIES OF E-PARTIAL DERIVATIVE OF LOGIC
More informationDEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018
DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Aoc. Prof. Dr. Burak Kllci Spring 08 OUTLINE Th Laplac Tranform Rgion of convrgnc for Laplac ranform Invr Laplac ranform Gomric valuaion
More informationInstitute of Actuaries of India
Insiu of Acuaris of India ubjc CT3 Probabiliy and Mahmaical aisics Novmbr Examinaions INDICATIVE OLUTION Pag of IAI CT3 Novmbr ol. a sampl man = 35 sampl sandard dviaion = 36.6 b for = uppr bound = 35+*36.6
More informationChap 2: Reliability and Availability Models
Chap : lably ad valably Modls lably = prob{s s fully fucog [,]} Suppos from [,] m prod, w masur ou of N compos, of whch N : # of compos oprag corrcly a m N f : # of compos whch hav fald a m rlably of h
More informationOptimal Ordering Policy in a Two-Level Supply Chain with Budget Constraint
Optmal Ordrng Polcy n a Two-Lvl Supply Chan wth Budgt Constrant Rasoul aj Alrza aj Babak aj ABSTRACT Ths papr consdrs a two- lvl supply chan whch consst of a vndor and svral rtalrs. Unsatsfd dmands n rtalrs
More informationA CONVERGENCE MODEL OF THE TERM STRUCTURE OF INTEREST RATES
ISN 9984 676 68 4 VIKORS AJVSKIS KRISĪN VĪOLA A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS WORKING PAPR 9 Lavja anka 9 h orc o b ndcad whn rprodcd. A CONVRGNC MOL OF H RM SRUCUR OF INRS RAS CONNS Abrac Inrodcon
More informationCharging of capacitor through inductor and resistor
cur 4&: R circui harging of capacior hrough inducor and rsisor us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R, an inducor of inducanc and a y K in sris.
More informationValuing Energy Options in a One Factor Model Fitted to Forward Prices
Vag Enrgy Oons n a On acor Modl Clwlow and rcland Vag Enrgy Oons n a On acor Modl d o orward Prcs Ls Clwlow and Chrs rcland hs Vrson: 5 h Arl 999 chool of nanc and Economcs Unvrsy of chnology ydny Asrala
More information3.4 Properties of the Stress Tensor
cto.4.4 Proprts of th trss sor.4. trss rasformato Lt th compots of th Cauchy strss tsor a coordat systm wth bas vctors b. h compots a scod coordat systm wth bas vctors j,, ar gv by th tsor trasformato
More informationNAME: ANSWER KEY DATE: PERIOD. DIRECTIONS: MULTIPLE CHOICE. Choose the letter of the correct answer.
R A T T L E R S S L U G S NAME: ANSWER KEY DATE: PERIOD PREAP PHYSICS REIEW TWO KINEMATICS / GRAPHING FORM A DIRECTIONS: MULTIPLE CHOICE. Chs h r f h rr answr. Us h fgur bw answr qusns 1 and 2. 0 10 20
More informationTesting EBUASI Class of Life Distribution Based on Goodness of Fit Approach
Tsng BUAS Class f Lf Dsrbn Basd n Gdnss f Apprach.A.Rad a,..lashn b and.a.abass b a ns f Sascal Sds and Rsarch, Car Unrs, gp b Mahmacs Dp., acl f ngnrng, Tana Unrs, gp Absrac: Ts sascs fr sng pnnal agans
More informationDynamic Controllability with Overlapping Targets: Or Why Target Independence May Not be Good for You
Dynamc Conrollably wh Ovrlappng Targs: Or Why Targ Indpndnc May No b Good for You Ncola Acoclla Unvrsy of Rom La Sapnza Govann D Barolomo Unvrsy of Rom La Sapnza and Unvrsy of Tramo Andrw Hughs Hall Vandrbl
More informationFirst looking at the scalar potential term, suppose that the displacement is given by u = φ. If one can find a scalar φ such that u = φ. u x.
7.4 Eastodynams 7.4. Propagaton of Wavs n East Sods Whn a strss wav travs throgh a matra, t ass matra parts to dspa by. It an b shown that any vtor an b wrttn n th form φ + ra (7.4. whr φ s a saar potnta
More informationCIVL 8/ D Boundary Value Problems - Triangular Elements (T6) 1/8
CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () /8 SI-ODE RIAGULAR ELEMES () A quadracall nrpolad rangular lmn dfnd b nod, hr a h vrc and hr a h mddl a ach d. h mddl nod, dpndng on locaon, ma dfn a
More information4. Which of the following organs develops first?
Biology 4. Which of h following organs dvlops firs? (A) Livr (C) Kidny (B) Har (D) Noochord 12. During mbryonic priod, animals rpa mbryonic sags of hir ancsors. This law is calld (A) Flokin s law (B) Biognic
More informationCopyright 2000, Kevin Wayne 1
Rcap: Maxmum 3-Sasfably Maxmum 3-Sasfably: Analyss CS 580: Algorhm Dsgn and Analyss Jrmah Block Purdu Unvrsy Sprng 2018 Announcmns: Homwork 6 dadln xndd o Aprl 24 h a 11:59 PM Cours Evaluaon Survy: Lv
More informationThe Hyperelastic material is examined in this section.
4. Hyprlastcty h Hyprlastc matral s xad n ths scton. 4..1 Consttutv Equatons h rat of chang of ntrnal nrgy W pr unt rfrnc volum s gvn by th strss powr, whch can b xprssd n a numbr of dffrnt ways (s 3.7.6):
More informationLecture 08 Multiple View Geometry 2. Prof. Dr. Davide Scaramuzza
Lctr 8 Mltpl V Gomtry Prof. Dr. Dad Scaramzza sdad@f.zh.ch Cors opcs Prncpls of mag formaton Imag fltrng Fatr dtcton Mlt- gomtry 3D Rconstrcton Rcognton Mltpl V Gomtry San Marco sqar, Vnc 4,79 mags, 4,55,57
More informationOne dimensional steady state heat transfer of composite slabs
BUILDING PHYSICS On dmnsonal sady sa a ransfr of compos slas Par 2 Ass. Prof. Dr. Norr Harmay Budaps Unvrsy of Tcnology and Economcs Dparmn of Buldng Enrgcs and Buldng Srvc Engnrng Inroducon - Buldng Pyscs
More informationMath 656 March 10, 2011 Midterm Examination Solutions
Math 656 March 0, 0 Mdtrm Eamnaton Soltons (4pts Dr th prsson for snh (arcsnh sng th dfnton of snh w n trms of ponntals, and s t to fnd all als of snh (. Plot ths als as ponts n th compl plan. Mak sr or
More informationContinous system: differential equations
/6/008 Coious sysm: diffrial quaios Drmiisic modls drivaivs isad of (+)-( r( compar ( + ) R( + r ( (0) ( R ( 0 ) ( Dcid wha hav a ffc o h sysm Drmi whhr h paramrs ar posiiv or gaiv, i.. giv growh or rducio
More informationBoyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors
Boc/DiPrima 9 h d, Ch.: Linar Equaions; Mhod of Ingraing Facors Elmnar Diffrnial Equaions and Boundar Valu Problms, 9 h diion, b William E. Boc and Richard C. DiPrima, 009 b John Wil & Sons, Inc. A linar
More informationGaussian Random Process and Its Application for Detecting the Ionospheric Disturbances Using GPS
Journal of Global Posonng Sysms (005) Vol. 4, No. 1-: 76-81 Gaussan Random Procss and Is Applcaon for Dcng h Ionosphrc Dsurbancs Usng GPS H.. Zhang 1,, J. Wang 3, W. Y. Zhu 1, C. Huang 1 (1) Shangha Asronomcal
More informationFourier Transform: Overview. The Fourier Transform. Why Fourier Transform? What is FT? FT of a pulse function. FT maps a function to its frequencies
.5.3..9.7.5.3. -. -.3 -.5.8.6.4. -. -.4 -.6 -.8 -. 8. 6. 4. -. -. 4 -. 6 -. 8 -.8.6.4. -. -.4 -.6 -.8 - orr Transform: Ovrvw Th orr Transform Wh T s sfl D T, DT, D DT T proprts Lnar ltrs Wh orr Transform?
More information