Supplementary Material to: IMU Preintegration on Manifold for E cient Visual-Inertial Maximum-a-Posteriori Estimation
|
|
- Silvester Jefferson
- 5 years ago
- Views:
Transcription
1 Supplemenary Maeral o: IMU Prenegraon on Manfold for E cen Vsual-Ineral Maxmum-a-Poseror Esmaon echncal Repor G-IRIM-CP&R Chrsan Forser, Luca Carlone, Fran Dellaer, and Davde Scaramuzza May 0, 05 hs repor provdes addonal dervaons and mplemenaon deals o suppor he paper [4] herefore, should no be consdered a self-conaned documen, bu raher regarded as an appendx of [4], and ced as: C Forser, L Carlone, F Dellaer, and D Scaramuzza, IMU prenegraon on manfold for e cen vsual-neral maxmum-a-poseror esmaon, (supplemenary maeral, G-IRIM-CP&R-05-00), In Robocs: Scence and Sysems (RSS), 05 Across hs repor, references n he form (x), eg, (), recall equaons from he man paper [4], whle references (Ax), eg, (A), refer o equaons whn hs appendx Conens IMU Prenegraon: Nose Propagaon and Bas Updaes Ierave Nose Propagaon Bas Correcon va Frs-Order Updaes IMU Facors: Resdual Errors and Jacobans 5 Jacobans of r pj 6 Jacobans of r vj 6 Jacobans of r Rj 7 Vson facors: Schur Complemen and Null Space Projecon 8 4 Angular Velocy and Rgh Jacobans for SO() 9
2 IMU Prenegraon: Nose Propagaon and Bas Updaes Ierave Nose Propagaon In hs secon we provde a complee dervaon of he prenegraed measuremens covarance j (cf Secon V- B n [4]), whch s such ha: j [ j, v j, p j] N (0 9, j ) (A) We call j he prenegraon nose vecor Le us rewre explcly wha he prenegraon errors j, v j, p j are (cf Eqs (8)-(9)-(0)): j ' v j ' R +jj r gd R (ã b a )^ p j ' R (ã b a )^ + R ad + R ad (A) where he relaons are vald up o he frs order Now, one way o derve he covarance of j s o subsue he expresson of j bac no v j and p j he resul of hs would be a lnear expresson ha relaes j o he raw measuremens nose gd, ad, on whch a (edous) nose propagaon can be carred ou o avod hs long procedure, we prefer o wre (A) n erave form, and hen o carry ou nose propagaon on he resulng (smpler) expressons In order o wre j n erave form we frs noe ha: j ' Xj R +jj r gd R +jj r gd + I z } { R jj J j r gd j Xj Xj ( R +j R j j ) J r gd + J j r gd j R j j R +j J r gd + J j r gd j R j j j + J j r gd j (A) where we smply oo he las erm ( j Repeang he same process for v j : ) ou of he sum and convenenly rearranged he erms v j Xj R (ã b a )^ R (ã b a )^ v j R j (ã j b a )^ + R ad + R ad R j (ã j b a )^ j + R j ad j j + R j ad j (A4) Dong he same for p j, and nong ha p j can be wren as a funcon of v j (hs can be easly seen from he expresson of v j n (A)): p j Xj v v + v j p j + v j R (ã b a )^ R (ã b a )^ R j (ã j b a )^ R j (ã j b a )^ + R ad + R ad j + R j ad j j + R j ad j (A5)
3 From (A), (A4), and (A5), follows ha ha (A) can be wren n erave form: + R + + J r gd v + v R (ã b a )^ + R ad p + p + v R (ã b a )^ + R ad (A6) for,, j, wh nal condons v p Eq (A6) can be convenenly wren n marx form: + R v R (ã b a)^ J r 0 I v R 5 gd p + R (ã b a)^ I I p R ad (A7) or more smply: 0 + A + B d (A8) where d [ gd ad ] From he lnear model (A8) and gven he covarance R 66 of he raw IMU measuremens nose d, s now possble o compue he covarance eravely: + A + A + B B sarng from nal condons 99 Noe ha he fac ha he covarance can be compued eravely s convenen, compuaonally, as means ha we can easly updae he covarance afer negrang a new measuremen he same erave compuaon s possble for he prenegraed measuremens hemselves: R + R Exp ((! b g ) ) ṽ + ṽ + R (ã b a ) (A9) p + p + ṽ + R (ã b a ) (A0) whch easly follows from Eqs (8)-(9)-(0) of he man documen Bas Correcon va Frs-Order Updaes In hs secon we provde a complee dervaon of he frs-order bas correcon proposed n Secon V-C of [4] Le us sar by recallng he expresson of he prenegraed measuremens R j, ṽ j, p j, gven n Eqs (8)-(9)-(0) of he man documen: jy R j Exp ((! b g ) ) ṽ j p j R (ã b a ) R (ã b a ) (A) [ bg b a ], and le Assume now ha we have compued he prenegraed varables a a gven bas esmae b us denoe he correspondng prenegraed measuremens as R j ( b ), ṽ j ( b ), p j ( b ) In hs secon we wan o devse an expresson o updae R j ( b ), ṽ j ( b ), p j ( b ) when our bas esmae changes Consder he case n whch we ge a new esmae ˆb b + b,where b s a small correcon wr he prevous esmae b Usng he new bas esmae ˆb n (A) we ge he updaed prenegraed measuremens: jy R j (ˆb ) ṽ j (ˆb ) p j (ˆb ) Exp! ˆbg R (ˆb ) ã ˆba R ã ˆba jy Exp! bg b g R (ˆb ) ã ba b a R (ˆb ) ã ba b a (A)
4 A nave soluon would be o recompue he prenegraed measuremens a he new bas esmae as prescrbed n (A) In hs secon, nsead, we show how o updae he prenegraed measuremens whou repeang he negraon Le us sar from he prenegraed roaon measuremen R j (ˆb ) We assumed ha he bas correcon s small, hence we use he frs-order approxmaon (7) for each erm n he produc: jy h R j (ˆb ) ' Exp! bg Exp J r b gd where we defned J r J r (! bg )(J r s he rgh Jacoban for SO() gven n Eq(8) of he paper) We rearrange he erms n he produc, by movng he erms ncludng b o he end, usng he relaon (): jy R j (ˆb ) R j ( b ) Exp R +j ( b ) J r b g (A) where we used he fac ha R j ( b ) Q j Exp! bg by defnon Repeaed applcaon of he frs-order approxmaon (9) (recall ha b g s small, hence he rgh Jacobans are close o he deny) produces: R j (ˆb )' R j ( b )Exp R +j ( b ) J r R j b g! R j R j ( b )Exp b g (A4) whch corresponds o eq (6) n [4] he Jacoban can be precompued durng prenegraon hs can be done n analogy wh Secon snce he srucure of R j s essenally he same as he one mulplyng he nose n (A) Usng (A4) we can updae he prevous prenegraed measuremen R j ( b ) o ge R j (ˆb ) Le us now focus on he prenegraed velocy ṽ j (ˆb ) We subsue R j (ˆb ) bac no (A): ṽ j (ˆb ) R j ( b )Exp R j b g ã ba b a (A5) Recallng ha he correcon b g ṽ j (ˆb ) ' R j R j ( b ) I + s small, we use he frs-order approxmaon (4): b g ^ Developng he prevous expresson and droppng hgher-order erms: ã ba b a (A6) ṽ j (ˆb ) ' R j ( b ) ã ba + R j ( b )( b a ) + R j ( b ) R j b g ^ ã ba (A7) Recallng ha ṽ j ( b ) P j R j ( b ) ã ba and usng propery (): ṽ j (ˆb ) ṽ j ( b )+ R j ( b ) b a + R j ( b ) ã ba ^ R j b g ṽ j ( b )+ b a b a + b g (A8) whch corresponds o he second expresson n Eq (6) of he paper 4
5 Fnally, repeang he same dervaon for p j (ˆb ) p j (ˆb ) Eq (A4) Eq (4) ' R (ˆb ) ã ba b a R j ( b )Exp R j ( b ) p j ( b )+ Eq () p j ( b )+ I + R j R j b g b g ^ ã ba b a ã ba b a R j ( b ) b a + R R j j ( b ) R j ( b ) b a + b g ^ ã ba R j ( b ) ã ba ^ R j b g p j ( b g, b a )+ p j b a b a + p j b g (A9) whch corresponds o he las expresson n Eq(6) o summarze, he Jacobans used for a-poseror bas updae are (cf eqs (A4)-(A8)-(A9)): R j b a p j b a p j R +j ( b ) J r R j ( b ) R j ã ba ^ R j R j ( b ) R j ( b ) ã ba ^ R j (A0) Repeang he same dervaon of Secon, s possble o show ha (A0) can be compued ncremenally, as new measuremens arrve IMU Facors: Resdual Errors and Jacobans In hs secon we provde analyc expressons for he Jacoban marces of he resdual errors nroduced n Secon V-D of [4] We sar from he expresson of he resdual errors for he prenegraed IMU measuremens (cf, Eq(7)):! r Rj Log R j ( b g )Exp R j b g R R j r vj R (v j v g j ) ṽ j ( b g, b a )+ b g + b a b a r pj R p j p v j g j Lfng he cos funcon consss n subsung he followng reracon: p j p j ( b g, b a )+ b g + p j b a (A) b a p p + R p, R R Exp( ), p j p j + R j p j, R j R j Exp( j), (A) whle, snce velocy and bases already lve n a vecor space he correspondng reracon reduces o: v v + v, v j v j + v, b g bg + b g, b a b a + b a, (A) he process of lfng maes he resdual errors a funcon defned on a vecor space, on whch s easy o compue Jacobans We derve he Jacobans wr he vecors, p, v, j, p j, v j, b g, b a n he followng secons 5
6 Jacobans of r pj r pj (p + R p )R p j p R p v j g j p j + p j b g + p j b a b a (A4) r pj (p )+( I ) p r pj (p j + R j p j )R p j + R j p j p v j g j p j + p j b g + p j b a b a (A5) r pj (p j )+(R R j ) p j r pj (v + v )R p j p v j v j g j p j + p j b g + p j b a b a (A6) r pj (v )+( R j ) v r pj (R Exp( )) (R Exp( )) p j p v j Eq (4) ' (I r pj (R )+ ^ )R p j p v j R p j p v j g j g j p j + g j p j + ^ p j p j b g + b g + p j b a b a p j b a b a (we used a^b b^a) (A7) In summary, he jacobans of r pj are: r pj r pj R p j p v j g j p I r pj v R j r pj r j pj p j R R j r pj v j r pj b a r pj b g p j b a p j ^ Jacobans of r vj r vj (v + v )R (v j v v g j ) ṽ j + b g + r v (v ) R v (A8) r vj (v j + v j )R (v j + v j v g j ) ṽ j + b g + r v (v j )+R v j (A9) b a b a b a b a r vj (R Exp( )) (R Exp( )) (v j v g j ) Eq (4) ' (I ^ )R (v j v g j ) r v (R )+ R (v j v g j ) ^ ṽ j + ṽ j + b g + b g + (we used a^b b^a) b a b a b a b a (A0) 6
7 In summary, he jacobans of r vj are: r vj r vj R (v j v g j ) ^ p r vj v R vj r r j vj p j r vj v j R r vj b a r vj b g b a Jacobans of r Rj r Rj (R Exp( )) Log R j ( b g )Exp R j Log R j ( b g )Exp R j Eq() Log R j ( b g )Exp R j Eq(9) ' Log R j ( b g )Exp R j + Jr Log R j ( b g )Exp r R (R ) J r (r R (R ))R j R! b g (R Exp( )) R j b g Exp( R j )R R j! b g R R j Exp( R j R )! b g R R j!!! b g R R j ( R j R ) (A) r Rj (R j Exp( j)) Log R j R j ( b g )Exp Eq(9) ' r R (R j )+J r (r R (R j )) j b g R (R j Exp( j))! (A) r Rj ( b g + b g ) Log R j ( b g )Exp R j b g + b! g R R j! Eq(7) ' Log R j ( b g )Exp R j b g R j Exp J r b g R j b b g g R R j! R j Log Exp J r b g R j b b g g R j ( b g )Exp R j b g R R j R j Log Exp J r b g R j b b g g Exp r Rj ( b g ) Eq() Log Exp r Rj ( b g ) Exp Exp r Rj ( b g ) R j J r b g R j b b g g Eq(9) ' r Rj ( b g ) J r r Rj ( b g ) Exp r R j ( b g ) R j J r b g R j b b g g (A) 7
8 In summary, he jacobans of r Rj are: Jr (r R (R ))R j R p v j J r (r R (R j )) p j v j b a J b g r r Rj ( b g ) Exp r R j ( b g ) J r R j b g R j Vson facors: Schur Complemen and Null Space Projecon Le us sar from he lnearzed vson facors, gven n Eq (44) of he man documen: X F l + E l l b l (A4) l X (l) where [ p ] R 6 s a perurbaon wr he lnearzaon pon of he pose a eyframe and l s a perurbaon wr he lnerzaon pon of landmar l he vecor b l R s he resdual error a he lnearzaon pon For brevy, we do no ener n he deals of he Jacobans F l R 6, E l R Now, as done n he man documen, we denoe wh X (l) R 6n he vecor sacng he perurbaons for each of he n l cameras observng landmar l Wh hs noaon (A4) can be wren n marx F l X (l) + E l l b l (A5) l where: F l F l R n l6n l, E l E l 7 5 Rn l, b l 6 4 b l R n l 7 5 (A6) Snce a landmar l appears n a sngle erm of he sum (A5), for any gven choce of he pose perurbaon X (l), he landmar perurbaon l ha mnmzes he quadrac cos F l X (l) + E l l b l s: l (E l E l ) E l (F l X (l) b l ) (A7) Subsung (A7) bac no (A5) we can elmnae he landmars from he opmzaon problem: (I E l (E l E l ) E l ) F l X (l) b l (A8) l whch corresponds o eq (46) of he man documen he srucureless facors (A8) only nvolve poses and allow o perform he opmzaon dsregardng landmar posons Compuaon can be furher mproved by he followng lnear algebra consderaons Frs, we noe ha Q (I E l (E l E l) E l ) Rn ln l s an orhogonal projecor of E l Roughly speang, Q projecs any vecor n R n l o he null space of he marx E l Moreover, any bass E? l R n ln l of he null space of E l sasfes he followng relaon [5]: E? l (E? l ) E? l (E? l ) I E l (E l E l ) E l (A9) 8
9 A bass for he null space can be easly compued from E l usng SVD Such bass s unary, e, sasfes (E? l ) E? l I Subsung (A9) no (A8), and recallng ha E? l s a unary marx, we oban: E? l (E? l ) F l X (l) b l l l l l F l X (l) b l E? l (E? l ) E? l (E? l ) F l X (l) b l F l X (l) b l E? l (E? l ) F l X (l) b l E? l (E? l ) F l X (l) b l E? l (E? l ) F l X (l) b l (E? l ) F l X (l) b l (A40) whch s an alernave represenaon of our vson facors (A8), and s usually preferable from a compuaonal sandpon A smlar null space projecon s used n [6] whn a Kalman fler archecure, whle a facor graph vew on Schur complemen s gven n [] 4 Angular Velocy and Rgh Jacobans for SO() he rgh Jacoban marx J r ( ) (also called body Jacoban [7]) relaes raes of change n he parameer vecor o he nsananeous body angular velocy B! WB : l B! WB J r ( ) (A4) A closed-form expresson of he rgh Jacoban s gven n []: J r ( )I cos( ) ^ + sn( ) ( ^) (A4) Noe ha he Jacoban becomes he deny marx for Consder a drec cosne marx R WB ( ) SO(), ha roaes a pon from body coordnaes B o world coordnaes W, and ha s paramerzed by he roaon vecor he relaon beween angular velocy and he dervave of a roaon marx s [7] R WB ṘWB B!^WB (A4) Hence, usng (A4) we can wre he dervave of a roaon marx a : Ṙ WB ( )R WB ( )(J r ( ) )^ (A44) Gven a mulplcave perurbaon Exp( ) on he rgh hand sde of an elemen of he group SO(), we may as wha s he equvalen addve perurbaon n he angen space so() ha resuls n he same compound roaon: Exp( )Exp( )Exp( + ) (A45) Compung he dervave wh respec o he ncremens on boh sdes, usng (A44), and assumng ha he ncremens are small, we fnd J r ( ), (A46) leadng o: Exp( + ) Exp( )Exp(J r ( ) ) (A47) A smlar frs-order approxmaon holds for he logarhm: Log Exp( )Exp( ) + J r ( ) (A48) hs propery follows drecly from he Baer Campbell Hausdor (BCH) formula under he assumpon ha s small [] An explc expresson for he nverse of he rgh Jacoban s gven n []: J r ( )I + ^ + cos( ) + + ( ^) sn( ) 9
10 References [] D Barfoo and P Furgale Assocang uncerany wh hree-dmensonal poses for use n esmaon problems IEEE rans Robocs, 0():679 69, 04 [] L Carlone, P F Alcanarlla, HP Chu, K Zsol, and F Dellaer Mnng srucure fragmens for smar bundle adjusmen In Brsh Machne Vson Conf (BMVC), 04 [] G S Chrjan Sochasc Models, Informaon heory, and Le Groups, Volume : Analyc Mehods and Modern Applcaons (Appled and Numercal Harmonc Analyss) Brhauser, 0 [4] C Forser, L Carlone, F Dellaer, and D Scaramuzza IMU prenegraon on manfold for e cen vsualneral maxmum-a-poseror esmaon In Robocs: Scence and Sysems (RSS), 05 [5] CD Meyer Marx Analyss and Appled Lnear Algebra SIAM, 000 [6] AI Mours and SI Roumelos A mul-sae consran Kalman fler for vson-aded neral navgaon In IEEE Inl Conf on Robocs and Auomaon (ICRA), pages , Aprl 007 [7] RM Murray, Z L, and S Sasry A Mahemacal Inroducon o Roboc Manpulaon CRC Press, 994 0
( ) () we define the interaction representation by the unitary transformation () = ()
Hgher Order Perurbaon Theory Mchael Fowler 3/7/6 The neracon Represenaon Recall ha n he frs par of hs course sequence, we dscussed he chrödnger and Hesenberg represenaons of quanum mechancs here n he chrödnger
More informationJohn Geweke a and Gianni Amisano b a Departments of Economics and Statistics, University of Iowa, USA b European Central Bank, Frankfurt, Germany
Herarchcal Markov Normal Mxure models wh Applcaons o Fnancal Asse Reurns Appendx: Proofs of Theorems and Condonal Poseror Dsrbuons John Geweke a and Gann Amsano b a Deparmens of Economcs and Sascs, Unversy
More informationHEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD
Journal of Appled Mahemacs and Compuaonal Mechancs 3, (), 45-5 HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD Sansław Kukla, Urszula Sedlecka Insue of Mahemacs,
More informationScattering at an Interface: Oblique Incidence
Course Insrucor Dr. Raymond C. Rumpf Offce: A 337 Phone: (915) 747 6958 E Mal: rcrumpf@uep.edu EE 4347 Appled Elecromagnecs Topc 3g Scaerng a an Inerface: Oblque Incdence Scaerng These Oblque noes may
More informationLi An-Ping. Beijing , P.R.China
A New Type of Cpher: DICING_csb L An-Png Bejng 100085, P.R.Chna apl0001@sna.com Absrac: In hs paper, we wll propose a new ype of cpher named DICING_csb, whch s derved from our prevous sream cpher DICING.
More information( t) Outline of program: BGC1: Survival and event history analysis Oslo, March-May Recapitulation. The additive regression model
BGC1: Survval and even hsory analyss Oslo, March-May 212 Monday May 7h and Tuesday May 8h The addve regresson model Ørnulf Borgan Deparmen of Mahemacs Unversy of Oslo Oulne of program: Recapulaon Counng
More informationShould Exact Index Numbers have Standard Errors? Theory and Application to Asian Growth
Should Exac Index umbers have Sandard Errors? Theory and Applcaon o Asan Growh Rober C. Feensra Marshall B. Rensdorf ovember 003 Proof of Proposon APPEDIX () Frs, we wll derve he convenonal Sao-Vara prce
More informationOn One Analytic Method of. Constructing Program Controls
Appled Mahemacal Scences, Vol. 9, 05, no. 8, 409-407 HIKARI Ld, www.m-hkar.com hp://dx.do.org/0.988/ams.05.54349 On One Analyc Mehod of Consrucng Program Conrols A. N. Kvko, S. V. Chsyakov and Yu. E. Balyna
More informationIncluding the ordinary differential of distance with time as velocity makes a system of ordinary differential equations.
Soluons o Ordnary Derenal Equaons An ordnary derenal equaon has only one ndependen varable. A sysem o ordnary derenal equaons consss o several derenal equaons each wh he same ndependen varable. An eample
More informationMechanics Physics 151
Mechancs Physcs 5 Lecure 9 Hamlonan Equaons of Moon (Chaper 8) Wha We Dd Las Tme Consruced Hamlonan formalsm H ( q, p, ) = q p L( q, q, ) H p = q H q = p H = L Equvalen o Lagrangan formalsm Smpler, bu
More informationCS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 4
CS434a/54a: Paern Recognon Prof. Olga Veksler Lecure 4 Oulne Normal Random Varable Properes Dscrmnan funcons Why Normal Random Varables? Analycally racable Works well when observaon comes form a corruped
More informationCH.3. COMPATIBILITY EQUATIONS. Continuum Mechanics Course (MMC) - ETSECCPB - UPC
CH.3. COMPATIBILITY EQUATIONS Connuum Mechancs Course (MMC) - ETSECCPB - UPC Overvew Compably Condons Compably Equaons of a Poenal Vecor Feld Compably Condons for Infnesmal Srans Inegraon of he Infnesmal
More informationMechanics Physics 151
Mechancs Physcs 5 Lecure 9 Hamlonan Equaons of Moon (Chaper 8) Wha We Dd Las Tme Consruced Hamlonan formalsm Hqp (,,) = qp Lqq (,,) H p = q H q = p H L = Equvalen o Lagrangan formalsm Smpler, bu wce as
More informationSolution in semi infinite diffusion couples (error function analysis)
Soluon n sem nfne dffuson couples (error funcon analyss) Le us consder now he sem nfne dffuson couple of wo blocks wh concenraon of and I means ha, n a A- bnary sysem, s bondng beween wo blocks made of
More informationNATIONAL UNIVERSITY OF SINGAPORE PC5202 ADVANCED STATISTICAL MECHANICS. (Semester II: AY ) Time Allowed: 2 Hours
NATONAL UNVERSTY OF SNGAPORE PC5 ADVANCED STATSTCAL MECHANCS (Semeser : AY 1-13) Tme Allowed: Hours NSTRUCTONS TO CANDDATES 1. Ths examnaon paper conans 5 quesons and comprses 4 prned pages.. Answer all
More informationLinear Response Theory: The connection between QFT and experiments
Phys540.nb 39 3 Lnear Response Theory: The connecon beween QFT and expermens 3.1. Basc conceps and deas Q: ow do we measure he conducvy of a meal? A: we frs nroduce a weak elecrc feld E, and hen measure
More informationDynamic Team Decision Theory. EECS 558 Project Shrutivandana Sharma and David Shuman December 10, 2005
Dynamc Team Decson Theory EECS 558 Proec Shruvandana Sharma and Davd Shuman December 0, 005 Oulne Inroducon o Team Decson Theory Decomposon of he Dynamc Team Decson Problem Equvalence of Sac and Dynamc
More informationLecture 18: The Laplace Transform (See Sections and 14.7 in Boas)
Lecure 8: The Lalace Transform (See Secons 88- and 47 n Boas) Recall ha our bg-cure goal s he analyss of he dfferenal equaon, ax bx cx F, where we emloy varous exansons for he drvng funcon F deendng on
More informationGENERATING CERTAIN QUINTIC IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS. Youngwoo Ahn and Kitae Kim
Korean J. Mah. 19 (2011), No. 3, pp. 263 272 GENERATING CERTAIN QUINTIC IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS Youngwoo Ahn and Kae Km Absrac. In he paper [1], an explc correspondence beween ceran
More informationP R = P 0. The system is shown on the next figure:
TPG460 Reservor Smulaon 08 page of INTRODUCTION TO RESERVOIR SIMULATION Analycal and numercal soluons of smple one-dmensonal, one-phase flow equaons As an nroducon o reservor smulaon, we wll revew he smples
More informationApproximate Analytic Solution of (2+1) - Dimensional Zakharov-Kuznetsov(Zk) Equations Using Homotopy
Arcle Inernaonal Journal of Modern Mahemacal Scences, 4, (): - Inernaonal Journal of Modern Mahemacal Scences Journal homepage: www.modernscenfcpress.com/journals/jmms.aspx ISSN: 66-86X Florda, USA Approxmae
More informationOn computing differential transform of nonlinear non-autonomous functions and its applications
On compung dfferenal ransform of nonlnear non-auonomous funcons and s applcaons Essam. R. El-Zahar, and Abdelhalm Ebad Deparmen of Mahemacs, Faculy of Scences and Humanes, Prnce Saam Bn Abdulazz Unversy,
More informationLecture 6: Learning for Control (Generalised Linear Regression)
Lecure 6: Learnng for Conrol (Generalsed Lnear Regresson) Conens: Lnear Mehods for Regresson Leas Squares, Gauss Markov heorem Recursve Leas Squares Lecure 6: RLSC - Prof. Sehu Vjayakumar Lnear Regresson
More informationComb Filters. Comb Filters
The smple flers dscussed so far are characered eher by a sngle passband and/or a sngle sopband There are applcaons where flers wh mulple passbands and sopbands are requred Thecomb fler s an example of
More informationRelative controllability of nonlinear systems with delays in control
Relave conrollably o nonlnear sysems wh delays n conrol Jerzy Klamka Insue o Conrol Engneerng, Slesan Techncal Unversy, 44- Glwce, Poland. phone/ax : 48 32 37227, {jklamka}@a.polsl.glwce.pl Keywor: Conrollably.
More informationGraduate Macroeconomics 2 Problem set 5. - Solutions
Graduae Macroeconomcs 2 Problem se. - Soluons Queson 1 To answer hs queson we need he frms frs order condons and he equaon ha deermnes he number of frms n equlbrum. The frms frs order condons are: F K
More informationLecture VI Regression
Lecure VI Regresson (Lnear Mehods for Regresson) Conens: Lnear Mehods for Regresson Leas Squares, Gauss Markov heorem Recursve Leas Squares Lecure VI: MLSC - Dr. Sehu Vjayakumar Lnear Regresson Model M
More informationIn the complete model, these slopes are ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL. (! i+1 -! i ) + [(!") i+1,q - [(!
ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL The frs hng o es n wo-way ANOVA: Is here neracon? "No neracon" means: The man effecs model would f. Ths n urn means: In he neracon plo (wh A on he horzonal
More informationFTCS Solution to the Heat Equation
FTCS Soluon o he Hea Equaon ME 448/548 Noes Gerald Reckenwald Porland Sae Unversy Deparmen of Mechancal Engneerng gerry@pdxedu ME 448/548: FTCS Soluon o he Hea Equaon Overvew Use he forward fne d erence
More informationAn introduction to Support Vector Machine
An nroducon o Suppor Vecor Machne 報告者 : 黃立德 References: Smon Haykn, "Neural Neworks: a comprehensve foundaon, second edon, 999, Chaper 2,6 Nello Chrsann, John Shawe-Tayer, An Inroducon o Suppor Vecor Machnes,
More informationPart II CONTINUOUS TIME STOCHASTIC PROCESSES
Par II CONTINUOUS TIME STOCHASTIC PROCESSES 4 Chaper 4 For an advanced analyss of he properes of he Wener process, see: Revus D and Yor M: Connuous marngales and Brownan Moon Karazas I and Shreve S E:
More informationFall 2010 Graduate Course on Dynamic Learning
Fall 200 Graduae Course on Dynamc Learnng Chaper 4: Parcle Flers Sepember 27, 200 Byoung-Tak Zhang School of Compuer Scence and Engneerng & Cognve Scence and Bran Scence Programs Seoul aonal Unversy hp://b.snu.ac.kr/~bzhang/
More informationThe Finite Element Method for the Analysis of Non-Linear and Dynamic Systems
Swss Federal Insue of Page 1 The Fne Elemen Mehod for he Analyss of Non-Lnear and Dynamc Sysems Prof. Dr. Mchael Havbro Faber Dr. Nebojsa Mojslovc Swss Federal Insue of ETH Zurch, Swzerland Mehod of Fne
More informationCS286.2 Lecture 14: Quantum de Finetti Theorems II
CS286.2 Lecure 14: Quanum de Fne Theorems II Scrbe: Mara Okounkova 1 Saemen of he heorem Recall he las saemen of he quanum de Fne heorem from he prevous lecure. Theorem 1 Quanum de Fne). Le ρ Dens C 2
More informationDEEP UNFOLDING FOR MULTICHANNEL SOURCE SEPARATION SUPPLEMENTARY MATERIAL
DEEP UNFOLDING FOR MULTICHANNEL SOURCE SEPARATION SUPPLEMENTARY MATERIAL Sco Wsdom, John Hershey 2, Jonahan Le Roux 2, and Shnj Waanabe 2 Deparmen o Elecrcal Engneerng, Unversy o Washngon, Seale, WA, USA
More informationLet s treat the problem of the response of a system to an applied external force. Again,
Page 33 QUANTUM LNEAR RESPONSE FUNCTON Le s rea he problem of he response of a sysem o an appled exernal force. Agan, H() H f () A H + V () Exernal agen acng on nernal varable Hamlonan for equlbrum sysem
More information. The geometric multiplicity is dim[ker( λi. number of linearly independent eigenvectors associated with this eigenvalue.
Lnear Algebra Lecure # Noes We connue wh he dscusson of egenvalues, egenvecors, and dagonalzably of marces We wan o know, n parcular wha condons wll assure ha a marx can be dagonalzed and wha he obsrucons
More informationHow about the more general "linear" scalar functions of scalars (i.e., a 1st degree polynomial of the following form with a constant term )?
lmcd Lnear ransformaon of a vecor he deas presened here are que general hey go beyond he radonal mar-vecor ype seen n lnear algebra Furhermore, hey do no deal wh bass and are equally vald for any se of
More information3. OVERVIEW OF NUMERICAL METHODS
3 OVERVIEW OF NUMERICAL METHODS 3 Inroducory remarks Ths chaper summarzes hose numercal echnques whose knowledge s ndspensable for he undersandng of he dfferen dscree elemen mehods: he Newon-Raphson-mehod,
More informationAdvanced Machine Learning & Perception
Advanced Machne Learnng & Percepon Insrucor: Tony Jebara SVM Feaure & Kernel Selecon SVM Eensons Feaure Selecon (Flerng and Wrappng) SVM Feaure Selecon SVM Kernel Selecon SVM Eensons Classfcaon Feaure/Kernel
More informationVariants of Pegasos. December 11, 2009
Inroducon Varans of Pegasos SooWoong Ryu bshboy@sanford.edu December, 009 Youngsoo Cho yc344@sanford.edu Developng a new SVM algorhm s ongong research opc. Among many exng SVM algorhms, we wll focus on
More informationMechanics Physics 151
Mechancs Physcs 5 Lecure 0 Canoncal Transformaons (Chaper 9) Wha We Dd Las Tme Hamlon s Prncple n he Hamlonan formalsm Dervaon was smple δi δ Addonal end-pon consrans pq H( q, p, ) d 0 δ q ( ) δq ( ) δ
More informationV.Abramov - FURTHER ANALYSIS OF CONFIDENCE INTERVALS FOR LARGE CLIENT/SERVER COMPUTER NETWORKS
R&RATA # Vol.) 8, March FURTHER AALYSIS OF COFIDECE ITERVALS FOR LARGE CLIET/SERVER COMPUTER ETWORKS Vyacheslav Abramov School of Mahemacal Scences, Monash Unversy, Buldng 8, Level 4, Clayon Campus, Wellngon
More informationOn elements with index of the form 2 a 3 b in a parametric family of biquadratic elds
On elemens wh ndex of he form a 3 b n a paramerc famly of bquadrac elds Bora JadrevĆ Absrac In hs paper we gve some resuls abou prmve negral elemens p(c p n he famly of bcyclc bquadrac elds L c = Q ) c;
More informationNotes on the stability of dynamic systems and the use of Eigen Values.
Noes on he sabl of dnamc ssems and he use of Egen Values. Source: Macro II course noes, Dr. Davd Bessler s Tme Seres course noes, zarads (999) Ineremporal Macroeconomcs chaper 4 & Techncal ppend, and Hamlon
More informationChapter Lagrangian Interpolation
Chaper 5.4 agrangan Inerpolaon Afer readng hs chaper you should be able o:. dere agrangan mehod of nerpolaon. sole problems usng agrangan mehod of nerpolaon and. use agrangan nerpolans o fnd deraes and
More information. The geometric multiplicity is dim[ker( λi. A )], i.e. the number of linearly independent eigenvectors associated with this eigenvalue.
Mah E-b Lecure #0 Noes We connue wh he dscusson of egenvalues, egenvecors, and dagonalzably of marces We wan o know, n parcular wha condons wll assure ha a marx can be dagonalzed and wha he obsrucons are
More informationJ i-1 i. J i i+1. Numerical integration of the diffusion equation (I) Finite difference method. Spatial Discretization. Internal nodes.
umercal negraon of he dffuson equaon (I) Fne dfference mehod. Spaal screaon. Inernal nodes. R L V For hermal conducon le s dscree he spaal doman no small fne spans, =,,: Balance of parcles for an nernal
More informationCHAPTER 10: LINEAR DISCRIMINATION
CHAPER : LINEAR DISCRIMINAION Dscrmnan-based Classfcaon 3 In classfcaon h K classes (C,C,, C k ) We defned dscrmnan funcon g j (), j=,,,k hen gven an es eample, e chose (predced) s class label as C f g
More information[ ] 2. [ ]3 + (Δx i + Δx i 1 ) / 2. Δx i-1 Δx i Δx i+1. TPG4160 Reservoir Simulation 2018 Lecture note 3. page 1 of 5
TPG460 Reservor Smulaon 08 page of 5 DISCRETIZATIO OF THE FOW EQUATIOS As we already have seen, fne dfference appromaons of he paral dervaves appearng n he flow equaons may be obaned from Taylor seres
More informationRobust and Accurate Cancer Classification with Gene Expression Profiling
Robus and Accurae Cancer Classfcaon wh Gene Expresson Proflng (Compuaonal ysems Bology, 2005) Auhor: Hafeng L, Keshu Zhang, ao Jang Oulne Background LDA (lnear dscrmnan analyss) and small sample sze problem
More informationPHYS 705: Classical Mechanics. Canonical Transformation
PHYS 705: Classcal Mechancs Canoncal Transformaon Canoncal Varables and Hamlonan Formalsm As we have seen, n he Hamlonan Formulaon of Mechancs,, are ndeenden varables n hase sace on eual foong The Hamlon
More informationMachine Learning Linear Regression
Machne Learnng Lnear Regresson Lesson 3 Lnear Regresson Bascs of Regresson Leas Squares esmaon Polynomal Regresson Bass funcons Regresson model Regularzed Regresson Sascal Regresson Mamum Lkelhood (ML)
More informationM. Y. Adamu Mathematical Sciences Programme, AbubakarTafawaBalewa University, Bauchi, Nigeria
IOSR Journal of Mahemacs (IOSR-JM e-issn: 78-578, p-issn: 9-765X. Volume 0, Issue 4 Ver. IV (Jul-Aug. 04, PP 40-44 Mulple SolonSoluons for a (+-dmensonalhroa-sasuma shallow waer wave equaon UsngPanlevé-Bӓclund
More informationPendulum Dynamics. = Ft tangential direction (2) radial direction (1)
Pendulum Dynams Consder a smple pendulum wh a massless arm of lengh L and a pon mass, m, a he end of he arm. Assumng ha he fron n he sysem s proporonal o he negave of he angenal veloy, Newon s seond law
More informationHandout # 6 (MEEN 617) Numerical Integration to Find Time Response of SDOF mechanical system Y X (2) and write EOM (1) as two first-order Eqs.
Handou # 6 (MEEN 67) Numercal Inegraon o Fnd Tme Response of SDOF mechancal sysem Sae Space Mehod The EOM for a lnear sysem s M X DX K X F() () X X X X V wh nal condons, a 0 0 ; 0 Defne he followng varables,
More informationVolatility Interpolation
Volaly Inerpolaon Prelmnary Verson March 00 Jesper Andreasen and Bran Huge Danse Mares, Copenhagen wan.daddy@danseban.com brno@danseban.com Elecronc copy avalable a: hp://ssrn.com/absrac=69497 Inro Local
More informationOrdinary Differential Equations in Neuroscience with Matlab examples. Aim 1- Gain understanding of how to set up and solve ODE s
Ordnary Dfferenal Equaons n Neuroscence wh Malab eamples. Am - Gan undersandng of how o se up and solve ODE s Am Undersand how o se up an solve a smple eample of he Hebb rule n D Our goal a end of class
More informationDensity Matrix Description of NMR BCMB/CHEM 8190
Densy Marx Descrpon of NMR BCMBCHEM 89 Operaors n Marx Noaon Alernae approach o second order specra: ask abou x magnezaon nsead of energes and ranson probables. If we say wh one bass se, properes vary
More informationA HIERARCHICAL KALMAN FILTER
A HIERARCHICAL KALMAN FILER Greg aylor aylor Fry Consulng Acuares Level 8, 3 Clarence Sree Sydney NSW Ausrala Professoral Assocae, Cenre for Acuaral Sudes Faculy of Economcs and Commerce Unversy of Melbourne
More informationTHE PREDICTION OF COMPETITIVE ENVIRONMENT IN BUSINESS
THE PREICTION OF COMPETITIVE ENVIRONMENT IN BUSINESS INTROUCTION The wo dmensonal paral dfferenal equaons of second order can be used for he smulaon of compeve envronmen n busness The arcle presens he
More informationChapter 2 Linear dynamic analysis of a structural system
Chaper Lnear dynamc analyss of a srucural sysem. Dynamc equlbrum he dynamc equlbrum analyss of a srucure s he mos general case ha can be suded as akes no accoun all he forces acng on. When he exernal loads
More informationChapter 6: AC Circuits
Chaper 6: AC Crcus Chaper 6: Oulne Phasors and he AC Seady Sae AC Crcus A sable, lnear crcu operang n he seady sae wh snusodal excaon (.e., snusodal seady sae. Complee response forced response naural response.
More informationOutline. Probabilistic Model Learning. Probabilistic Model Learning. Probabilistic Model for Time-series Data: Hidden Markov Model
Probablsc Model for Tme-seres Daa: Hdden Markov Model Hrosh Mamsuka Bonformacs Cener Kyoo Unversy Oulne Three Problems for probablsc models n machne learnng. Compung lkelhood 2. Learnng 3. Parsng (predcon
More informationClustering (Bishop ch 9)
Cluserng (Bshop ch 9) Reference: Daa Mnng by Margare Dunham (a slde source) 1 Cluserng Cluserng s unsupervsed learnng, here are no class labels Wan o fnd groups of smlar nsances Ofen use a dsance measure
More informationF-Tests and Analysis of Variance (ANOVA) in the Simple Linear Regression Model. 1. Introduction
ECOOMICS 35* -- OTE 9 ECO 35* -- OTE 9 F-Tess and Analyss of Varance (AOVA n he Smple Lnear Regresson Model Inroducon The smple lnear regresson model s gven by he followng populaon regresson equaon, or
More informationLecture 11 SVM cont
Lecure SVM con. 0 008 Wha we have done so far We have esalshed ha we wan o fnd a lnear decson oundary whose margn s he larges We know how o measure he margn of a lnear decson oundary Tha s: he mnmum geomerc
More information( ) [ ] MAP Decision Rule
Announcemens Bayes Decson Theory wh Normal Dsrbuons HW0 due oday HW o be assgned soon Proec descrpon posed Bomercs CSE 90 Lecure 4 CSE90, Sprng 04 CSE90, Sprng 04 Key Probables 4 ω class label X feaure
More informationAdvanced time-series analysis (University of Lund, Economic History Department)
Advanced me-seres analss (Unvers of Lund, Economc Hsor Dearmen) 3 Jan-3 Februar and 6-3 March Lecure 4 Economerc echnues for saonar seres : Unvarae sochasc models wh Box- Jenns mehodolog, smle forecasng
More informationOnline Appendix for. Strategic safety stocks in supply chains with evolving forecasts
Onlne Appendx for Sraegc safey socs n supply chans wh evolvng forecass Tor Schoenmeyr Sephen C. Graves Opsolar, Inc. 332 Hunwood Avenue Hayward, CA 94544 A. P. Sloan School of Managemen Massachuses Insue
More informationResponse of MDOF systems
Response of MDOF syses Degree of freedo DOF: he nu nuber of ndependen coordnaes requred o deerne copleely he posons of all pars of a syse a any nsan of e. wo DOF syses hree DOF syses he noral ode analyss
More informationNormal Random Variable and its discriminant functions
Noral Rando Varable and s dscrnan funcons Oulne Noral Rando Varable Properes Dscrnan funcons Why Noral Rando Varables? Analycally racable Works well when observaon coes for a corruped snle prooype 3 The
More informatione-journal Reliability: Theory& Applications No 2 (Vol.2) Vyacheslav Abramov
June 7 e-ournal Relably: Theory& Applcaons No (Vol. CONFIDENCE INTERVALS ASSOCIATED WITH PERFORMANCE ANALYSIS OF SYMMETRIC LARGE CLOSED CLIENT/SERVER COMPUTER NETWORKS Absrac Vyacheslav Abramov School
More information(,,, ) (,,, ). In addition, there are three other consumers, -2, -1, and 0. Consumer -2 has the utility function
MACROECONOMIC THEORY T J KEHOE ECON 87 SPRING 5 PROBLEM SET # Conder an overlappng generaon economy le ha n queon 5 on problem e n whch conumer lve for perod The uly funcon of he conumer born n perod,
More informationChapters 2 Kinematics. Position, Distance, Displacement
Chapers Knemacs Poson, Dsance, Dsplacemen Mechancs: Knemacs and Dynamcs. Knemacs deals wh moon, bu s no concerned wh he cause o moon. Dynamcs deals wh he relaonshp beween orce and moon. The word dsplacemen
More informationFI 3103 Quantum Physics
/9/4 FI 33 Quanum Physcs Aleander A. Iskandar Physcs of Magnesm and Phooncs Research Grou Insu Teknolog Bandung Basc Conces n Quanum Physcs Probably and Eecaon Value Hesenberg Uncerany Prncle Wave Funcon
More information10. A.C CIRCUITS. Theoretically current grows to maximum value after infinite time. But practically it grows to maximum after 5τ. Decay of current :
. A. IUITS Synopss : GOWTH OF UNT IN IUIT : d. When swch S s closed a =; = d. A me, curren = e 3. The consan / has dmensons of me and s called he nducve me consan ( τ ) of he crcu. 4. = τ; =.63, n one
More informationA NEW TECHNIQUE FOR SOLVING THE 1-D BURGERS EQUATION
S19 A NEW TECHNIQUE FOR SOLVING THE 1-D BURGERS EQUATION by Xaojun YANG a,b, Yugu YANG a*, Carlo CATTANI c, and Mngzheng ZHU b a Sae Key Laboraory for Geomechancs and Deep Underground Engneerng, Chna Unversy
More informationOn complete system of invariants for the binary form of degree 7
Journal of Symbolc Compuaon 42 (2007 935 947 www.elsever.com/locae/jsc On complee sysem of nvarans for he bnary form of degree 7 Leond Bedrayuk Khmelnysky Naonal Unversy, Appled Mahemacs, Insyus ka s.,
More informationMethod of upper lower solutions for nonlinear system of fractional differential equations and applications
Malaya Journal of Maemak, Vol. 6, No. 3, 467-472, 218 hps://do.org/1.26637/mjm63/1 Mehod of upper lower soluons for nonlnear sysem of fraconal dfferenal equaons and applcaons D.B. Dhagude1 *, N.B. Jadhav2
More informationBorn Oppenheimer Approximation and Beyond
L Born Oppenhemer Approxmaon and Beyond aro Barba A*dex Char Professor maro.barba@unv amu.fr Ax arselle Unversé, nsu de Chme Radcalare LGHT AD Adabac x dabac x nonadabac LGHT AD From Gree dabaos: o be
More informationMotion in Two Dimensions
Phys 1 Chaper 4 Moon n Two Dmensons adzyubenko@csub.edu hp://www.csub.edu/~adzyubenko 005, 014 A. Dzyubenko 004 Brooks/Cole 1 Dsplacemen as a Vecor The poson of an objec s descrbed by s poson ecor, r The
More information2/20/2013. EE 101 Midterm 2 Review
//3 EE Mderm eew //3 Volage-mplfer Model The npu ressance s he equalen ressance see when lookng no he npu ermnals of he amplfer. o s he oupu ressance. I causes he oupu olage o decrease as he load ressance
More informationAppendix H: Rarefaction and extrapolation of Hill numbers for incidence data
Anne Chao Ncholas J Goell C seh lzabeh L ander K Ma Rober K Colwell and Aaron M llson 03 Rarefacon and erapolaon wh ll numbers: a framewor for samplng and esmaon n speces dversy sudes cology Monographs
More informationSurvival Analysis and Reliability. A Note on the Mean Residual Life Function of a Parallel System
Communcaons n Sascs Theory and Mehods, 34: 475 484, 2005 Copyrgh Taylor & Francs, Inc. ISSN: 0361-0926 prn/1532-415x onlne DOI: 10.1081/STA-200047430 Survval Analyss and Relably A Noe on he Mean Resdual
More informationDensity Matrix Description of NMR BCMB/CHEM 8190
Densy Marx Descrpon of NMR BCMBCHEM 89 Operaors n Marx Noaon If we say wh one bass se, properes vary only because of changes n he coeffcens weghng each bass se funcon x = h< Ix > - hs s how we calculae
More informationMath 128b Project. Jude Yuen
Mah 8b Proec Jude Yuen . Inroducon Le { Z } be a sequence of observed ndependen vecor varables. If he elemens of Z have a on normal dsrbuon hen { Z } has a mean vecor Z and a varancecovarance marx z. Geomercally
More informationON THE WEAK LIMITS OF SMOOTH MAPS FOR THE DIRICHLET ENERGY BETWEEN MANIFOLDS
ON THE WEA LIMITS OF SMOOTH MAPS FOR THE DIRICHLET ENERGY BETWEEN MANIFOLDS FENGBO HANG Absrac. We denfy all he weak sequenal lms of smooh maps n W (M N). In parcular, hs mples a necessary su cen opologcal
More informationDual Approximate Dynamic Programming for Large Scale Hydro Valleys
Dual Approxmae Dynamc Programmng for Large Scale Hydro Valleys Perre Carpener and Jean-Phlppe Chanceler 1 ENSTA ParsTech and ENPC ParsTech CMM Workshop, January 2016 1 Jon work wh J.-C. Alas, suppored
More informationA Simulation Based Optimal Control System For Water Resources
Cy Unversy of New York (CUNY) CUNY Academc Works Inernaonal Conference on Hydronformacs 8--4 A Smulaon Based Opmal Conrol Sysem For Waer Resources Aser acasa Maro Morales-Hernández Plar Brufau Plar García-Navarro
More informationCubic Bezier Homotopy Function for Solving Exponential Equations
Penerb Journal of Advanced Research n Compung and Applcaons ISSN (onlne: 46-97 Vol. 4, No.. Pages -8, 6 omoopy Funcon for Solvng Eponenal Equaons S. S. Raml *,,. Mohamad Nor,a, N. S. Saharzan,b and M.
More informationPHYS 1443 Section 001 Lecture #4
PHYS 1443 Secon 001 Lecure #4 Monda, June 5, 006 Moon n Two Dmensons Moon under consan acceleraon Projecle Moon Mamum ranges and heghs Reerence Frames and relae moon Newon s Laws o Moon Force Newon s Law
More informationChapter 6 DETECTION AND ESTIMATION: Model of digital communication system. Fundamental issues in digital communications are
Chaper 6 DEECIO AD EIMAIO: Fundamenal ssues n dgal communcaons are. Deecon and. Esmaon Deecon heory: I deals wh he desgn and evaluaon of decson makng processor ha observes he receved sgnal and guesses
More informationAdvanced Macroeconomics II: Exchange economy
Advanced Macroeconomcs II: Exchange economy Krzyszof Makarsk 1 Smple deermnsc dynamc model. 1.1 Inroducon Inroducon Smple deermnsc dynamc model. Defnons of equlbrum: Arrow-Debreu Sequenal Recursve Equvalence
More informationRobustness Experiments with Two Variance Components
Naonal Insue of Sandards and Technology (NIST) Informaon Technology Laboraory (ITL) Sascal Engneerng Dvson (SED) Robusness Expermens wh Two Varance Componens by Ana Ivelsse Avlés avles@ns.gov Conference
More informationComputer Robot Vision Conference 2010
School of Compuer Scence McGll Unversy Compuer Robo Vson Conference 2010 Ioanns Rekles Fundamenal Problems In Robocs How o Go From A o B? (Pah Plannng) Wha does he world looks lke? (mappng) sense from
More informationTechnical Appendix to The Equivalence of Wage and Price Staggering in Monetary Business Cycle Models
Techncal Appendx o The Equvalence of Wage and Prce Saggerng n Moneary Busness Cycle Models Rochelle M. Edge Dvson of Research and Sascs Federal Reserve Board Sepember 24, 2 Absrac Ths appendx deals he
More informationMachine Learning 2nd Edition
INTRODUCTION TO Lecure Sldes for Machne Learnng nd Edon ETHEM ALPAYDIN, modfed by Leonardo Bobadlla and some pars from hp://www.cs.au.ac.l/~aparzn/machnelearnng/ The MIT Press, 00 alpaydn@boun.edu.r hp://www.cmpe.boun.edu.r/~ehem/mle
More informationExistence and Uniqueness Results for Random Impulsive Integro-Differential Equation
Global Journal of Pure and Appled Mahemacs. ISSN 973-768 Volume 4, Number 6 (8), pp. 89-87 Research Inda Publcaons hp://www.rpublcaon.com Exsence and Unqueness Resuls for Random Impulsve Inegro-Dfferenal
More informationBayes rule for a classification problem INF Discriminant functions for the normal density. Euclidean distance. Mahalanobis distance
INF 43 3.. Repeon Anne Solberg (anne@f.uo.no Bayes rule for a classfcaon problem Suppose we have J, =,...J classes. s he class label for a pxel, and x s he observed feaure vecor. We can use Bayes rule
More information