Modeling and Predicting Sequences: HMM and (may be) CRF. Amr Ahmed Feb 25

Size: px
Start display at page:

Download "Modeling and Predicting Sequences: HMM and (may be) CRF. Amr Ahmed Feb 25"

Transcription

1 Modelg d redcg Sequeces: HMM d m be CRF Amr Ahmed 070 Feb 25

2 Bg cure redcg Sgle Lbel Ipu : A se of feures: - Bg of words docume - Oupu : Clss lbel - Topc of he docume - redcg Sequece of Lbels Noo Noe: I use orml fce leers for sclr s d b fce leers for vecors le d Ipu : A se of feures wh order/srucure mog hem Sequece of words seece Oupu r of speech OS g of ech word

3 redcg Sequeces Emple: OS NN VBD DD NN JJ Sudes foud he HW es Emple N chug : B I E O B E O B E X: Rocwell Ierol Corp sged greeme wh Boeg Co.

4 Bc To bg cure Sgle Oupu Geerve: Models redc usg Bes rule rgm Nïve Bes Dscrmve: Model redc usg rgm Logsc Regresso

5 Bc To bg cure Sequece of Oupu Geerve: Models redc usg Bes rule rgm HMM Dscrmve: Model redc usg rgm CRF

6 HMM Defes geerve model over Ech hs M opos d ech hs opos You eed bg ble of sze M X Y We eed o dd some codol depedece ssumpo o me hgs mgeble We hve doe h Nïve Bes 2 3 T 2 3 T

7 HMM 2 3 T Wh we eed o defe #pr. shorhd Il se: - p Trso: - *- + Emsso: *M- b Fcorzo: 2 3 T ; : : : ; T ; ; : : : ; T p p 2 : : : 2 T T ; : : : ; T ; ; : : : ; T Q

8 Tss Iferece Fd MA: Veerb: X Lerg Lerg model prmeers usg MLE p b Full Observed:» cou d ormlze Usupervsed:» EM

9 Fd Iferece: MA rgm We eed o compue frs p ; : : : ; T p ; ; : : : ; T p ; : : : ; T p ; ; : : : ; p + ; : : : ; T ; ; : : : ; p ; : : : ; T p ; ; : : : ; p + ; : : : ; T p ; : : : ; T p ; : : : ; T 2 3 T 2 3 T

10 A rc h we wll use ofe: dd vrble d mrglze over o be ble o ppl recurso MA We eed o do h for. ; 2 ; :::; T A... A Defe recursve progrm p ; ; : : : ; p ; ; : : : ; A - Dvde vrble o hree ses: {X } o be ble o see - { }{ } he ppl ch rule Summg over - s us summg Over -...

11 p 2 Forwrd Algorhm 2 2 T T p 2 T 2 T

12 Fd Iferece: MA rgm We eed o compue frs p ; : : : ; T p ; ; : : : ; T p ; : : : ; T p ; ; : : : ; p + ; : : : ; T ; ; : : : ; p ; : : : ; T p ; ; : : : ; p + ; : : : ; T p ; : : : ; T p ; : : : ; T 2 3 T 2 3 T

13 Bcwrd Algorhm We eed o do h for. dd d mrglze rc ; 2; : : : ; T Defe recursve progrm p + ; : : : ; T + p +2 ; : : : ; T + T T p 2... T p 2... T p A + A T A Dvde vrble o hree ses: { + } { + }{X +2.. T } o be ble o see + he ppl ch rule T

14 Bcwrd Algorhm p T T T T T p T T T T T 2 T 2 T

15 Fd Iferece: MA rgm We eed o compue frs p ; : : : ; T p ; ; : : : ; T p ; : : : ; T p ; ; : : : ; p + ; : : : ; T ; ; : : : ; p ; : : : ; T p ; ; : : : ; p + ; : : : ; T p ; : : : ; T p ; : : : ; T 2 3 T 2 3 T

16 Evluo ; : : : ; T X T ; : : : ; T ; T X ; : : : ; T ; T X T Now we hve everhg o compue: p ; : : : ; T p ; : : : ; T

17 rccl Cosdero re produc of m erms Lel o ru d ou wll o uderflow for sequece > 0 C we use s? I geerl we dd ge o he rgh hd sde bu ou c use echque clled dd h I dd dscuss he reco. Soluo: resclg --- ormlze fer ech sep!

18 Sclg Normlze fer ech sep! c s ormlzo cos eep rc of c for ll c ˆ

19 Sclg: Ierpreo How o erpre c d he ormlzed Clm: remember roof b duco: ssume s rue for - c ˆ c c c c c c ' ' ' ' ' ' ˆ B defo of c Subs. - from hpohess

20 Sclg: Compuo C we sll clcule.. T Yes! Bu ou rell eed o do spce: T T T T T T c c... ˆ T T c...

21 Sclg: Bcwrd You c use he sme rc wh Now how o compue MA The fll ormlze Noe h he cos he h verso of boh d s ol fuco of sme for ll

22 Tss Iferece Fd MA: Veerb: X Lerg Lerg model prmeers usg MLE p b Full Observed:» cou d ormlze Usupervsed:» EM

23 Verb Fd he globll mml poseror sequece rgm..t.. T T Sme s rgm..t.. T T wh? Develop dmc recursve progrm m..-.. d rele o m We cll hs qu V whch s vecor: V m... } { - I mes he mml prob of edg se me where we re mmzg over.. -

24 Verb: he mh You should be bored of h b ow? No hs s dffere rc pushg m V m {... } m {... } m... } Also eep rc of he mmzg { m m {... } m m V V

25 Verb Algorhm m V V 2 V 2 2 m V T V T V T V T z V V m V V T V T z rg m T T zvt 2 T 2 T

26 V Verb Algorhm V 2 2 m V 2 V T V T m 4V T V T2 4 V T z V V 2 V T V T 2 2 m V z rg m T T z zvt 2 T 2 T

27 You c use here V m V Verb: sclg V V p m

28 Tss Iferece Fd MA: Veerb: X Lerg Lerg model prmeers usg MLE p b Full Observed:» cou d ormlze Usupervsed:» EM

29 Lerg For full observed dd { } :N LL s -l For prll observed d mssg Y Usg Jese LL θ LL θ p X Y p θ p θ Q θ θ p θ p θ

30 Lerg: Observed For gve sequece C : umber of mes frs se ws 0 or B o : umber of mes se ems o A : umber of me se moves o se M o o o M o o B o A C T T b B A C b p p p p LL o 2 p p Y X θ O he bord see e pge

31 dgresso Te The Whch p * 2 * 23 * 32 * 2 *b *b 23 *b 35 *b *b 2 p * 2 2 * 23 * 3 *b 2 *b 23 *b 35 *b 2 -Noe h f he cou of em C or A or B s zero he smpl he erm h volves wll be. T T p p p p 2 M o o B o A C o b p

32 Lerg: observed M LL θ C p A Bo o b o Noe h ll prmeers re decoupled Te grde d solve for ever oe seprel Smpl cou d ormlze for emple: ML # # A A ' '

33 Lerg: usupervsed Recll LL θ p θ Q θ θ p θ p θ p M C p A Bo o b o So we hve: Q θ θ θ C p A M Bo o b o

34 All epecos re uder We ow how o compue h F-B C p C θ θ θ M o o o b B A C Q p θ θ

35 Q θ θ Where C p A M Bo o b o B o : p o θ θ B o We lso ow how o compue h F-B

36 Q θ θ Where C p A M Bo o b o A : T p θ θ A Do we ow how o compue h? Sor of

37 Recll Bu: Whch we ow how o compue : T A p A θ θ θ θ θ θ You should be ble o rove he bove sep

38 M-Sep Now we hve ll wh we eed Q θ θ C p A M Bo o b o Jus s before solve for MLE for e: ML # # A A ' '

39 EM Summr for HMM Ilze HMM model prmeers Repe E-Sep Ru forwrd-bcwrd over ever sequece Compue ecessr epecos usg d or her ormlzed versos MSep Re-esme model prmeers Smpl cou d ormlze

40 Fl Noe bou Resclg Recll Remember he uderflow soluo Sme hg here compue usg ormlzed vecors d he fll ormlze θ θ

Interval Estimation. Consider a random variable X with a mean of X. Let X be distributed as X X

Interval Estimation. Consider a random variable X with a mean of X. Let X be distributed as X X ECON 37: Ecoomercs Hypohess Tesg Iervl Esmo Wh we hve doe so fr s o udersd how we c ob esmors of ecoomcs reloshp we wsh o sudy. The queso s how comforble re we wh our esmors? We frs exme how o produce

More information

Decompression diagram sampler_src (source files and makefiles) bin (binary files) --- sh (sample shells) --- input (sample input files)

Decompression diagram sampler_src (source files and makefiles) bin (binary files) --- sh (sample shells) --- input (sample input files) . Iroduco Probblsc oe-moh forecs gudce s mde b 50 esemble members mproved b Model Oupu scs (MO). scl equo s mde b usg hdcs d d observo d. We selec some prmeers for modfg forecs o use mulple regresso formul.

More information

Unscented Transformation Unscented Kalman Filter

Unscented Transformation Unscented Kalman Filter Usceed rsformo Usceed Klm Fler Usceed rcle Fler Flerg roblem Geerl roblem Seme where s he se d s he observo Flerg s he problem of sequell esmg he ses (prmeers or hdde vrbles) of ssem s se of observos become

More information

Introduction to Neural Networks Computing. CMSC491N/691N, Spring 2001

Introduction to Neural Networks Computing. CMSC491N/691N, Spring 2001 Iroduco o Neurl Neorks Compug CMSC49N/69N, Sprg 00 us: cvo/oupu: f eghs: X, Y j X Noos, j s pu u, for oher us, j pu sgl here f. s he cvo fuco for j from u o u j oher books use Y f _ j j j Y j X j Y j bs:

More information

Laplace Transform. Definition of Laplace Transform: f(t) that satisfies The Laplace transform of f(t) is defined as.

Laplace Transform. Definition of Laplace Transform: f(t) that satisfies The Laplace transform of f(t) is defined as. Lplce Trfor The Lplce Trfor oe of he hecl ool for olvg ordry ler dfferel equo. - The hoogeeou equo d he prculr Iegrl re olved oe opero. - The Lplce rfor cover he ODE o lgerc eq. σ j ple do. I he pole o

More information

The ray paths and travel times for multiple layers can be computed using ray-tracing, as demonstrated in Lab 3.

The ray paths and travel times for multiple layers can be computed using ray-tracing, as demonstrated in Lab 3. C. Trael me cures for mulple reflecors The ray pahs ad rael mes for mulple layers ca be compued usg ray-racg, as demosraed Lab. MATLAB scrp reflec_layers_.m performs smple ray racg. (m) ref(ms) ref(ms)

More information

Least Squares Fitting (LSQF) with a complicated function Theexampleswehavelookedatsofarhavebeenlinearintheparameters

Least Squares Fitting (LSQF) with a complicated function Theexampleswehavelookedatsofarhavebeenlinearintheparameters Leas Squares Fg LSQF wh a complcaed fuco Theeampleswehavelookedasofarhavebeelearheparameers ha we have bee rg o deerme e.g. slope, ercep. For he case where he fuco s lear he parameers we ca fd a aalc soluo

More information

Week 8 Lecture 3: Problems 49, 50 Fourier analysis Courseware pp (don t look at French very confusing look in the Courseware instead)

Week 8 Lecture 3: Problems 49, 50 Fourier analysis Courseware pp (don t look at French very confusing look in the Courseware instead) Week 8 Lecure 3: Problems 49, 5 Fourier lysis Coursewre pp 6-7 (do look Frech very cofusig look i he Coursewre ised) Fourier lysis ivolves ddig wves d heir hrmoics, so i would hve urlly followed fer he

More information

Integral Equations and their Relationship to Differential Equations with Initial Conditions

Integral Equations and their Relationship to Differential Equations with Initial Conditions Scece Refleco SR Vol 6 wwwscecereflecocom Geerl Leers Mhemcs GLM 6 3-3 Geerl Leers Mhemcs GLM Wese: hp://wwwscecereflecocom/geerl-leers--mhemcs/ Geerl Leers Mhemcs Scece Refleco Iegrl Equos d her Reloshp

More information

Chapter Simpson s 1/3 Rule of Integration. ( x)

Chapter Simpson s 1/3 Rule of Integration. ( x) Cper 7. Smpso s / Rule o Iegro Aer redg s per, you sould e le o. derve e ormul or Smpso s / rule o egro,. use Smpso s / rule o solve egrls,. develop e ormul or mulple-segme Smpso s / rule o egro,. use

More information

EE 6885 Statistical Pattern Recognition

EE 6885 Statistical Pattern Recognition EE 6885 Sascal Paer Recogo Fall 005 Prof. Shh-Fu Chag hp://www.ee.columba.edu/~sfchag Lecure 5 (9//05 4- Readg Model Parameer Esmao ML Esmao, Chap. 3. Mure of Gaussa ad EM Referece Boo, HTF Chap. 8.5 Teboo,

More information

Chapter Unary Matrix Operations

Chapter Unary Matrix Operations Chpter 04.04 Ury trx Opertos After redg ths chpter, you should be ble to:. kow wht ury opertos mes,. fd the trspose of squre mtrx d t s reltoshp to symmetrc mtrces,. fd the trce of mtrx, d 4. fd the ermt

More information

Available online through

Available online through Avlble ole through wwwmfo FIXED POINTS FOR NON-SELF MAPPINGS ON CONEX ECTOR METRIC SPACES Susht Kumr Moht* Deprtmet of Mthemtcs West Begl Stte Uverst Brst 4 PrgsNorth) Kolt 76 West Begl Id E-ml: smwbes@yhoo

More information

STOCHASTIC CALCULUS I STOCHASTIC DIFFERENTIAL EQUATION

STOCHASTIC CALCULUS I STOCHASTIC DIFFERENTIAL EQUATION The Bk of Thld Fcl Isuos Polcy Group Que Models & Fcl Egeerg Tem Fcl Mhemcs Foudo Noe 8 STOCHASTIC CALCULUS I STOCHASTIC DIFFERENTIAL EQUATION. ก Through he use of ordry d/or prl deres, ODE/PDE c rele

More information

ME 501A Seminar in Engineering Analysis Page 1

ME 501A Seminar in Engineering Analysis Page 1 Mtr Trsformtos usg Egevectors September 8, Mtr Trsformtos Usg Egevectors Lrry Cretto Mechcl Egeerg A Semr Egeerg Alyss September 8, Outle Revew lst lecture Trsformtos wth mtr of egevectors: = - A ermt

More information

CS344: Introduction to Artificial Intelligence

CS344: Introduction to Artificial Intelligence C344: Iroduco o Arfcal Iellgece Puhpa Bhaacharyya CE Dep. IIT Bombay Lecure 3 3 32 33: Forward ad bacward; Baum elch 9 h ad 2 March ad 2 d Aprl 203 Lecure 27 28 29 were o EM; dae 2 h March o 8 h March

More information

Lecture 3 summary. C4 Lecture 3 - Jim Libby 1

Lecture 3 summary. C4 Lecture 3 - Jim Libby 1 Lecue su Fes of efeece Ivce ude sfoos oo of H wve fuco: d-fucos Eple: e e - µ µ - Agul oeu s oo geeo Eule gles Geec slos cosevo lws d Noehe s heoe C4 Lecue - Lbb Fes of efeece Cosde fe of efeece O whch

More information

4.8 Improper Integrals

4.8 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 information

Cyclone. Anti-cyclone

Cyclone. Anti-cyclone Adveco Cycloe A-cycloe Lorez (963) Low dmesoal aracors. Uclear f hey are a good aalogy o he rue clmae sysem, bu hey have some appealg characerscs. Dscusso Is he al codo balaced? Is here a al adjusme

More information

(1) Cov(, ) E[( E( ))( E( ))]

(1) Cov(, ) E[( E( ))( E( ))] Impac of Auocorrelao o OLS Esmaes ECON 3033/Evas Cosder a smple bvarae me-seres model of he form: y 0 x The four key assumpos abou ε hs model are ) E(ε ) = E[ε x ]=0 ) Var(ε ) =Var(ε x ) = ) Cov(ε, ε )

More information

Solution set Stat 471/Spring 06. Homework 2

Solution set Stat 471/Spring 06. Homework 2 oluo se a 47/prg 06 Homework a Whe he upper ragular elemes are suppressed due o smmer b Le Y Y Y Y A weep o he frs colum o oba: A ˆ b chagg he oao eg ad ec YY weep o he secod colum o oba: Aˆ YY weep o

More information

Graphing Review Part 3: Polynomials

Graphing Review Part 3: Polynomials Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)

More information

Average & instantaneous velocity and acceleration Motion with constant acceleration

Average & 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 information

Chapter Linear Regression

Chapter Linear Regression Chpte 6.3 Le Regesso Afte edg ths chpte, ou should be ble to. defe egesso,. use sevel mmzg of esdul cte to choose the ght cteo, 3. deve the costts of le egesso model bsed o lest sques method cteo,. use

More information

Density estimation III. Linear regression.

Density estimation III. Linear regression. Lecure 6 Mlos Hauskrec mlos@cs.p.eu 539 Seo Square Des esmao III. Lear regresso. Daa: Des esmao D { D D.. D} D a vecor of arbue values Obecve: r o esmae e uerlg rue probabl srbuo over varables X px usg

More information

INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY

INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY [Mjuh, : Jury, 0] ISSN: -96 Scefc Jourl Impc Fcr: 9 ISRA, Impc Fcr: IJESRT INTERNATIONAL JOURNAL OF ENINEERIN SCIENCES & RESEARCH TECHNOLOY HAMILTONIAN LACEABILITY IN MIDDLE RAPHS Mjuh*, MurlR, B Shmukh

More information

Density estimation III.

Density estimation III. Lecure 4 esy esmao III. Mlos Hauskrec mlos@cs..edu 539 Seo Square Oule Oule: esy esmao: Mamum lkelood ML Bayesa arameer esmaes MP Beroull dsrbuo. Bomal dsrbuo Mulomal dsrbuo Normal dsrbuo Eoeal famly Eoeal

More information

Chapter 2 Intro to Math Techniques for Quantum Mechanics

Chapter 2 Intro to Math Techniques for Quantum Mechanics Wter 3 Chem 356: Itroductory Qutum Mechcs Chpter Itro to Mth Techques for Qutum Mechcs... Itro to dfferetl equtos... Boudry Codtos... 5 Prtl dfferetl equtos d seprto of vrbles... 5 Itroducto to Sttstcs...

More information

Chapter 2: Evaluative Feedback

Chapter 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 information

Density estimation. Density estimations. CS 2750 Machine Learning. Lecture 5. Milos Hauskrecht 5329 Sennott Square

Density estimation. Density estimations. CS 2750 Machine Learning. Lecture 5. Milos Hauskrecht 5329 Sennott Square Lecure 5 esy esmao Mlos Hauskrec mlos@cs..edu 539 Seo Square esy esmaos ocs: esy esmao: Mamum lkelood ML Bayesa arameer esmaes M Beroull dsrbuo. Bomal dsrbuo Mulomal dsrbuo Normal dsrbuo Eoeal famly Noaramerc

More information

PHYSICS 1210 Exam 1 University of Wyoming 14 February points

PHYSICS 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 information

Chapter Trapezoidal Rule of Integration

Chapter Trapezoidal Rule of Integration Cper 7 Trpezodl Rule o Iegro Aer redg s per, you sould e le o: derve e rpezodl rule o egro, use e rpezodl rule o egro o solve prolems, derve e mulple-segme rpezodl rule o egro, 4 use e mulple-segme rpezodl

More information

The z-transform. LTI System description. Prof. Siripong Potisuk

The z-transform. LTI System description. Prof. Siripong Potisuk The -Trsform Prof. Srpog Potsuk LTI System descrpto Prevous bss fucto: ut smple or DT mpulse The put sequece s represeted s ler combto of shfted DT mpulses. The respose s gve by covoluto sum of the put

More information

Physics 101 Lecture 4 Motion in 2D and 3D

Physics 101 Lecture 4 Motion in 2D and 3D Phsics 11 Lecure 4 Moion in D nd 3D Dr. Ali ÖVGÜN EMU Phsics Deprmen www.ogun.com Vecor nd is componens The componens re he legs of he righ ringle whose hpoenuse is A A A A A n ( θ ) A Acos( θ) A A A nd

More information

Continuous Time Markov Chains

Continuous Time Markov Chains Couous me Markov chas have seay sae probably soluos f a oly f hey are ergoc, us lke scree me Markov chas. Fg he seay sae probably vecor for a couous me Markov cha s o more ffcul ha s he scree me case,

More information

Taylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best

Taylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =

More information

The Poisson Process Properties of the Poisson Process

The Poisson Process Properties of the Poisson Process Posso Processes Summary The Posso Process Properes of he Posso Process Ierarrval mes Memoryless propery ad he resdual lfeme paradox Superposo of Posso processes Radom seleco of Posso Pos Bulk Arrvals ad

More information

Sequences and summations

Sequences and summations Lecture 0 Sequeces d summtos Istructor: Kgl Km CSE) E-ml: kkm0@kokuk.c.kr Tel. : 0-0-9 Room : New Mleum Bldg. 0 Lb : New Egeerg Bldg. 0 All sldes re bsed o CS Dscrete Mthemtcs for Computer Scece course

More information

4. Runge-Kutta Formula For Differential Equations. A. Euler Formula B. Runge-Kutta Formula C. An Example for Fourth-Order Runge-Kutta Formula

4. Runge-Kutta Formula For Differential Equations. A. Euler Formula B. Runge-Kutta Formula C. An Example for Fourth-Order Runge-Kutta Formula NCTU Deprme o Elecrcl d Compuer Egeerg Seor Course By Pro. Yo-Pg Ce. Ruge-Ku Formul For Derel Equos A. Euler Formul B. Ruge-Ku Formul C. A Emple or Four-Order Ruge-Ku Formul

More information

Physics 2A HW #3 Solutions

Physics 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 information

Algebra Of Matrices & Determinants

Algebra Of Matrices & Determinants lgebr Of Mtrices & Determinnts Importnt erms Definitions & Formule 0 Mtrix - bsic introduction: mtrix hving m rows nd n columns is clled mtrix of order m n (red s m b n mtrix) nd mtrix of order lso in

More information

Physic 231 Lecture 4. Mi it ftd l t. Main points of today s lecture: Example: addition of velocities Trajectories of objects in 2 = =

Physic 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 information

Motion. Part 2: Constant Acceleration. Acceleration. October Lab Physics. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.

Motion. 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 information

Maximum likelihood estimate of phylogeny. BIOL 495S/ CS 490B/ MATH 490B/ STAT 490B Introduction to Bioinformatics April 24, 2002

Maximum likelihood estimate of phylogeny. BIOL 495S/ CS 490B/ MATH 490B/ STAT 490B Introduction to Bioinformatics April 24, 2002 Mmm lkelhood eme of phylogey BIO 9S/ S 90B/ MH 90B/ S 90B Iodco o Bofomc pl 00 Ovevew of he pobblc ppoch o phylogey o k ee ccodg o he lkelhood d ee whee d e e of eqece d ee by ee wh leve fo he eqece. he

More information

Introduction to mathematical Statistics

Introduction to mathematical Statistics Itroducto to mthemtcl ttstcs Fl oluto. A grou of bbes ll of whom weghed romtely the sme t brth re rdomly dvded to two grous. The bbes smle were fed formul A; those smle were fed formul B. The weght gs

More information

4. Runge-Kutta Formula For Differential Equations

4. Runge-Kutta Formula For Differential Equations NCTU Deprme o Elecrcl d Compuer Egeerg 5 Sprg Course by Pro. Yo-Pg Ce. Ruge-Ku Formul For Derel Equos To solve e derel equos umerclly e mos useul ormul s clled Ruge-Ku ormul

More information

Stat 6863-Handout 5 Fundamentals of Interest July 2010, Maurice A. Geraghty

Stat 6863-Handout 5 Fundamentals of Interest July 2010, Maurice A. Geraghty S 6863-Hou 5 Fuels of Ieres July 00, Murce A. Gerghy The pror hous resse beef cl occurreces, ous, ol cls e-ulero s ro rbles. The fl copoe of he curl oel oles he ecooc ssupos such s re of reur o sses flo.

More information

PubH 7405: REGRESSION ANALYSIS REGRESSION IN MATRIX TERMS

PubH 7405: REGRESSION ANALYSIS REGRESSION IN MATRIX TERMS PubH 745: REGRESSION ANALSIS REGRESSION IN MATRIX TERMS A mtr s dspl of umbers or umercl quttes ld out rectgulr rr of rows d colums. The rr, or two-w tble of umbers, could be rectgulr or squre could be

More information

Solution. The straightforward approach is surprisingly difficult because one has to be careful about the limits.

Solution. The straightforward approach is surprisingly difficult because one has to be careful about the limits. ose ad Varably Homewor # (8), aswers Q: Power spera of some smple oses A Posso ose A Posso ose () s a sequee of dela-fuo pulses, eah ourrg depedely, a some rae r (More formally, s a sum of pulses of wdh

More information

Generalisation on the Zeros of a Family of Complex Polynomials

Generalisation on the Zeros of a Family of Complex Polynomials Ieol Joul of hemcs esech. ISSN 976-584 Volume 6 Numbe 4. 93-97 Ieol esech Publco House h://www.house.com Geelso o he Zeos of Fmly of Comlex Polyomls Aee sgh Neh d S.K.Shu Deme of hemcs Lgys Uvesy Fdbd-

More information

Preliminary Examinations: Upper V Mathematics Paper 1

Preliminary Examinations: Upper V Mathematics Paper 1 relmr Emtos: Upper V Mthemtcs per Jul 03 Emer: G Evs Tme: 3 hrs Modertor: D Grgortos Mrks: 50 INSTRUCTIONS ND INFORMTION Ths questo pper sts of 0 pges, cludg swer Sheet pge 8 d Iformto Sheet pges 9 d 0

More information

BEST PATTERN OF MULTIPLE LINEAR REGRESSION

BEST PATTERN OF MULTIPLE LINEAR REGRESSION ERI COADA GERMAY GEERAL M.R. SEFAIK AIR FORCE ACADEMY ARMED FORCES ACADEMY ROMAIA SLOVAK REPUBLIC IERAIOAL COFERECE of SCIEIFIC PAPER AFASES Brov 6-8 M BES PAER OF MULIPLE LIEAR REGRESSIO Corel GABER PEROLEUM-GAS

More information

Three Main Questions on HMMs

Three Main Questions on HMMs Mache Learg 0-70/5-78 78 Srg 00 Hdde Marov Model II Erc Xg Lecure Februar 4 00 Readg: Cha. 3 CB Three Ma Quesos o HMMs. Evaluao GIVEN a HMM M ad a sequece FIND Prob M ALGO. Forward. Decodg GIVEN a HMM

More information

A 1.3 m 2.5 m 2.8 m. x = m m = 8400 m. y = 4900 m 3200 m = 1700 m

A 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 information

Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]

Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ] Are d the Defte Itegrl 1 Are uder Curve We wt to fd the re uder f (x) o [, ] y f (x) x The Prtto We eg y prttog the tervl [, ] to smller su-tervls x 0 x 1 x x - x -1 x 1 The Bsc Ide We the crete rectgles

More information

e t dt e t dt = lim e t dt T (1 e T ) = 1

e 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 information

Learning of Graphical Models Parameter Estimation and Structure Learning

Learning of Graphical Models Parameter Estimation and Structure Learning Learg of Grahal Models Parameer Esmao ad Sruure Learg e Fukumzu he Isue of Sasal Mahemas Comuaoal Mehodology Sasal Iferee II Work wh Grahal Models Deermg sruure Sruure gve by modelg d e.g. Mxure model

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.

(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 information

12 Iterative Methods. Linear Systems: Gauss-Seidel Nonlinear Systems Case Study: Chemical Reactions

12 Iterative Methods. Linear Systems: Gauss-Seidel Nonlinear Systems Case Study: Chemical Reactions HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 Itertve Methods er Systems: Guss-Sedel Noler Systems Cse Study: Chemcl Rectos Itertve or ppromte methods or systems o equtos cosst o guessg vlue d the

More information

QR factorization. Let P 1, P 2, P n-1, be matrices such that Pn 1Pn 2... PPA

QR factorization. Let P 1, P 2, P n-1, be matrices such that Pn 1Pn 2... PPA QR facorzao Ay x real marx ca be wre as AQR, where Q s orhogoal ad R s upper ragular. To oba Q ad R, we use he Householder rasformao as follows: Le P, P, P -, be marces such ha P P... PPA ( R s upper ragular.

More information

Chapter Gauss-Seidel Method

Chapter Gauss-Seidel Method Chpter 04.08 Guss-Sedel Method After redg ths hpter, you should be ble to:. solve set of equtos usg the Guss-Sedel method,. reogze the dvtges d ptflls of the Guss-Sedel method, d. determe uder wht odtos

More information

Chapter 7. Bounds for weighted sums of Random Variables

Chapter 7. Bounds for weighted sums of Random Variables Chpter 7. Bouds for weghted sums of Rdom Vrbles 7. Itroducto Let d 2 be two depedet rdom vrbles hvg commo dstrbuto fucto. Htczeko (998 d Hu d L (2000 vestgted the Rylegh dstrbuto d obted some results bout

More information

Machine Learning. Hidden Markov Model. Eric Xing / /15-781, 781, Fall Lecture 17, March 24, 2008

Machine Learning. Hidden Markov Model. Eric Xing / /15-781, 781, Fall Lecture 17, March 24, 2008 Mache Learg 0-70/5 70/5-78 78 Fall 2008 Hdde Marov Model Erc Xg Lecure 7 March 24 2008 Readg: Cha. 3 C.B boo Erc Xg Erc Xg 2 Hdde Marov Model: from sac o damc mure models Sac mure Damc mure Y Y Y 2 Y 3

More information

Chapter 2. Review of Hydrodynamics and Vector Analysis

Chapter 2. Review of Hydrodynamics and Vector Analysis her. Ree o Hdrodmcs d Vecor Alss. Tlor seres L L L L ' ' L L " " " M L L! " ' L " ' I s o he c e romed he Tlor seres. O he oher hd ' " L . osero o mss -dreco: L L IN ] OUT [mss l [mss l] mss ccmled h me

More information

Technical Appendix for Inventory Management for an Assembly System with Product or Component Returns, DeCroix and Zipkin, Management Science 2005.

Technical Appendix for Inventory Management for an Assembly System with Product or Component Returns, DeCroix and Zipkin, Management Science 2005. Techc Appedx fo Iveoy geme fo Assemy Sysem wh Poduc o Compoe eus ecox d Zp geme Scece 2005 Lemm µ µ s c Poof If J d µ > µ he ˆ 0 µ µ µ µ µ µ µ µ Sm gumes essh he esu f µ ˆ > µ > µ > µ o K ˆ If J he so

More information

Speech, NLP and the Web

Speech, NLP and the Web peech NL ad he Web uhpak Bhaacharyya CE Dep. IIT Bombay Lecure 38: Uuperved learg HMM CFG; Baum Welch lecure 37 wa o cogve NL by Abh Mhra Baum Welch uhpak Bhaacharyya roblem HMM arg emac ar of peech Taggg

More information

ENGR 1990 Engineering Mathematics The Integral of a Function as a Function

ENGR 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 information

ICS141: Discrete Mathematics for Computer Science I

ICS141: Discrete Mathematics for Computer Science I Uversty o Hw ICS: Dscrete Mthemtcs or Computer Scece I Dept. Iormto & Computer Sc., Uversty o Hw J Stelovsy bsed o sldes by Dr. Be d Dr. Stll Orgls by Dr. M. P. Fr d Dr. J.L. Gross Provded by McGrw-Hll

More information

Hidden Markov Model. a ij. Observation : O1,O2,... States in time : q1, q2,... All states : s1, s2,..., sn

Hidden Markov Model. a ij. Observation : O1,O2,... States in time : q1, q2,... All states : s1, s2,..., sn Hdden Mrkov Model S S servon : 2... Ses n me : 2... All ses : s s2... s 2 3 2 3 2 Hdden Mrkov Model Con d Dscree Mrkov Model 2 z k s s s s s s Degree Mrkov Model Hdden Mrkov Model Con d : rnson roly from

More information

MTH 146 Class 7 Notes

MTH 146 Class 7 Notes 7.7- Approxmte Itegrto Motvto: MTH 46 Clss 7 Notes I secto 7.5 we lered tht some defte tegrls, lke x e dx, cot e wrtte terms of elemetry fuctos. So, good questo to sk would e: How c oe clculte somethg

More information

Chapter 2. Motion along a straight line. 9/9/2015 Physics 218

Chapter 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 information

Linear Regression Linear Regression with Shrinkage

Linear Regression Linear Regression with Shrinkage Lear Regresso Lear Regresso h Shrkage Iroduco Regresso meas predcg a couous (usuall scalar oupu from a vecor of couous pus (feaures x. Example: Predcg vehcle fuel effcec (mpg from 8 arbues: Lear Regresso

More information

Some Probability Inequalities for Quadratic Forms of Negatively Dependent Subgaussian Random Variables

Some Probability Inequalities for Quadratic Forms of Negatively Dependent Subgaussian Random Variables Joural of Sceces Islamc epublc of Ira 6(: 63-67 (005 Uvers of ehra ISSN 06-04 hp://scecesuacr Some Probabl Iequales for Quadrac Forms of Negavel Depede Subgaussa adom Varables M Am A ozorga ad H Zare 3

More information

3. REVIEW OF PROPERTIES OF EIGENVALUES AND EIGENVECTORS

3. REVIEW OF PROPERTIES OF EIGENVALUES AND EIGENVECTORS . REVIEW OF PROPERTIES OF EIGENVLUES ND EIGENVECTORS. EIGENVLUES ND EIGENVECTORS We hll ow revew ome bc fct from mtr theory. Let be mtr. clr clled egevlue of f there et ozero vector uch tht Emle: Let 9

More information

Differential Entropy 吳家麟教授

Differential Entropy 吳家麟教授 Deretl Etropy 吳家麟教授 Deto Let be rdom vrble wt cumultve dstrbuto ucto I F s cotuous te r.v. s sd to be cotuous. Let = F we te dervtve s deed. I te s clled te pd or. Te set were > 0 s clled te support set

More information

Isotropic Non-Heisenberg Magnet for Spin S=1

Isotropic Non-Heisenberg Magnet for Spin S=1 Ierol Jourl of Physcs d Applcos. IN 974- Volume, Number (, pp. 7-4 Ierol Reserch Publco House hp://www.rphouse.com Isoropc No-Heseberg Mge for p = Y. Yousef d Kh. Kh. Mumov.U. Umrov Physcl-Techcl Isue

More information

Remember: Project Proposals are due April 11.

Remember: Project Proposals are due April 11. Bonformtcs ecture Notes Announcements Remember: Project Proposls re due Aprl. Clss 22 Aprl 4, 2002 A. Hdden Mrov Models. Defntons Emple - Consder the emple we tled bout n clss lst tme wth the cons. However,

More information

2D Motion WS. A horizontally launched projectile s initial vertical velocity is zero. Solve the following problems with this information.

2D Motion WS. A horizontally launched projectile s initial vertical velocity is zero. Solve the following problems with this information. Nme D Moion WS The equions of moion h rele o projeciles were discussed in he Projecile Moion Anlsis Acii. ou found h projecile moes wih consn eloci in he horizonl direcion nd consn ccelerion in he ericl

More information

In the complete model, these slopes are ANALYSIS OF VARIANCE FOR THE COMPLETE TWO-WAY MODEL. (! i+1 -! i ) + [(!") i+1,q - [(!

In 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 information

Key words: Fractional difference equation, oscillatory solutions,

Key words: Fractional difference equation, oscillatory solutions, OSCILLATION PROPERTIES OF SOLUTIONS OF FRACTIONAL DIFFERENCE EQUATIONS Musafa BAYRAM * ad Ayd SECER * Deparme of Compuer Egeerg, Isabul Gelsm Uversy Deparme of Mahemacal Egeerg, Yldz Techcal Uversy * Correspodg

More information

1.B Appendix to Chapter 1

1.B Appendix to Chapter 1 Secon.B.B Append o Chper.B. The Ordnr Clcl Here re led ome mporn concep rom he ordnr clcl. The Dervve Conder ncon o one ndependen vrble. The dervve o dened b d d lm lm.b. where he ncremen n de o n ncremen

More information

FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),

FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx), FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To -periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the

More information

Review for the Midterm Exam.

Review for the Midterm Exam. Review for he iderm Exm Rememer! Gross re e re Vriles suh s,, /, p / p, r, d R re gross res 2 You should kow he disiio ewee he fesile se d he udge se, d kow how o derive hem The Fesile Se Wihou goverme

More information

rank Additionally system of equation only independent atfect Gawp (A) possible ( Alb ) easily process form rang A. Proposition with Definition

rank Additionally system of equation only independent atfect Gawp (A) possible ( Alb ) easily process form rang A. Proposition with Definition Defiion nexivnol numer ler dependen rows mrix sid row Gwp elimion mehod does no fec h numer end process i possile esily red rng fc for mrix form der zz rn rnk wih m dcussion i holds rr o Proposiion ler

More information

PROGRESSIONS AND SERIES

PROGRESSIONS AND SERIES PROGRESSIONS AND SERIES A sequece is lso clled progressio. We ow study three importt types of sequeces: () The Arithmetic Progressio, () The Geometric Progressio, () The Hrmoic Progressio. Arithmetic Progressio.

More information

1 4 6 is symmetric 3 SPECIAL MATRICES 3.1 SYMMETRIC MATRICES. Defn: A matrix A is symmetric if and only if A = A, i.e., a ij =a ji i, j. Example 3.1.

1 4 6 is symmetric 3 SPECIAL MATRICES 3.1 SYMMETRIC MATRICES. Defn: A matrix A is symmetric if and only if A = A, i.e., a ij =a ji i, j. Example 3.1. SPECIAL MATRICES SYMMETRIC MATRICES Def: A mtr A s symmetr f d oly f A A, e,, Emple A s symmetr Def: A mtr A s skew symmetr f d oly f A A, e,, Emple A s skew symmetr Remrks: If A s symmetr or skew symmetr,

More information

Review: Transformations. Transformations - Viewing. Transformations - Modeling. world CAMERA OBJECT WORLD CSE 681 CSE 681 CSE 681 CSE 681

Review: Transformations. Transformations - Viewing. Transformations - Modeling. world CAMERA OBJECT WORLD CSE 681 CSE 681 CSE 681 CSE 681 Revew: Trnsforons Trnsforons Modelng rnsforons buld cople odels b posonng (rnsforng sple coponens relve o ech oher ewng rnsforons plcng vrul cer n he world rnsforon fro world coordnes o cer coordnes Perspecve

More information

Northwest High School s Algebra 2

Northwest High School s Algebra 2 Northwest High School s Algebr Summer Review Pcket 0 DUE August 8, 0 Studet Nme This pcket hs bee desiged to help ou review vrious mthemticl topics tht will be ecessr for our success i Algebr. Istructios:

More information

The Linear Regression Of Weighted Segments

The Linear Regression Of Weighted Segments The Lear Regresso Of Weghed Segmes George Dael Maeescu Absrac. We proposed a regresso model where he depede varable s made o up of pos bu segmes. Ths suao correspods o he markes hroughou he da are observed

More information

Determination of Antoine Equation Parameters. December 4, 2012 PreFEED Corporation Yoshio Kumagae. Introduction

Determination of Antoine Equation Parameters. December 4, 2012 PreFEED Corporation Yoshio Kumagae. Introduction refeed Soluos for R&D o Desg Deermao of oe Equao arameers Soluos for R&D o Desg December 4, 0 refeed orporao Yosho Kumagae refeed Iroduco hyscal propery daa s exremely mpora for performg process desg ad

More information

Integration and Differentiation

Integration and Differentiation ome Clculus bckgroud ou should be fmilir wih, or review, for Mh 404 I will be, for he mos pr, ssumed ou hve our figerips he bsics of (mulivrible) fucios, clculus, d elemer differeil equios If here hs bee

More information

We partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.

We partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b. Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn

More information

A Brief Introduction to Olympiad Inequalities

A Brief Introduction to Olympiad Inequalities Ev Che Aprl 0, 04 The gol of ths documet s to provde eser troducto to olympd equltes th the stdrd exposto Olympd Iequltes, by Thoms Mldorf I ws motvted to wrte t by feelg gulty for gettg free 7 s o problems

More information

Here we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.

Here we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system. Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous

More information

Optimality of Strategies for Collapsing Expanded Random Variables In a Simple Random Sample Ed Stanek

Optimality of Strategies for Collapsing Expanded Random Variables In a Simple Random Sample Ed Stanek Optmlt of Strteges for Collpsg Expe Rom Vrles Smple Rom Smple E Stek troucto We revew the propertes of prectors of ler comtos of rom vrles se o rom vrles su-spce of the orgl rom vrles prtculr, we ttempt

More information

Density estimation II

Density estimation II CS 750 Mche Lerg Lecture 6 esty estmto II Mlos Husrecht mlos@tt.edu 539 Seott Squre t: esty estmto {.. } vector of ttrute vlues Ojectve: estmte the model of the uderlyg rolty dstruto over vrles X X usg

More information

A Kalman filtering simulation

A 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 information

b a 2 ((g(x))2 (f(x)) 2 dx

b a 2 ((g(x))2 (f(x)) 2 dx Clc II Fll 005 MATH Nme: T3 Istructios: Write swers to problems o seprte pper. You my NOT use clcultors or y electroic devices or otes of y kid. Ech st rred problem is extr credit d ech is worth 5 poits.

More information

Solution of Impulsive Differential Equations with Boundary Conditions in Terms of Integral Equations

Solution of Impulsive Differential Equations with Boundary Conditions in Terms of Integral Equations Joural of aheacs ad copuer Scece (4 39-38 Soluo of Ipulsve Dffereal Equaos wh Boudary Codos Ters of Iegral Equaos Arcle hsory: Receved Ocober 3 Acceped February 4 Avalable ole July 4 ohse Rabba Depare

More information

P-Convexity Property in Musielak-Orlicz Function Space of Bohner Type

P-Convexity Property in Musielak-Orlicz Function Space of Bohner Type J N Sce & Mh Res Vol 3 No (7) -7 Alble ole h://orlwlsogocd/deh/sr P-Coey Proery Msel-Orlcz Fco Sce o Boher ye Yl Rodsr Mhecs Edco Deree Fcly o Ss d echology Uerss sl Neger Wlsogo Cerl Jdoes Absrcs Corresodg

More information