Political Economy of Institutions and Development: Problem Set 2 Due Date: Thursday, March 15, 2019.

Size: px
Start display at page:

Download "Political Economy of Institutions and Development: Problem Set 2 Due Date: Thursday, March 15, 2019."

Transcription

1 Polcal Economy of Insuons and Developmen: Problem Se 2 Due Dae: Thursday, March 15, Please answer Quesons 1, 2 and 3. Queson 1 Consder an nfne-horzon dynamc game beween wo groups, an ele and a group of czens. The ele s currenly n power n a non-democrac regme, N. In he nondemocrac regme, he ele receves ncome y N, whle czens receve ncome w N. A he begnnng of each perod, he ele (acng as a sngle agen n hs game) chooses how much o spend o defend he regme, denoed by x 0. If x v, he ele are able o manan he non-democrac regme. Here v s a random erm represenng he sochasc power of he czens o cones he regme. Assume ha v s dsrbued accordng o connuous, smooh densy f, whch s n addon sngle peaked and symmerc around zero. If he czens are able o change he regme, hen a new democrac regme, D, s nsued, and n hs regme, he ele receves ncome y D < y N, whle each czen receves w D > w N (so ha he ele prefers non-democracy and czens prefer democracy). In democracy, he mng of evens s smlar: a he begnnng of each perod, he ele decde how much o nves n acves o subver democracy back o non democracy, agan denoed by x. The only dfference s ha because n democracy he czens are more powerful, he ele wll succeed only f x v + γ, where γ > 0, and v have he same dsrbuon, f. 1. Wre down he dynamc maxmzaon problem of he ele n non-democracy and separaely n democracy as a dynamc programmng problem (wh maxmzaon over x ). 1

2 2. Assume ha he maxmzaon problem of he ele has an neror soluon n boh non-democracy and democracy, and derve he frs-order condon for boh problems. Inerpre he condons. 3. Show, agan under he assumpon of neror soluon, ha he equlbrum probably ha nex perod here wll be non-democracy s he same sarng from democracy and nondemocracy. Inerpre hs resul. 4. Wha happens f γ ncreases (sll assumng an neror soluon)? Inerpre hs resul. 5. Wha are he mplcaons of hese resuls for welfare (Hn: compue he expeced dscouned ne presen value of boh he ele and czens sarng from he wo regmes and compare hem, and hen deermne he mplcaons of a greaer γ for welfare). 6. Explan why he resul obaned n hs model s specal and provde hree dfferen modfcaons of he model (whou dong he mah) ha wll change hs resul. Do you hnk here wll be an overlap n erms of he qualave effec of nsuons beween hese modfed models and he one n hs queson? Queson 2 Ideas dang back o de Tocquevlle sugges ha greaer socal mobly makes democracy work beer or become more sable. Consder a smple wo-perod model wh hree groups, poor, mddle-class and rch (ndvduals whn each group are compleely homogeneous). Preferences sasfy sngle crossng and sngle peakedness, and he deal polces of he hree groups are dsnc. Suppose here are also hree polcal nsuons: democracy n whch he mddle class s he medan voer, a lef dcaorshp n whch he poor make all he decsons, and an ele dcaorshp n whch he rch make all he decsons. In he second perod, he relevan decson-maker jus decdes polcy. In he frs perod, he relevan decson-maker decdes boh polcy and wha omorrow s nsuons should be. Suppose ha we sar n democracy, so ha he frs-perod decsons wll be made by he mddle class. Suppose also ha here are 200 rch ndvduals, 150 mddle-class ndvduals and 300 poor ndvduals, so ha n democracy a member of he mddle class s he medan voer. Suppose ha here s he followng ype of socal mobly: K 150 mddle-class ndvduals become poor n he second perod, whle K poor ndvduals become mddle-class (leavng he overall dsrbuon of socey across he hree groups unchanged, wh no mobly for he rch). All ndvduals know her new socal group before he vong sage n he second perod. Show ha f K < 75, democracy s sable (n he sense ha he socey remans democrac n he second perod), whle f K > 75, democracy s unsable. Explan he nuon for hs resul and dscuss de Tocquevlle s hypohess n hs lgh. 2

3 Queson 3 Consder an economy populaed by λ rch agens who nally hold power, and 1 λ poor agens who are excluded from power, wh λ < 1/2. All agens are nfnely lved and dscoun he fuure a he rae β (0, 1). Each rch agen has ncome θ/λ whle each poor agen has ncome (1 θ) / (1 λ) where θ > λ. The polcal sysem deermnes a lnear ax rae, τ, he proceeds of whch are redsrbued lump-sum. Each agen can hde her money n an alernave non-axable producon echnology, and n he process hey lose a fracon φ of her ncome. There are no oher coss of axaon. The poor can underake a revoluon, and f hey do so, n all fuure perods, hey oban a fracon µ () of he oal ncome of he socey (.e., an ncome of µ () /(1 λ) per poor agen). The poor canno revol agans democracy. The rch lose everyhng and receve zero payoff afer a revoluon. A he begnnng of every perod, he rch can also decde o exend he franchse o he poor, and hs s rreversble. If he franchse s exended, he poor decde he ax rae n all fuure perods. 1. Defne MPE n hs game. 2. Frs suppose ha µ () = µ l a all mes. Also assume ha 0 < µ l < 1 θ. Show ha n he MPE, here wll be no axaon when he rch are n power, and he ax rae wll be τ = φ when he poor are n power. Show ha n he MPE, here s no exenson of he franchse and no axaon. 3. Suppose ha µ l (1 θ, (1 φ) (1 θ) + φ (1 λ)). Characerze he MPE n hs case. Why s he resrcon µ l < (1 φ) (1 θ) + φ (1 λ) necessary? 4. Now consder he SPE of hs game when µ l > 1 θ. Consruc an equlbrum where here s exenson of he franchse along he equlbrum pah. [Hn: frs, o smplfy, ake β 1, and hen consder a sraegy profle where he rch are always expeced o se τ = 0 n he fuure; show ha n hs case he poor would underake a revoluon; also explan why he connuaon sraegy of τ = 0 by he rch n all fuure perods could be par of a SPE]. Why s here exenson of he franchse now? Can you consruc a smlar non-markovan equlbrum when µ l < 1 θ? 5. Explan why he MPE led o dfferen predcons han he non-markovan equlbra. Whch one s more sasfacory? 6. Now suppose ha µ () = µ l wh probably 1 q, and µ () = µ h wh probably q, where µ h > 1 θ > µ l. Consruc a MPE where he rch exend he franchse, and from here on, a poor agen ses ha ax rae. Deermne he parameer values ha are necessary for such an equlbrum o exs. Explan why exenson of he franchse s useful for rch agens? 3

4 7. Now consder non-markovan equlbra agan. Suppose ha he unque MPE has franchse exenson. Can you consruc a SPE equlbrum, as β 1, where here s no franchse exenson? 8. Conras he role of resrcng sraeges o be Markovan n he wo cases above [Hn: why s hs resrcon rulng ou franchse exenson n he frs case, whle ensurng ha franchse exenson s he unque equlbrum n he second?]. Queson 4 Consder he followng nfne horzon economy. Tme s dscree and ndexed by. There s a se of czens, wh mass normalzed o 1 and a ruler. Czens dscoun he fuure wh he dscoun facor β, and have he uly funcon [ ] c 1 θ u = β j +j 1 θ e +j, j= where θ > 0, c +j s consumpon and e +j s effor (he ruler exers no effor). Each czen has access o he followng Cobb-Douglas producon echnology: y = A α ( e ) 1 α, where A denoes he sae of echnology and nfrasrucure a me, whch wll be deermned by he ruler. In addon, he ruler ses a ax rae τ on ncome. Also, each czen can decde o hde a fracon z of hs oupu, whch s no axable, bu hdng oupu s cosly, so a fracon δ of s los n he process. So consumpon of agen s: c [ (1 τ ) ( 1 z ) + (1 δ) z ] y, and ax revenues are T = τ (1 z ) y d. The ruler a me decdes how much o spend on A +1 wh echnology: A +1 = G where G denoes governmen spendng on nfrasrucure, and φ < 1. Ths mples ha he consumpon of he ruler s c R = T G. The ruler s assumed o maxmze he ne presen dscouned value of hs or her consumpon. The mng of evens whn every perod s as follows: 4

5 The economy nhers A from governmen spendng a me 1. Czens choose nvesmen, { e }. The ruler ses ax rae τ. Czens decde how much of her oupu o hde, { z }. The ruler decdes how much spen on nex perod s nfrasrucure G. 1. Fnd he frs-bes allocaon n hs economy. 2. Defne a Markov Perfec Equlbrum (MPE) where sraeges a me depend on he payoff-relevan sae of he game a me. Be specfc abou wha he sraeges are and wha he sae s. 3. Show ha n a MPE τ = δ for all. Inerpre wha δ corresponds o. 4. Gven hs, fnd he equlbrum level of effor by czens and ax revenues as funcons of A, T (A ). 5. Explan why we could wre he value funcon of he ruler as V (A ) = max A +1 {T (A ) A +1 + βv (A +1 )}, and usng hs, fnd he equlbrum spendng on nfrasrucure, G. 6. Wha s he effec of δ on spendng on nfrasrucure and equlbrum nvesmens. Does a declne n δ always mply greaer effor by czens? If no, why no? Calculae he equlbrum level oupu and fnd he oupu maxmzng level of δ. Dscuss wha hs means, and how you could map hs o realy. 7. How would you generalze/modfy hs model so ha some counres ax a lo and use mos of he proceeds for nfrasrucure and publc good nvesmens? 5

UNIVERSITAT AUTÒNOMA DE BARCELONA MARCH 2017 EXAMINATION

UNIVERSITAT AUTÒNOMA DE BARCELONA MARCH 2017 EXAMINATION INTERNATIONAL TRADE T. J. KEHOE UNIVERSITAT AUTÒNOMA DE BARCELONA MARCH 27 EXAMINATION Please answer wo of he hree quesons. You can consul class noes, workng papers, and arcles whle you are workng on he

More information

Graduate Macroeconomics 2 Problem set 5. - Solutions

Graduate 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 information

V.Abramov - FURTHER ANALYSIS OF CONFIDENCE INTERVALS FOR LARGE CLIENT/SERVER COMPUTER NETWORKS

V.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 information

Solution in semi infinite diffusion couples (error function analysis)

Solution 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 information

2 Aggregate demand in partial equilibrium static framework

2 Aggregate demand in partial equilibrium static framework Unversy of Mnnesoa 8107 Macroeconomc Theory, Sprng 2009, Mn 1 Fabrzo Perr Lecure 1. Aggregaon 1 Inroducon Probably so far n he macro sequence you have deal drecly wh represenave consumers and represenave

More information

(,,, ) (,,, ). In addition, there are three other consumers, -2, -1, and 0. Consumer -2 has the utility function

(,,, ) (,,, ). 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 information

CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 4

CS434a/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 information

Let s treat the problem of the response of a system to an applied external force. Again,

Let 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

Ordinary Differential Equations in Neuroscience with Matlab examples. Aim 1- Gain understanding of how to set up and solve ODE s

Ordinary 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 information

2 Aggregate demand in partial equilibrium static framework

2 Aggregate demand in partial equilibrium static framework Unversy of Mnnesoa 8107 Macroeconomc Theory, Sprng 2012, Mn 1 Fabrzo Perr Lecure 1. Aggregaon 1 Inroducon Probably so far n he macro sequence you have deal drecly wh represenave consumers and represenave

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

[ ] 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 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

Linear Response Theory: The connection between QFT and experiments

Linear 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 information

Advanced Macroeconomics II: Exchange economy

Advanced 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 information

Notes on the stability of dynamic systems and the use of Eigen Values.

Notes 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 information

Online Supplement for Dynamic Multi-Technology. Production-Inventory Problem with Emissions Trading

Online Supplement for Dynamic Multi-Technology. Production-Inventory Problem with Emissions Trading Onlne Supplemen for Dynamc Mul-Technology Producon-Invenory Problem wh Emssons Tradng by We Zhang Zhongsheng Hua Yu Xa and Baofeng Huo Proof of Lemma For any ( qr ) Θ s easy o verfy ha he lnear programmng

More information

Mechanics Physics 151

Mechanics 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 information

January Examinations 2012

January Examinations 2012 Page of 5 EC79 January Examnaons No. of Pages: 5 No. of Quesons: 8 Subjec ECONOMICS (POSTGRADUATE) Tle of Paper EC79 QUANTITATIVE METHODS FOR BUSINESS AND FINANCE Tme Allowed Two Hours ( hours) Insrucons

More information

Dynamic Team Decision Theory. EECS 558 Project Shrutivandana Sharma and David Shuman December 10, 2005

Dynamic 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 information

Mechanics Physics 151

Mechanics 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 information

Fall 2009 Social Sciences 7418 University of Wisconsin-Madison. Problem Set 2 Answers (4) (6) di = D (10)

Fall 2009 Social Sciences 7418 University of Wisconsin-Madison. Problem Set 2 Answers (4) (6) di = D (10) Publc Affars 974 Menze D. Chnn Fall 2009 Socal Scences 7418 Unversy of Wsconsn-Madson Problem Se 2 Answers Due n lecure on Thursday, November 12. " Box n" your answers o he algebrac quesons. 1. Consder

More information

Internet Appendix for Political Cycles and Stock Returns

Internet Appendix for Political Cycles and Stock Returns Inerne Appendx for Polcal Cycles and Sock Reurns ĽUBOŠ PÁSTOR and PIETRO VERONESI November 6, 07 Ths Inerne Appendx provdes he proofs of all proposons n Pásor and Verones 07 Proof of Proposon a Governmen

More information

Technical Appendix to The Equivalence of Wage and Price Staggering in Monetary Business Cycle Models

Technical 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 information

ECON 8105 FALL 2017 ANSWERS TO MIDTERM EXAMINATION

ECON 8105 FALL 2017 ANSWERS TO MIDTERM EXAMINATION MACROECONOMIC THEORY T. J. KEHOE ECON 85 FALL 7 ANSWERS TO MIDTERM EXAMINATION. (a) Wh an Arrow-Debreu markes sruure fuures markes for goods are open n perod. Consumers rade fuures onras among hemselves.

More information

EP2200 Queuing theory and teletraffic systems. 3rd lecture Markov chains Birth-death process - Poisson process. Viktoria Fodor KTH EES

EP2200 Queuing theory and teletraffic systems. 3rd lecture Markov chains Birth-death process - Poisson process. Viktoria Fodor KTH EES EP Queung heory and eleraffc sysems 3rd lecure Marov chans Brh-deah rocess - Posson rocess Vora Fodor KTH EES Oulne for oday Marov rocesses Connuous-me Marov-chans Grah and marx reresenaon Transen and

More information

TSS = SST + SSE An orthogonal partition of the total SS

TSS = SST + SSE An orthogonal partition of the total SS ANOVA: Topc 4. Orhogonal conrass [ST&D p. 183] H 0 : µ 1 = µ =... = µ H 1 : The mean of a leas one reamen group s dfferen To es hs hypohess, a basc ANOVA allocaes he varaon among reamen means (SST) equally

More information

Trade Patterns and Perpetual Youth in A Dynamic Small Open Economy

Trade Patterns and Perpetual Youth in A Dynamic Small Open Economy Econ. J. of Hokkado Unv., Vol. 40 (2011), pp. 29-40 Trade Paerns and Perpeual Youh n A Dynamc Small Open Economy Naoshge Kanamor n hs paper, examne he long-run specalzaon paerns ha arse n a small open

More information

Online Appendix for. Strategic safety stocks in supply chains with evolving forecasts

Online 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 information

Problem Set 3 EC2450A. Fall ) Write the maximization problem of the individual under this tax system and derive the first-order conditions.

Problem Set 3 EC2450A. Fall ) Write the maximization problem of the individual under this tax system and derive the first-order conditions. Problem Se 3 EC450A Fall 06 Problem There are wo ypes of ndvduals, =, wh dfferen ables w. Le be ype s onsumpon, l be hs hours worked and nome y = w l. Uly s nreasng n onsumpon and dereasng n hours worked.

More information

Midterm Exam. Thursday, April hour, 15 minutes

Midterm Exam. Thursday, April hour, 15 minutes Economcs of Grow, ECO560 San Francsco Sae Unvers Mcael Bar Sprng 04 Mderm Exam Tursda, prl 0 our, 5 mnues ame: Insrucons. Ts s closed boo, closed noes exam.. o calculaors of an nd are allowed. 3. Sow all

More information

Demographics in Dynamic Heckscher-Ohlin Models: Overlapping Generations versus Infinitely Lived Consumers*

Demographics in Dynamic Heckscher-Ohlin Models: Overlapping Generations versus Infinitely Lived Consumers* Federal Reserve Ban of Mnneapols Research Deparmen Saff Repor 377 Sepember 6 Demographcs n Dynamc Hecscher-Ohln Models: Overlappng Generaons versus Infnely Lved Consumers* Clausre Bajona Unversy of Mam

More information

CS 268: Packet Scheduling

CS 268: Packet Scheduling Pace Schedulng Decde when and wha pace o send on oupu ln - Usually mplemened a oupu nerface CS 68: Pace Schedulng flow Ion Soca March 9, 004 Classfer flow flow n Buffer managemen Scheduler soca@cs.bereley.edu

More information

Mechanics Physics 151

Mechanics 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 information

II. Light is a Ray (Geometrical Optics)

II. Light is a Ray (Geometrical Optics) II Lgh s a Ray (Geomercal Opcs) IIB Reflecon and Refracon Hero s Prncple of Leas Dsance Law of Reflecon Hero of Aleandra, who lved n he 2 nd cenury BC, posulaed he followng prncple: Prncple of Leas Dsance:

More information

Variants of Pegasos. December 11, 2009

Variants 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 information

( ) () we define the interaction representation by the unitary transformation () = ()

( ) () 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 information

Department of Economics University of Toronto

Department of Economics University of Toronto Deparmen of Economcs Unversy of Torono ECO408F M.A. Economercs Lecure Noes on Heeroskedascy Heeroskedascy o Ths lecure nvolves lookng a modfcaons we need o make o deal wh he regresson model when some of

More information

Existence and Uniqueness Results for Random Impulsive Integro-Differential Equation

Existence 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 information

Macroeconomic Theory Ph.D. Qualifying Examination Fall 2005 ANSWER EACH PART IN A SEPARATE BLUE BOOK. PART ONE: ANSWER IN BOOK 1 WEIGHT 1/3

Macroeconomic Theory Ph.D. Qualifying Examination Fall 2005 ANSWER EACH PART IN A SEPARATE BLUE BOOK. PART ONE: ANSWER IN BOOK 1 WEIGHT 1/3 Macroeconomic Theory Ph.D. Qualifying Examinaion Fall 2005 Comprehensive Examinaion UCLA Dep. of Economics You have 4 hours o complee he exam. There are hree pars o he exam. Answer all pars. Each par has

More information

e-journal Reliability: Theory& Applications No 2 (Vol.2) Vyacheslav Abramov

e-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

NATIONAL UNIVERSITY OF SINGAPORE PC5202 ADVANCED STATISTICAL MECHANICS. (Semester II: AY ) Time Allowed: 2 Hours

NATIONAL 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 information

Math 128b Project. Jude Yuen

Math 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 information

Problem Set 5. Graduate Macro II, Spring 2017 The University of Notre Dame Professor Sims

Problem Set 5. Graduate Macro II, Spring 2017 The University of Notre Dame Professor Sims Problem Se 5 Graduae Macro II, Spring 2017 The Universiy of Nore Dame Professor Sims Insrucions: You may consul wih oher members of he class, bu please make sure o urn in your own work. Where applicable,

More information

Epistemic Game Theory: Online Appendix

Epistemic Game Theory: Online Appendix Epsemc Game Theory: Onlne Appendx Edde Dekel Lucano Pomao Marcano Snscalch July 18, 2014 Prelmnares Fx a fne ype srucure T I, S, T, β I and a probably µ S T. Le T µ I, S, T µ, βµ I be a ype srucure ha

More information

The Intertemporal Keynesian Cross

The Intertemporal Keynesian Cross he Ineremporal Keynesan Cross Adren Aucler Mahew Rognle Ludwg Sraub January 207 Absrac We derve a mcrofounded, dynamc verson of he radonal Keynesan cross, whch we call he neremporal Keynesan cross. I characerzes

More information

John Geweke a and Gianni Amisano b a Departments of Economics and Statistics, University of Iowa, USA b European Central Bank, Frankfurt, Germany

John 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 information

Part II CONTINUOUS TIME STOCHASTIC PROCESSES

Part 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 information

. The geometric multiplicity is dim[ker( λi. number of linearly independent eigenvectors associated with this eigenvalue.

. 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 information

CS286.2 Lecture 14: Quantum de Finetti Theorems II

CS286.2 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 information

P R = P 0. The system is shown on the next figure:

P 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 information

Bundling with Customer Self-Selection: A Simple Approach to Bundling Low Marginal Cost Goods On-Line Appendix

Bundling with Customer Self-Selection: A Simple Approach to Bundling Low Marginal Cost Goods On-Line Appendix Bundlng wh Cusomer Self-Selecon: A Smple Approach o Bundlng Low Margnal Cos Goods On-Lne Appendx Lorn M. H Unversy of Pennsylvana, Wharon School 57 Jon M. Hunsman Hall Phladelpha, PA 94 lh@wharon.upenn.edu

More information

National Exams December 2015 NOTES: 04-BS-13, Biology. 3 hours duration

National Exams December 2015 NOTES: 04-BS-13, Biology. 3 hours duration Naonal Exams December 205 04-BS-3 Bology 3 hours duraon NOTES: f doub exss as o he nerpreaon of any queson he canddae s urged o subm wh he answer paper a clear saemen of any assumpons made 2 Ths s a CLOSED

More information

Density Matrix Description of NMR BCMB/CHEM 8190

Density 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 information

CH.3. COMPATIBILITY EQUATIONS. Continuum Mechanics Course (MMC) - ETSECCPB - UPC

CH.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 information

Solutions Problem Set 3 Macro II (14.452)

Solutions Problem Set 3 Macro II (14.452) Soluions Problem Se 3 Macro II (14.452) Francisco A. Gallego 04/27/2005 1 Q heory of invesmen in coninuous ime and no uncerainy Consider he in nie horizon model of a rm facing adjusmen coss o invesmen.

More information

There are a total of two problems, each with multiple subparts.

There are a total of two problems, each with multiple subparts. eparmen of Economcs Boson College Economcs 0 (Secon 05) acroeconomc Theory Problem Se Suggesed Soluons Professor Sanjay Chugh Fall 04 ue: ecember 9, 04 (no laer han :30pm) Insrucons: Clearly-wren (yped

More information

Capital Income Taxation and Economic Growth in Open Economies

Capital Income Taxation and Economic Growth in Open Economies WP/04/91 Capal Income Taxaon and Economc Growh n Open Economes Gerema Palomba 2004 Inernaonal Moneary Fund WP/04/91 IMF Workng Paper Fscal Affars Deparmen Capal Income Taxaon and Economc Growh n Open Economes

More information

Density Matrix Description of NMR BCMB/CHEM 8190

Density 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 information

Appendix H: Rarefaction and extrapolation of Hill numbers for incidence data

Appendix 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 information

Should we internalize intertemporal production externalities in the case of pest resistance? Elsa Martin* AgroSup Dijon, CESAER (UMR 1041 of INRA)

Should we internalize intertemporal production externalities in the case of pest resistance? Elsa Martin* AgroSup Dijon, CESAER (UMR 1041 of INRA) Should we nernalze neremporal producon exernales n he case of pes ressance? Elsa Marn AgroSup Djon CESAER (UMR 04 of IRA) Conrbued Paper prepared for presenaon a he 88h Annual Conference of he Agrculural

More information

Modeling and Solving of Multi-Product Inventory Lot-Sizing with Supplier Selection under Quantity Discounts

Modeling and Solving of Multi-Product Inventory Lot-Sizing with Supplier Selection under Quantity Discounts nernaonal ournal of Appled Engneerng Research SSN 0973-4562 Volume 13, Number 10 (2018) pp. 8708-8713 Modelng and Solvng of Mul-Produc nvenory Lo-Szng wh Suppler Selecon under Quany Dscouns Naapa anchanaruangrong

More information

Final Exam. Tuesday, December hours

Final Exam. Tuesday, December hours San Francisco Sae Universiy Michael Bar ECON 560 Fall 03 Final Exam Tuesday, December 7 hours Name: Insrucions. This is closed book, closed noes exam.. No calculaors of any kind are allowed. 3. Show all

More information

. The geometric multiplicity is dim[ker( λi. A )], i.e. the number of linearly independent eigenvectors associated with this eigenvalue.

. 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 information

Modern Dynamic Asset Pricing Models

Modern Dynamic Asset Pricing Models Modern Dynamc Asse Prcng Models Lecure Noes 2. Equlbrum wh Complee Markes 1 Pero Verones The Unversy of Chcago Booh School of Busness CEPR, NBER 1 These eachng noes draw heavly on Duffe (1996, Chapers

More information

Chapter 9: Factor pricing models. Asset Pricing Zheng Zhenlong

Chapter 9: Factor pricing models. Asset Pricing Zheng Zhenlong Chaper 9: Facor prcng models Asse Prcng Conens Asse Prcng Inroducon CAPM ICAPM Commens on he CAPM and ICAPM APT APT vs. ICAPM Bref nroducon Asse Prcng u β u ( c + 1 ) a + b f + 1 ( c ) Bref nroducon Asse

More information

Knowing What Others Know: Coordination Motives in Information Acquisition Additional Notes

Knowing What Others Know: Coordination Motives in Information Acquisition Additional Notes Knowng Wha Ohers Know: Coordnaon Moves n nformaon Acquson Addonal Noes Chrsan Hellwg Unversy of Calforna, Los Angeles Deparmen of Economcs Laura Veldkamp New York Unversy Sern School of Busness March 1,

More information

Lecture 18: The Laplace Transform (See Sections and 14.7 in Boas)

Lecture 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 information

Lecture Notes 4: Consumption 1

Lecture Notes 4: Consumption 1 Leure Noes 4: Consumpon Zhwe Xu (xuzhwe@sju.edu.n) hs noe dsusses households onsumpon hoe. In he nex leure, we wll dsuss rm s nvesmen deson. I s safe o say ha any propagaon mehansm of maroeonom model s

More information

Teaching Notes #2 Equilibrium with Complete Markets 1

Teaching Notes #2 Equilibrium with Complete Markets 1 Teachng Noes #2 Equlbrum wh Complee Markes 1 Pero Verones Graduae School of Busness Unversy of Chcago Busness 35909 Sprng 2005 c by Pero Verones Ths Verson: November 17, 2005 1 These eachng noes draw heavly

More information

Lecture 2 M/G/1 queues. M/G/1-queue

Lecture 2 M/G/1 queues. M/G/1-queue Lecure M/G/ queues M/G/-queue Posson arrval process Arbrary servce me dsrbuon Sngle server To deermne he sae of he sysem a me, we mus now The number of cusomers n he sysems N() Tme ha he cusomer currenly

More information

Chapter Lagrangian Interpolation

Chapter 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

Normal Random Variable and its discriminant functions

Normal 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 information

Imperfect Information

Imperfect Information Imerfec Informaon Comlee Informaon - all layers know: Se of layers Se of sraeges for each layer Oucomes as a funcon of he sraeges Payoffs for each oucome (.e. uly funcon for each layer Incomlee Informaon

More information

Example: MOSFET Amplifier Distortion

Example: MOSFET Amplifier Distortion 4/25/2011 Example MSFET Amplfer Dsoron 1/9 Example: MSFET Amplfer Dsoron Recall hs crcu from a prevous handou: ( ) = I ( ) D D d 15.0 V RD = 5K v ( ) = V v ( ) D o v( ) - K = 2 0.25 ma/v V = 2.0 V 40V.

More information

J i-1 i. J i i+1. Numerical integration of the diffusion equation (I) Finite difference method. Spatial Discretization. Internal nodes.

J 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 information

FI 3103 Quantum Physics

FI 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 information

10. A.C CIRCUITS. Theoretically current grows to maximum value after infinite time. But practically it grows to maximum after 5τ. Decay of current :

10. 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 information

Outline. Probabilistic Model Learning. Probabilistic Model Learning. Probabilistic Model for Time-series Data: Hidden Markov Model

Outline. 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 information

Time-interval analysis of β decay. V. Horvat and J. C. Hardy

Time-interval analysis of β decay. V. Horvat and J. C. Hardy Tme-nerval analyss of β decay V. Horva and J. C. Hardy Work on he even analyss of β decay [1] connued and resuled n he developmen of a novel mehod of bea-decay me-nerval analyss ha produces hghly accurae

More information

F-Tests and Analysis of Variance (ANOVA) in the Simple Linear Regression Model. 1. Introduction

F-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 information

How about the more general "linear" scalar functions of scalars (i.e., a 1st degree polynomial of the following form with a constant term )?

How 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 information

Should we internalize intertemporal production externalities in the case of pest resistance?

Should we internalize intertemporal production externalities in the case of pest resistance? Should we nernalze neremporal producon exernales n he case of pes ressance? Elsa MARI AgroSup Djon, CESAER (UMR 04 of IRA), elsa.marn@djon.nra.fr Paper prepared for presenaon a he EAAE 04 Congress Agr-Food

More information

The topology and signature of the regulatory interactions predict the expression pattern of the segment polarity genes in Drosophila m elanogaster

The topology and signature of the regulatory interactions predict the expression pattern of the segment polarity genes in Drosophila m elanogaster The opology and sgnaure of he regulaory neracons predc he expresson paern of he segmen polary genes n Drosophla m elanogaser Hans Ohmer and Réka Alber Deparmen of Mahemacs Unversy of Mnnesoa Complex bologcal

More information

THEORETICAL AUTOCORRELATIONS. ) if often denoted by γ. Note that

THEORETICAL AUTOCORRELATIONS. ) if often denoted by γ. Note that THEORETICAL AUTOCORRELATIONS Cov( y, y ) E( y E( y))( y E( y)) ρ = = Var( y) E( y E( y)) =,, L ρ = and Cov( y, y ) s ofen denoed by whle Var( y ) f ofen denoed by γ. Noe ha γ = γ and ρ = ρ and because

More information

HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD

HEAT 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 information

Polymerization Technology Laboratory Course

Polymerization Technology Laboratory Course Prakkum Polymer Scence/Polymersaonsechnk Versuch Resdence Tme Dsrbuon Polymerzaon Technology Laboraory Course Resdence Tme Dsrbuon of Chemcal Reacors If molecules or elemens of a flud are akng dfferen

More information

On One Analytic Method of. Constructing Program Controls

On 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 information

Comb Filters. Comb Filters

Comb 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 information

Robust and Accurate Cancer Classification with Gene Expression Profiling

Robust 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 information

EEL 6266 Power System Operation and Control. Chapter 5 Unit Commitment

EEL 6266 Power System Operation and Control. Chapter 5 Unit Commitment EEL 6266 Power Sysem Operaon and Conrol Chaper 5 Un Commmen Dynamc programmng chef advanage over enumeraon schemes s he reducon n he dmensonaly of he problem n a src prory order scheme, here are only N

More information

Tight results for Next Fit and Worst Fit with resource augmentation

Tight results for Next Fit and Worst Fit with resource augmentation Tgh resuls for Nex F and Wors F wh resource augmenaon Joan Boyar Leah Epsen Asaf Levn Asrac I s well known ha he wo smple algorhms for he classc n packng prolem, NF and WF oh have an approxmaon rao of

More information

SUPPLEMENT TO EFFICIENT DYNAMIC MECHANISMS IN ENVIRONMENTS WITH INTERDEPENDENT VALUATIONS: THE ROLE OF CONTINGENT TRANSFERS

SUPPLEMENT TO EFFICIENT DYNAMIC MECHANISMS IN ENVIRONMENTS WITH INTERDEPENDENT VALUATIONS: THE ROLE OF CONTINGENT TRANSFERS SUPPLEMENT TO EFFICIENT DYNAMIC MECHANISMS IN ENVIRONMENTS WITH INTERDEPENDENT VALUATIONS: THE ROLE OF CONTINGENT TRANSFERS HENG LIU In hs onlne appendx, we dscuss budge balance and surplus exracon n dynamc

More information

Chapter 6: AC Circuits

Chapter 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 information

Midterm Exam. Tuesday, September hour, 15 minutes

Midterm Exam. Tuesday, September hour, 15 minutes Ecoomcs of Growh, ECON560 Sa Fracsco Sae Uvers Mchael Bar Fall 203 Mderm Exam Tuesda, Sepember 24 hour, 5 mues Name: Isrucos. Ths s closed boo, closed oes exam. 2. No calculaors of a d are allowed. 3.

More information

Advanced Machine Learning & Perception

Advanced 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 information

GENERATING CERTAIN QUINTIC IRREDUCIBLE POLYNOMIALS OVER FINITE FIELDS. Youngwoo Ahn and Kitae Kim

GENERATING 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 information

Approximate Analytic Solution of (2+1) - Dimensional Zakharov-Kuznetsov(Zk) Equations Using Homotopy

Approximate 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 information

( ) [ ] MAP Decision Rule

( ) [ ] 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 information

1 Constant Real Rate C 1

1 Constant Real Rate C 1 Consan Real Rae. Real Rae of Inees Suppose you ae equally happy wh uns of he consumpon good oday o 5 uns of he consumpon good n peod s me. C 5 Tha means you ll be pepaed o gve up uns oday n eun fo 5 uns

More information

Oligopoly with exhaustible resource input

Oligopoly with exhaustible resource input Olgopoly wh exhausble resource npu e, P-Y. 78 Olgopoly wh exhausble resource npu Recebmeno dos orgnas: 25/03/202 Aceação para publcação: 3/0/203 Pu-yan e PhD em Scences pela Chnese Academy of Scence Insução:

More information