1 Bertrand duopoly with incomplete information
|
|
- Suzanna Gibbs
- 6 years ago
- Views:
Transcription
1 Game Theory Solution to Problem Set 5 1 Bertrand duopoly ith incomplete information The game i de ned by I = f1; g ; et of player A i = [0; 1) T i = fb L ; b H g, ith p(b L ) = u i (b i ; p i ; p j ) = (a pi b i p j )p i ; if p i 6 a b i p j 0 ; otherie : In addition, the trategy et S i (8i = 1; ) are the et of all function i : T i i, i.e. mapping from type into action. The ymmetric puretrategy BNE are the BNE i hich i (b i = b L ) = j (b j = b L ) and i (b i = b H ) = j (b j = b H ): No, in order to nd the ymmetric pure trategy BNE, rt uppoe that p i 6 a b i p j ; 8i = 1;. Since e are focuing on ymmetric equilibria, e have that: p i (b i ) = arg max p i (a pi b i p L j )p i + (1 )(a p i b i p H j )p i F OC : (a p i b i p L j ) p i + (1 )(a p i b i p H j ) (1 )p i = 0 a b i p L j p i (1 )b i p H j = 0 p i (b i ) = a b ip L j (1 )b i p H j No, impoing ymmetry and denoting p L and p H the equilibrium price of the lo type and high type repectively, e have that: p L = a b Lp L (1 )b L p H p L( + b L ) = a (1 )b L p H p L = a (1 )b Lp H ( + b L ) 1
2 Similarly, for the high type: p H = a b Hp L ( + (1 )b H ) Combining the expreion for p L and p H, e get: p L = a ( + (1 )(b H b L )) ( + (1 )b H + b L ) p H = a ( (b H b L )) ( + (1 )b H + b L ) (1) () If the equilibrium price and quantitie are all poitive (i.e. if e in fact have an interior olution), p L and p H a de ned above contitute a ymmetric BNE. Note that p L >0 alay hold, o the condition that have to hold for both price to be poitive i: (b H b L ) > 0: The quantitie are poitive if: q(p H; p H) = a p H b H p H > 0 q(p H; p L) = a p H b H p L > 0 q(p L; p H) = a p L b L p H > 0 q(p L; p L) = a p L b L p L > 0 Remember that b H > b L, and notice that p L > p H :Thi implie that if the inequality q(p H ; p L ) > 0 hold, the other three inequalitie ill alo hold. Hence, in order to check that the equilibrium quantitie are poitive, it i enough to check that the folloing hold: q(p H; p L) = a p H b H p L > 0 p H < a b H p L a ( (b H b L )) ( + (1 )b H + b L ) < a b H p L a ( (b H b L )) < a( + b H (b H b L )) b H a ( + (1 )(b H b L )) ( (b H b L )) < ( + b H (b H b L )) b H ( + b H b L (b H b L )) = ( b H )( + b H (b H b L )) + b L b H = ( b H )( (b H b L )) + b H ( b H + b L ) 0 < (1 b H )( (b H b L )) + b H ( b H + b L ) 0 < (1 b H )( + b H (b H b L )) + b H (1 + b L ) (1 b H )A + B if b H < 1 then q(p H; p L) > 0.
3 No, note that the expreion A i alay poitive, and o i B. Hence, if e aume that b H < 1 (hich i u cient but not neceary), e have that q(p H ; p L ) > 0, and expreion (1) and () decribe a ymmetric pure trategy BNE. A ar game T 1 = T = f; g ; p() = p() = 1 A 1 = A = fa; Ng 1 : T 1 1 (pure trategie) : T (pure trategie) Payo : t 1 = ; t = A 8; 8 10; 0 t 1 = ; t = A 8; 4 10; 0 t 1 = ; t = A 4; 8 10; 0 t 1 = ; t = A 6; 6 10; 0 The approach e take here to nding the pure trategy BNE i to look at all poible trategie (AA, AN, NA, NN) for one of the player, and check for each if it can be an equilibrium trategy. Firt, uppoe player follo the folloing trategy: = Then, e can derive the bet-repone of player 1: 9 Eu 1 (A; ; ) = 8 >= Eu 1 (N; ; ) = 0 ) BR Eu 1 (A; ; ) = 1 1 ( ) = >; Eu 1 (N; ; ) = 0 Similarly, if e uppoe that 1 =, the bet-repone for player i =. Hence, by the ymmetry of the game e have found to xed-point. That i, e have the folloing to pure trategy BNE: 3
4 i = ; j =, i f1; g No, uppoe player play according to: = Then, e can derive the bet-repone of player 1: 9 Eu 1 (A; ; ) = 1 >= Eu 1 (N; ; ) = 0 ) BR Eu 1 (A; ; ) = 1 ( ) = >; Eu 1 (N; ; ) = 0 But if 1 =, the bet-repone of player i BR ( 1 ) =. Hence, thi i not a BNE. No, uppoe player play according to: = Then, e can derive the bet-repone of player 1: 9 Eu 1 (A; ) = 1 >= Eu 1 (N; ) = 0 ) BR Eu 1 (A; ) = 7 1 ( ) = >; Eu 1 (N; ) = 0 But if 1 =, the bet-repone of player i BR ( 1 ) =. Hence, thi i not a BNE. To ummarize: the to BNE in pure trategie are: i = ; j =,for i; j f1; g ; i 6= j: 4
5 3 I Information alay bene cial? The game i decribed by: and the payo : T 1 = fa; bg ; p(a) = p(b) = 1 A 1 = fu; Dg ; A = fl; C; Rg t 1 = a t 1 = b L C R L C R U 4; 1 3; 0 3; 3 U 4; 1 3; 3 3; 0 D 5; 4 ; 5 ; 0 D 5; 4 ; 0 ; 5 Finally, a pure trategy for player 1 i a function 1 : T 1 1, herea a pure trategy for player i an A. Firt note that if e had aumed that thee ere to eparate game of complete information, e could have ued repeated deletion of trictly dominated trategie to nd unique pure trategy NE - (U, C) and (U, R) - for the game correponding to "a" and "b" repectively. In both of thee game, the equilibrium payo for player ould have been 0 (and the payo for player 1 ould have been 3). No, let u return to the incomplete information game decribed above. Note rt that no L trictly dominate both C and R for player. Thi implie that in any BNE, it mut be the cae that player play trategy L. Then, if player play L, the bet-repone for player 1 i trategy DD, i.e. to play D he /he i type a, and D hen /he i type b. Hence, e have found a unique BNE: = ( 1 ; ) = (DD; L): Finally, note that the equilibrium payo for player in the incomplete information game i 4 > 0. That i, if e believe that the olution concept ued here (NE and BNE) are appropriate a predictor of real-orld outcome, player ould actually bene t from having le information. 3.1 Alternative approach In order to illutrate di erent ay of looking for BNE, thi ubection preent an approach to quetion 3 that i not baed on the fact that C and R are trictly dominated by L. In thi cae, e need to conider the di erent poible trategie for player. The trategie for player can take the folloing form: L C 5
6 R LC (i.e., a randomization beteen L and C), LR CR LCR We need to check each of thee and ee if e can have a BNE on each particular form. Firt, uppoe S = L D S 1 = D + if t1 = a if t 1 = b Eu (L) = 1 f4 + 4g = 4 Eu (C) = Eu (R) = :5 hich implie that ((D; D) ; L) i a BNE. No, uppoe S = C U S 1 = U + Eu (L) = 1 if t1 = a if t 1 = b Eu (C) = Eu (R) = 1:5 Hence, BR (S 1 ) = L 6= C, o thi i not a BNE. Similarly, e can ho that player cannot play S = R in any BNE. Suppoe player mixe beteen fc; Rg :Since for both S = C and S = R, the bet repone of player 1 (both type) i U, hich mean that thi i te bet-repone alo to a trategy in hich player mixe beteen fc; Rg : But if player 1 play U, the bet repone of player i L: Thu, a mix beteen fc; Rg cannot be played in a BNE. Suppoe player mixe beteen fl; Cg ith probabilitie f; 1 g. For player to mix it mut be the cae that her expected payo of playing L and C are equal and greater than the expected utility of playing R. To compute the expected utility, e need to pecify hat player believe that the to type of player 1 ill do. Suppoe player believe that type a mixe beteen fu; Dg ith probability fp 1 ; 1 p 1 g and type b mixe beteen fu; Dg ith probability fp ; 1 p g :Then: 6
7 Eu (L) = 1 [p 1( 1) + (1 p 1 )4] + 1 [p ( 1) + (1 p )4] = 4 :5(p 1 + p ) Eu (C) = 1 ( p 1 3p ) = :5 :5p 1 1:5p Eu (R) = 1 ( 3p p ) = :5 :5p 1:5p 1 Hoever, then for Eu (L) = Eu (C) e mut have that p = 1:5; hich i impoible. A imilar analyi rule out a mix beteen fl; Rg. Finally, uppoe player i mixing among fl; C; Rg. A e have hoed, for player to mix beteen fl; Cg or fl; Rg, it mut be the cae that ome of the probabilitie de ned above exceed 1. Hence, it i impoible to induce player to mix beteen thee action. To ummarize, the unique BNE i ((D; D) ; L), here: Eu (DD; L) = 4 u 1 (DD; L j a) = 5 u 1 (DD; L j b) = 5: 7
CMSC 474, Introduction to Game Theory Maxmin and Minmax Strategies
CMSC 474, Introduction to Game Theory Maxmin and Minmax Strategie Mohammad T. Hajiaghayi Univerity of Maryland Wort-Cae Expected Utility For agent i, the wort-cae expected utility of a trategy i i the
More informationEconS Advanced Microeconomics II Handout on Bayesian Nash Equilibrium
EconS 503 - Avance icroeconomic II Hanout on Bayeian Nah Equilibrium 1. WG 8.E.1 Conier the following trategic ituation. Two oppoe armie are poie to eize an ilan. Each army general can chooe either "attack"
More information(b) Is the game below solvable by iterated strict dominance? Does it have a unique Nash equilibrium?
14.1 Final Exam Anwer all quetion. You have 3 hour in which to complete the exam. 1. (60 Minute 40 Point) Anwer each of the following ubquetion briefly. Pleae how your calculation and provide rough explanation
More informationSAMPLING. Sampling is the acquisition of a continuous signal at discrete time intervals and is a fundamental concept in real-time signal processing.
SAMPLING Sampling i the acquiition of a continuou ignal at dicrete time interval and i a fundamental concept in real-time ignal proceing. he actual ampling operation can alo be defined by the figure belo
More informationPhysics 741 Graduate Quantum Mechanics 1 Solutions to Final Exam, Fall 2014
Phyic 7 Graduate Quantum Mechanic Solution to inal Eam all 0 Each quetion i worth 5 point with point for each part marked eparately Some poibly ueful formula appear at the end of the tet In four dimenion
More informationSTOCHASTIC DIFFERENTIAL GAMES:THE LINEAR QUADRATIC ZERO SUM CASE
Sankhyā : he Indian Journal of Statitic 1995, Volume 57, Serie A, Pt. 1, pp.161 165 SOCHASIC DIFFERENIAL GAMES:HE LINEAR QUADRAIC ZERO SUM CASE By R. ARDANUY Univeridad de Salamanca SUMMARY. hi paper conider
More informationTechnical Appendix: Auxiliary Results and Proofs
A Technical Appendix: Auxiliary Reult and Proof Lemma A. The following propertie hold for q (j) = F r [c + ( ( )) ] de- ned in Lemma. (i) q (j) >, 8 (; ]; (ii) R q (j)d = ( ) q (j) + R q (j)d ; (iii) R
More informationConvex Hulls of Curves Sam Burton
Convex Hull of Curve Sam Burton 1 Introduction Thi paper will primarily be concerned with determining the face of convex hull of curve of the form C = {(t, t a, t b ) t [ 1, 1]}, a < b N in R 3. We hall
More informationGame Theory. Solutions to Problem Set 4
1 Hotelling s model 1.1 Two vendors Game Theory Solutions to Problem Set 4 Consider a strategy pro le (s 1 s ) with s 1 6= s Suppose s 1 < s In this case, it is pro table to for player 1 to deviate and
More informationLecture 21. The Lovasz splitting-off lemma Topics in Combinatorial Optimization April 29th, 2004
18.997 Topic in Combinatorial Optimization April 29th, 2004 Lecture 21 Lecturer: Michel X. Goeman Scribe: Mohammad Mahdian 1 The Lovaz plitting-off lemma Lovaz plitting-off lemma tate the following. Theorem
More informationPreemptive scheduling on a small number of hierarchical machines
Available online at www.ciencedirect.com Information and Computation 06 (008) 60 619 www.elevier.com/locate/ic Preemptive cheduling on a mall number of hierarchical machine György Dóa a, Leah Eptein b,
More informationNotes on Strategic Substitutes and Complements in Global Games
Note on Strategic Subtitute an Complement in Global Game Stephen Morri Cowle Founation, Yale Univerity, POBox 208281, New Haven CT 06520, U S A tephenmorri@yaleeu Hyun Song Shin Lonon School of Economic,
More informationOnline Appendix for Managerial Attention and Worker Performance by Marina Halac and Andrea Prat
Online Appendix for Managerial Attention and Worker Performance by Marina Halac and Andrea Prat Thi Online Appendix contain the proof of our reult for the undicounted limit dicued in Section 2 of the paper,
More informationTRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL
GLASNIK MATEMATIČKI Vol. 38583, 73 84 TRIPLE SOLUTIONS FOR THE ONE-DIMENSIONAL p-laplacian Haihen Lü, Donal O Regan and Ravi P. Agarwal Academy of Mathematic and Sytem Science, Beijing, China, National
More informationBogoliubov Transformation in Classical Mechanics
Bogoliubov Tranformation in Claical Mechanic Canonical Tranformation Suppoe we have a et of complex canonical variable, {a j }, and would like to conider another et of variable, {b }, b b ({a j }). How
More informationFlag-transitive non-symmetric 2-designs with (r, λ) = 1 and alternating socle
Flag-tranitive non-ymmetric -deign with (r, λ = 1 and alternating ocle Shenglin Zhou, Yajie Wang School of Mathematic South China Univerity of Technology Guangzhou, Guangdong 510640, P. R. China lzhou@cut.edu.cn
More informationSocial Studies 201 Notes for November 14, 2003
1 Social Studie 201 Note for November 14, 2003 Etimation of a mean, mall ample ize Section 8.4, p. 501. When a reearcher ha only a mall ample ize available, the central limit theorem doe not apply to the
More informationAn Inequality for Nonnegative Matrices and the Inverse Eigenvalue Problem
An Inequality for Nonnegative Matrice and the Invere Eigenvalue Problem Robert Ream Program in Mathematical Science The Univerity of Texa at Dalla Box 83688, Richardon, Texa 7583-688 Abtract We preent
More informationComputers and Mathematics with Applications. Sharp algebraic periodicity conditions for linear higher order
Computer and Mathematic with Application 64 (2012) 2262 2274 Content lit available at SciVere ScienceDirect Computer and Mathematic with Application journal homepage: wwweleviercom/locate/camwa Sharp algebraic
More informationAssignment for Mathematics for Economists Fall 2016
Due date: Mon. Nov. 1. Reading: CSZ, Ch. 5, Ch. 8.1 Aignment for Mathematic for Economit Fall 016 We now turn to finihing our coverage of concavity/convexity. There are two part: Jenen inequality for concave/convex
More informationQuantitative Information Leakage. Lecture 9
Quantitative Information Leakage Lecture 9 1 The baic model: Sytem = Information-Theoretic channel Secret Information Obervable 1 o1... Sytem... m on Input Output 2 Toward a quantitative notion of leakage
More informationLecture Notes on Game Theory
Lecture Notes on Game Theory Levent Koçkesen Strategic Form Games In this part we will analyze games in which the players choose their actions simultaneously (or without the knowledge of other players
More informationChapter 4. The Laplace Transform Method
Chapter 4. The Laplace Tranform Method The Laplace Tranform i a tranformation, meaning that it change a function into a new function. Actually, it i a linear tranformation, becaue it convert a linear combination
More informationFermi Distribution Function. n(e) T = 0 T > 0 E F
LECTURE 3 Maxwell{Boltzmann, Fermi, and Boe Statitic Suppoe we have a ga of N identical point particle in a box ofvolume V. When we ay \ga", we mean that the particle are not interacting with one another.
More information2 where. x 1 θ = 1 θ = 0.6 x 2 θ = 0 θ = 0.2
Convex et I a ne and convex et I ome important example I operation that preerve convexity I eparating and upporting hyperplane I generalized inequalitie I dual cone and generalized inequalitie IOE 6: Nonlinear
More informationMATEMATIK Datum: Tid: eftermiddag. A.Heintz Telefonvakt: Anders Martinsson Tel.:
MATEMATIK Datum: 20-08-25 Tid: eftermiddag GU, Chalmer Hjälpmedel: inga A.Heintz Telefonvakt: Ander Martinon Tel.: 073-07926. Löningar till tenta i ODE och matematik modellering, MMG5, MVE6. Define what
More informationSocial Studies 201 Notes for March 18, 2005
1 Social Studie 201 Note for March 18, 2005 Etimation of a mean, mall ample ize Section 8.4, p. 501. When a reearcher ha only a mall ample ize available, the central limit theorem doe not apply to the
More informationSuggested Answers To Exercises. estimates variability in a sampling distribution of random means. About 68% of means fall
Beyond Significance Teting ( nd Edition), Rex B. Kline Suggeted Anwer To Exercie Chapter. The tatitic meaure variability among core at the cae level. In a normal ditribution, about 68% of the core fall
More informationDigital Control System
Digital Control Sytem - A D D A Micro ADC DAC Proceor Correction Element Proce Clock Meaurement A: Analog D: Digital Continuou Controller and Digital Control Rt - c Plant yt Continuou Controller Digital
More informationLecture Notes on Game Theory
Lecture Notes on Game Theory Levent Koçkesen 1 Bayesian Games So far we have assumed that all players had perfect information regarding the elements of a game. These are called games with complete information.
More informationNCAAPMT Calculus Challenge Challenge #3 Due: October 26, 2011
NCAAPMT Calculu Challenge 011 01 Challenge #3 Due: October 6, 011 A Model of Traffic Flow Everyone ha at ome time been on a multi-lane highway and encountered road contruction that required the traffic
More informationTheoretical Computer Science. Optimal algorithms for online scheduling with bounded rearrangement at the end
Theoretical Computer Science 4 (0) 669 678 Content lit available at SciVere ScienceDirect Theoretical Computer Science journal homepage: www.elevier.com/locate/tc Optimal algorithm for online cheduling
More informationSOME RESULTS ON INFINITE POWER TOWERS
NNTDM 16 2010) 3, 18-24 SOME RESULTS ON INFINITE POWER TOWERS Mladen Vailev - Miana 5, V. Hugo Str., Sofia 1124, Bulgaria E-mail:miana@abv.bg Abtract To my friend Kratyu Gumnerov In the paper the infinite
More informationCodes Correcting Two Deletions
1 Code Correcting Two Deletion Ryan Gabry and Frederic Sala Spawar Sytem Center Univerity of California, Lo Angele ryan.gabry@navy.mil fredala@ucla.edu Abtract In thi work, we invetigate the problem of
More information7.2 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 281
72 INVERSE TRANSFORMS AND TRANSFORMS OF DERIVATIVES 28 and i 2 Show how Euler formula (page 33) can then be ued to deduce the reult a ( a) 2 b 2 {e at co bt} {e at in bt} b ( a) 2 b 2 5 Under what condition
More informationOne Class of Splitting Iterative Schemes
One Cla of Splitting Iterative Scheme v Ciegi and V. Pakalnytė Vilniu Gedimina Technical Univerity Saulėtekio al. 11, 2054, Vilniu, Lithuania rc@fm.vtu.lt Abtract. Thi paper deal with the tability analyi
More informationON THE APPROXIMATION ERROR IN HIGH DIMENSIONAL MODEL REPRESENTATION. Xiaoqun Wang
Proceeding of the 2008 Winter Simulation Conference S. J. Maon, R. R. Hill, L. Mönch, O. Roe, T. Jefferon, J. W. Fowler ed. ON THE APPROXIMATION ERROR IN HIGH DIMENSIONAL MODEL REPRESENTATION Xiaoqun Wang
More informationLecture 7: Testing Distributions
CSE 5: Sublinear (and Streaming) Algorithm Spring 014 Lecture 7: Teting Ditribution April 1, 014 Lecturer: Paul Beame Scribe: Paul Beame 1 Teting Uniformity of Ditribution We return today to property teting
More informationControl Systems Analysis and Design by the Root-Locus Method
6 Control Sytem Analyi and Deign by the Root-Locu Method 6 1 INTRODUCTION The baic characteritic of the tranient repone of a cloed-loop ytem i cloely related to the location of the cloed-loop pole. If
More informationNon-myopic Strategies in Prediction Markets
Non-myopic Strategie in Prediction Market Stanko Dimitrov Department of Indutrial and Operation Engineering Univerity of Michigan, 205 Beal Avenue, Ann Arbor, MI 4809-27, USA dimitro@umich.edu Rahul Sami
More informationLecture 10 Filtering: Applied Concepts
Lecture Filtering: Applied Concept In the previou two lecture, you have learned about finite-impule-repone (FIR) and infinite-impule-repone (IIR) filter. In thee lecture, we introduced the concept of filtering
More informationParametrization of the 511 kev respond in BGO/ LSO Crystals with respect to Spatial Resolution in PETR/CT Scans
Univerity of Tenneee, Knoxville Trace: Tenneee Reearch and Creative Exchange Univerity of Tenneee Honor Thei Project Univerity of Tenneee Honor Program 3-2005 Parametrization of the 511 kev repond in BGO/
More informationCHAPTER 8 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS
CHAPTER 8 OBSERVER BASED REDUCED ORDER CONTROLLER DESIGN FOR LARGE SCALE LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.1 INTRODUCTION 8.2 REDUCED ORDER MODEL DESIGN FOR LINEAR DISCRETE-TIME CONTROL SYSTEMS 8.3
More informationOnline Appendix for Corporate Control Activism
Online Appendix for Corporate Control Activim B Limited veto power and tender offer In thi ection we extend the baeline model by allowing the bidder to make a tender offer directly to target hareholder.
More informationExample: Amplifier Distortion
4/6/2011 Example Amplifier Ditortion 1/9 Example: Amplifier Ditortion Recall thi circuit from a previou handout: 15.0 R C =5 K v ( t) = v ( t) o R B =5 K β = 100 _ vi( t ) 58. R E =5 K CUS We found that
More informationThe Informativeness Principle Under Limited Liability
The Informativene Principle Under Limited Liability Pierre Chaigneau HEC Montreal Alex Edman LBS, Wharton, NBER, CEPR, and ECGI Daniel Gottlieb Wharton Augut 7, 4 Abtract Thi paper how that the informativene
More informationin a circular cylindrical cavity K. Kakazu Department of Physics, University of the Ryukyus, Okinawa , Japan Y. S. Kim
Quantization of electromagnetic eld in a circular cylindrical cavity K. Kakazu Department of Phyic, Univerity of the Ryukyu, Okinawa 903-0, Japan Y. S. Kim Department of Phyic, Univerity of Maryland, College
More informationSecretary problems with competing employers
Secretary problem with competing employer Nicole Immorlica 1, Robert Kleinberg 2, and Mohammad Mahdian 1 1 Microoft Reearch, One Microoft Way, Redmond, WA. {nickle,mahdian}@microoft.com 2 UC Berkeley Computer
More informationUNIQUE CONTINUATION FOR A QUASILINEAR ELLIPTIC EQUATION IN THE PLANE
UNIQUE CONTINUATION FOR A QUASILINEAR ELLIPTIC EQUATION IN THE PLANE SEPPO GRANLUND AND NIKO MAROLA Abtract. We conider planar olution to certain quailinear elliptic equation ubject to the Dirichlet boundary
More informationChapter K - Problems
Chapter K - Problem Blinn College - Phyic 2426 - Terry Honan Problem K. A He-Ne (helium-neon) laer ha a wavelength of 632.8 nm. If thi i hot at an incident angle of 55 into a gla block with index n =.52
More informationProblem Set 8 Solutions
Deign and Analyi of Algorithm April 29, 2015 Maachuett Intitute of Technology 6.046J/18.410J Prof. Erik Demaine, Srini Devada, and Nancy Lynch Problem Set 8 Solution Problem Set 8 Solution Thi problem
More informationSOLUTIONS TO ALGEBRAIC GEOMETRY AND ARITHMETIC CURVES BY QING LIU. I will collect my solutions to some of the exercises in this book in this document.
SOLUTIONS TO ALGEBRAIC GEOMETRY AND ARITHMETIC CURVES BY QING LIU CİHAN BAHRAN I will collect my olution to ome of the exercie in thi book in thi document. Section 2.1 1. Let A = k[[t ]] be the ring of
More informationIEOR 3106: Fall 2013, Professor Whitt Topics for Discussion: Tuesday, November 19 Alternating Renewal Processes and The Renewal Equation
IEOR 316: Fall 213, Profeor Whitt Topic for Dicuion: Tueday, November 19 Alternating Renewal Procee and The Renewal Equation 1 Alternating Renewal Procee An alternating renewal proce alternate between
More informationMarkus A. Hitz and Erich Kaltofen. Adaption to symmetric RNS is not dicult, but destroys. some of the clarity in the descriptions.
Integer Diviion in eidue Number Sytem arku A. Hitz and Erich Kaltofen Abtract Thi contribution to the ongoing dicuion of diviion algorithm for reidue number ytem (NS) i baed on Neton iteration for computing
More informationMAE140 Linear Circuits Fall 2012 Final, December 13th
MAE40 Linear Circuit Fall 202 Final, December 3th Intruction. Thi exam i open book. You may ue whatever written material you chooe, including your cla note and textbook. You may ue a hand calculator with
More informationOptimal Coordination of Samples in Business Surveys
Paper preented at the ICES-III, June 8-, 007, Montreal, Quebec, Canada Optimal Coordination of Sample in Buine Survey enka Mach, Ioana Şchiopu-Kratina, Philip T Rei, Jean-Marc Fillion Statitic Canada New
More informationUNIT # 07 (PART - II)
9. ph ph [H + 0. [H + 0.0 V 50 V 50 [H + of miture i[h + N V N V V V 50(0. 0.0) 00 [H + 0. 0.055 ph.6. ph 7 [H + 0 7, [H 0 7 ne ph fter ddition of be ph [H + 0 [H 0 [H + concentrtion incree 0 5 time..
More information1. The F-test for Equality of Two Variances
. The F-tet for Equality of Two Variance Previouly we've learned how to tet whether two population mean are equal, uing data from two independent ample. We can alo tet whether two population variance are
More informationEE 508 Lecture 16. Filter Transformations. Lowpass to Bandpass Lowpass to Highpass Lowpass to Band-reject
EE 508 Lecture 6 Filter Tranformation Lowpa to Bandpa Lowpa to Highpa Lowpa to Band-reject Review from Lat Time Theorem: If the perimeter variation and contact reitance are neglected, the tandard deviation
More informationON A CERTAIN FAMILY OF QUARTIC THUE EQUATIONS WITH THREE PARAMETERS. Volker Ziegler Technische Universität Graz, Austria
GLASNIK MATEMATIČKI Vol. 1(61)(006), 9 30 ON A CERTAIN FAMILY OF QUARTIC THUE EQUATIONS WITH THREE PARAMETERS Volker Ziegler Techniche Univerität Graz, Autria Abtract. We conider the parameterized Thue
More informationThe Secret Life of the ax + b Group
The Secret Life of the ax + b Group Linear function x ax + b are prominent if not ubiquitou in high chool mathematic, beginning in, or now before, Algebra I. In particular, they are prime exhibit in any
More informationFractional Hedonic Games: Individual and Group Stability
Fractional Hedonic Game: Individual and Group Stability Florian Brandl Intitut für Informatik TU München, Germany brandlfl@in.tum.de Felix Brandt Intitut für Informatik TU München, Germany brandtf@in.tum.de
More informationEE 508 Lecture 16. Filter Transformations. Lowpass to Bandpass Lowpass to Highpass Lowpass to Band-reject
EE 508 Lecture 6 Filter Tranformation Lowpa to Bandpa Lowpa to Highpa Lowpa to Band-reject Review from Lat Time Theorem: If the perimeter variation and contact reitance are neglected, the tandard deviation
More informationEXTENDED STABILITY MARGINS ON CONTROLLER DESIGN FOR NONLINEAR INPUT DELAY SYSTEMS. Otto J. Roesch, Hubert Roth, Asif Iqbal
EXTENDED STABILITY MARGINS ON CONTROLLER DESIGN FOR NONLINEAR INPUT DELAY SYSTEMS Otto J. Roech, Hubert Roth, Aif Iqbal Intitute of Automatic Control Engineering Univerity Siegen, Germany {otto.roech,
More informationDetermination of the local contrast of interference fringe patterns using continuous wavelet transform
Determination of the local contrat of interference fringe pattern uing continuou wavelet tranform Jong Kwang Hyok, Kim Chol Su Intitute of Optic, Department of Phyic, Kim Il Sung Univerity, Pyongyang,
More informationSuggestions - Problem Set (a) Show the discriminant condition (1) takes the form. ln ln, # # R R
Suggetion - Problem Set 3 4.2 (a) Show the dicriminant condition (1) take the form x D Ð.. Ñ. D.. D. ln ln, a deired. We then replace the quantitie. 3ß D3 by their etimate to get the proper form for thi
More informationMath 273 Solutions to Review Problems for Exam 1
Math 7 Solution to Review Problem for Exam True or Fale? Circle ONE anwer for each Hint: For effective tudy, explain why if true and give a counterexample if fale (a) T or F : If a b and b c, then a c
More informationZbigniew Dziong Department of Electrical Engineering, Ecole de Technologie Superieure 1100 Notre-Dame Street West, Montreal, Quebec, Canada H3C 1k3
Capacity Allocation in Serice Oerlay Network - Maximum Profit Minimum Cot Formulation Ngok Lam Department of Electrical and Computer Engineering, McGill Unierity 3480 Unierity Street, Montreal, Quebec,
More informationON THE EQUIVALENCE BETWEEN SUBGAME PERFECTION AND SEQUENTIALITY * J. Carlos González-Pimienta 1 y Cristian M. Litan 2
Working Paper 05-26 Economic Serie 16 April 2005 Departamento de Economía Univeridad Carlo III de Madrid Calle Madrid, 126 28903 Getafe (Spain) Fax (34) 91 624 98 75 ON THE EQUIVALENCE BETWEEN SUBGAME
More informationInference for Two Stage Cluster Sampling: Equal SSU per PSU. Projections of SSU Random Variables on Each SSU selection.
Inference for Two Stage Cluter Sampling: Equal SSU per PSU Projection of SSU andom Variable on Eac SSU election By Ed Stanek Introduction We review etimating equation for PSU mean in a two tage cluter
More informationRELIABILITY OF REPAIRABLE k out of n: F SYSTEM HAVING DISCRETE REPAIR AND FAILURE TIMES DISTRIBUTIONS
www.arpapre.com/volume/vol29iue1/ijrras_29_1_01.pdf RELIABILITY OF REPAIRABLE k out of n: F SYSTEM HAVING DISCRETE REPAIR AND FAILURE TIMES DISTRIBUTIONS Sevcan Demir Atalay 1,* & Özge Elmataş Gültekin
More informationDesign of Digital Filters
Deign of Digital Filter Paley-Wiener Theorem [ ] ( ) If h n i a caual energy ignal, then ln H e dω< B where B i a finite upper bound. One implication of the Paley-Wiener theorem i that a tranfer function
More informationChapter 4 Conflict Resolution and Sector Workload Formulations
Equation Chapter 4 Section Chapter 4 Conflict Reolution and Sector Workload Formulation 4.. Occupancy Workload Contraint Formulation We tart by examining the approach ued by Sherali, Smith, and Trani [47]
More informationME2142/ME2142E Feedback Control Systems
Root Locu Analyi Root Locu Analyi Conider the cloed-loop ytem R + E - G C B H The tranient repone, and tability, of the cloed-loop ytem i determined by the value of the root of the characteritic equation
More informationonline learning Unit Workbook 4 RLC Transients
online learning Pearon BTC Higher National in lectrical and lectronic ngineering (QCF) Unit 5: lectrical & lectronic Principle Unit Workbook 4 in a erie of 4 for thi unit Learning Outcome: RLC Tranient
More informationStochastic Neoclassical Growth Model
Stochatic Neoclaical Growth Model Michael Bar May 22, 28 Content Introduction 2 2 Stochatic NGM 2 3 Productivity Proce 4 3. Mean........................................ 5 3.2 Variance......................................
More informationarxiv: v3 [quant-ph] 23 Nov 2011
Generalized Bell Inequality Experiment and Computation arxiv:1108.4798v3 [quant-ph] 23 Nov 2011 Matty J. Hoban, 1, 2 Joel J. Wallman, 3 and Dan E. Browne 1 1 Department of Phyic and Atronomy, Univerity
More informationIntroduction to Laplace Transform Techniques in Circuit Analysis
Unit 6 Introduction to Laplace Tranform Technique in Circuit Analyi In thi unit we conider the application of Laplace Tranform to circuit analyi. A relevant dicuion of the one-ided Laplace tranform i found
More informationContracting with private knowledge of signal quality
RAND Journal of Economic Vol. 41, No. 2, Summer 2010 pp. 244 269 Contracting with private knowledge of ignal quality Leon Yang Chu and David E. M. Sappington We characterize the optimal procurement contract
More informationConvergence criteria and optimization techniques for beam moments
Pure Appl. Opt. 7 (1998) 1221 1230. Printed in the UK PII: S0963-9659(98)90684-5 Convergence criteria and optimization technique for beam moment G Gbur and P S Carney Department of Phyic and Atronomy and
More informationWeek 3 Statistics for bioinformatics and escience
Week 3 Statitic for bioinformatic and escience Line Skotte 28. november 2008 2.9.3-4) In thi eercie we conider microrna data from Human and Moue. The data et repreent 685 independent realiation of the
More informationLinear System Fundamentals
Linear Sytem Fundamental MEM 355 Performance Enhancement of Dynamical Sytem Harry G. Kwatny Department of Mechanical Engineering & Mechanic Drexel Univerity Content Sytem Repreentation Stability Concept
More informationON A CERTAIN FAMILY OF QUARTIC THUE EQUATIONS WITH THREE PARAMETERS
ON A CERTAIN FAMILY OF QUARTIC THUE EQUATIONS WITH THREE PARAMETERS VOLKER ZIEGLER Abtract We conider the parameterized Thue equation X X 3 Y (ab + (a + bx Y abxy 3 + a b Y = ±1, where a, b 1 Z uch that
More informationCorrection for Simple System Example and Notes on Laplace Transforms / Deviation Variables ECHE 550 Fall 2002
Correction for Simple Sytem Example and Note on Laplace Tranform / Deviation Variable ECHE 55 Fall 22 Conider a tank draining from an initial height of h o at time t =. With no flow into the tank (F in
More informationCHAPTER 6. Estimation
CHAPTER 6 Etimation Definition. Statitical inference i the procedure by which we reach a concluion about a population on the bai of information contained in a ample drawn from that population. Definition.
More informationLinear Motion, Speed & Velocity
Add Important Linear Motion, Speed & Velocity Page: 136 Linear Motion, Speed & Velocity NGSS Standard: N/A MA Curriculum Framework (006): 1.1, 1. AP Phyic 1 Learning Objective: 3.A.1.1, 3.A.1.3 Knowledge/Undertanding
More informationEE Control Systems LECTURE 14
Updated: Tueday, March 3, 999 EE 434 - Control Sytem LECTURE 4 Copyright FL Lewi 999 All right reerved ROOT LOCUS DESIGN TECHNIQUE Suppoe the cloed-loop tranfer function depend on a deign parameter k We
More informationEconS Microeconomic Theory II Midterm Exam #2 - Answer Key
EconS 50 - Microeconomic Theory II Midterm Exam # - Answer Key 1. Revenue comparison in two auction formats. Consider a sealed-bid auction with bidders. Every bidder i privately observes his valuation
More informationFeedback Control Systems (FCS)
Feedback Control Sytem (FCS) Lecture19-20 Routh-Herwitz Stability Criterion Dr. Imtiaz Huain email: imtiaz.huain@faculty.muet.edu.pk URL :http://imtiazhuainkalwar.weebly.com/ Stability of Higher Order
More informationResearch Article Fixed Points and Stability in Nonlinear Equations with Variable Delays
Hindawi Publihing Corporation Fixed Point Theory and Application Volume 21, Article ID 195916, 14 page doi:1.1155/21/195916 Reearch Article Fixed Point and Stability in Nonlinear Equation with Variable
More informationTHE IDENTIFICATION OF THE OPERATING REGIMES OF THE CONTROLLERS BY THE HELP OF THE PHASE TRAJECTORY
Mariu M. B LA Aurel Vlaicu Univerity of Arad, Engineering Faculty Bd. Revolu iei nr. 77, 3030, Arad, Romania, E-mail: mariu.bala@ieee.org THE IDENTIFICATION OF THE OPERATING REGIMES OF THE CONTROLLERS
More informationc n b n 0. c k 0 x b n < 1 b k b n = 0. } of integers between 0 and b 1 such that x = b k. b k c k c k
1. Exitence Let x (0, 1). Define c k inductively. Suppoe c 1,..., c k 1 are already defined. We let c k be the leat integer uch that x k An eay proof by induction give that and for all k. Therefore c n
More information5. Fuzzy Optimization
5. Fuzzy Optimization 1. Fuzzine: An Introduction 135 1.1. Fuzzy Memberhip Function 135 1.2. Memberhip Function Operation 136 2. Optimization in Fuzzy Environment 136 3. Fuzzy Set for Water Allocation
More informationPythagorean Triple Updated 08--5 Drlnoordzij@leennoordzijnl wwwleennoordzijme Content A Roadmap for generating Pythagorean Triple Pythagorean Triple 3 Dicuion Concluion 5 A Roadmap for generating Pythagorean
More informationMulti-dimensional Fuzzy Euler Approximation
Mathematica Aeterna, Vol 7, 2017, no 2, 163-176 Multi-dimenional Fuzzy Euler Approximation Yangyang Hao College of Mathematic and Information Science Hebei Univerity, Baoding 071002, China hdhyywa@163com
More informationReputation and Multiproduct-firm Behavior: Product line and Price Rivalry Among Retailers
Reputation and Multiproduct-firm Behavior: Product line and Price Rivalry Among Retailer Shaoyan Sun and Henry An Department of Reource Economic and Environmental Sociology, Univerity of Alberta, Canada
More informationIn-Class Problem 5: Newton s Laws of Motion
In-Cla Problem 5: Neton La of Motion Conider a trac ith a pulley located at the end. The force enor and cart have total ma m 1. They are connected by a inextenible rope of length l (paing over the pulley)
More informationThe Laplace Transform , Haynes Miller and Jeremy Orloff
The Laplace Tranform 8.3, Hayne Miller and Jeremy Orloff Laplace tranform baic: introduction An operator take a function a input and output another function. A tranform doe the ame thing with the added
More informationCOS 511: Theoretical Machine Learning. Lecturer: Rob Schapire Lecture # 12 Scribe: Indraneel Mukherjee March 12, 2008
COS 511: Theoretical Machine Learning Lecturer: Rob Schapire Lecture # 12 Scribe: Indraneel Mukherjee March 12, 2008 In the previous lecture, e ere introduced to the SVM algorithm and its basic motivation
More informationIf Y is normally Distributed, then and 2 Y Y 10. σ σ
ull Hypothei Significance Teting V. APS 50 Lecture ote. B. Dudek. ot for General Ditribution. Cla Member Uage Only. Chi-Square and F-Ditribution, and Diperion Tet Recall from Chapter 4 material on: ( )
More information