Monopoly Part III: Multiple Local Equilibria and Adaptive Search
|
|
- Brook Flynn
- 5 years ago
- Views:
Transcription
1 FH-Kiel University of Applie Sciences Prof Dr Anreas Thiemer, 00 Monopoly Part III: Multiple Local Equilibria an Aaptive Search Summary: The information about market eman neee by a profit maimizing monopolist is rather more comple an costly than that neee by a "price taker" in a perfect competitive market: The monopolist must know at least the entire eman curve instea of just one point on it In tet book cases the eman curve is a simple ownwar sloping straight line But in her pioneering work Joan Robinson 9 has alreay remarke that "cases of multiple equilibrium may arise when the eman curve changes its slope, being highly elastic for a stretch, then perhaps becoming relatively inelastic, then elastic again" This may happen if the eman is a composite of the eman from several subgroups of consumers, with ifferent elasticities an specific reservation prices Puu (997) further assume, that the monopolist oesn't know the eman function Hunting for a profit maimum, the firm follows an aaptive search proceure just using the information about recent reaction of profits to supply changes Such an output rule may cause perioic cycles an chaotic behaviour of the monopolistic supply This worksheet allows you to eplore the comple ynamics of Puu's moel in epth A Thiemer, 00 monopolymc,60006
2 I The case with one prouct Basic assumptions The (inverse) eman function is p( ) α 0 α α α where p enotes commoity price, enotes quantity emane an α i represents (positive) constants: α α 7 0 α 6 00 α 0 This function is own-sloping an thus invertible provie that the following conition hols (=): α < α α = Because the total revenue is R() = p(), the marginal revenue must be: R' ( ) p( ) vereinfachen To provie an opportunity of multiple equilibria, the conition for an upwar slope at the point of inflection of the marginal revenue curve must hol (=): 8 α α < α = Net we efine the total cost as a function of the quantity supplie: Κ ( ) β 0 β β β As positive constants we take: β 0 0 β β 0 β 0 Therefore, the marginal cost K/ must be: Κ' ( ) Κ ( ) 0 The profit function is compute by Ψ ( ) p( ) Κ ( ) vereinfachen A Thiemer, 00 monopolymc,60006
3 The first orer conition (FOC) of a profit maimum is Ψ' ( ) 0 We etermine the solution FOC for this equation: Compute the marginal profit: Ψ' ( ) Ψ ( ) Collect the coefficients of this polynom: coeff Ψ' ( ) koeff, Solve the polynom: FOC nullstellen( coeff) FOC = 47 Ψ FOC = 8 Let's check for the secon orer conition: Compute the secon erivative of the profit function: Ψ'' ( ) 4 Ψ ( ) 8 Search for local maima where Ψ'' FOC < 0 : Ψ'' FOC = 06 A Thiemer, 00 monopolymc,60006
4 Now take a look to the figures below to verify the results from above: ma 6 0, ma ma 000 Profit function Ψ( ) Deman, marginal revenue & cost p ( ) R' ( ) 4 Κ' ( ) A Thiemer, 00 4 monopolymc,60006
5 Introucing aaptive search Assume that the monopolist oesn't know more than a few points on the eman function This information woul be of local character an short lifetimethe monopolist might not even know that globally there are two istinct profit maima Given this, the monopolist follows a simple search algorithm (in the vein of Newton) for the maimum of the unknown profit function: He estimates the ifference of profits from the last two visite points an Then he uses a given step length δ > 0 to move his supply in the irection of increasing profits: step size: δ 6 initial values: time Τ maimum of perios compute: Τ 00 Ψ Ψ δ time Τ 0 Time series of supplie quantity time The search proceure from above escribes a ifference equation of secon orer We can ecompose this in a system of two first orer equations by efining: y y time Τ y time y time Ψ y time Ψ y time δ y time A Thiemer, 00 monopolymc,60006
6 Hence, we raw the trajectory for this iterate map: Trajectory y time Fie points, cycles an chaos Not unepecte, the maima an the minimum of the profit function are fie points of the iterate map given by y time an Ψ' 0 Of course the minimum must be an unstable fie point To fin out the stability of the maima we reformulate the iterate map to use Mathca's symbolic processor: fxy (, ) Y g( X, Y) Y δ Ψ ( Y) Ψ ( X) Y X Now we are able to evaluate the Jacobian: Jacobian( X) X fxy (, ) X g( X, Y) Y fxy (, ) Y g( X, Y) ersetzen, Y X vereinfachen 0 δ X 6X 8 A Thiemer, 00 6 monopolymc,60006
7 Loss of stability occurs when the Jacobian is unitary Thus, we get as the critical value δ c of the step with: δ c ( X) Jacobian( X) auflösen, δ 0 X 6X δ c FOC = Plotting the phase variable against the parameter δ in a Feigenbaum iagram, we see the alternative fie points coeist at δ< δ c After that they are replace by cycles (Note: This cycles are of perio 4 with one value taken on twice as often as are the other two!) Chaos takes over once this cycles lose stability Puu (997, p -6) proves that this happens if the value of the step parameter ecees 488 At first there appears chaotic bans, with each its own basin of attraction After a certain point the attractors merge in a single one To run the bifurcation plot you must enter a resolution number RES (Note: High resolution is very time consuming) Resolution (RES=,,,0): RES 0 range of δ: δ bot δ top range of : bot 0 top 6 6 Feigenbaum iagram 4 0 parameter of step with A Thiemer, 00 7 monopolymc,60006
8 4 Attractors The net figure plots attractors of this system into the phase plane Change δ using the ifferent values from above to see fie points, peroic cycles an two coeistent chaotic attractors merging to a single one Note the symmetry of the shapes of the attractors For symmetric systems any attractors that are not symmetric in themselves come pairwise, forming together a symmetric picture δ 8 6 Attractors 4 y A Thiemer, 00 8 monopolymc,60006
9 II The case with two complementary proucts Suppose the monopolist supplies two goos an y that are complementary in consumers perferences Both goos have eman functions similar to the one in case I But now the proucts of sales volumes y respectively y enter as a positive term in both eman functions This moels the "coupling effect" ue to complementary Puu uses the same cost function K(+y) from above, stating the moifie profit function Π(,y) with the "coupling" parameter κ as: Π (, y, κ) Ψ ( ) Ψ ( y) κ y y Again it will be assume, that the firm hunts with an aaptive search proceure for the profit maimum To simulate this behaviour we nee the graient of profits: Π (, y, κ) Π (, y, κ) κ y y Π y (, y, κ) y Π (, y, κ) 8 4 y 9 y y κ y Given a small value of κ (for eample 000) we obtain a profit function with four ifferent local maima: κ 00 Profit function Iso profit lines profit profit A Thiemer, 00 9 monopolymc,60006
10 To start the aaptive search proceure we set step size: δ 7 initial values: 9 y time Τ maimum of perios compute: Τ 0000 y time δπ, y time, κ y time δπ y, y time, κ time 00 8 Supply of an y 6 y time t start 000 time t start Τ time Attractor 6 y time 4 y , A Thiemer, 00 0 monopolymc,60006
11 Now run this system with increasing step length δ First you will fin coeistent stable fie points After their loss of stability these fie points are replace by perioic cycles an then chaotic attractors You will also fin coeistent cycles an chaotic attractors at once It epens on the initial conitions embee in ifferent basins of attraction, to which one the process settles Here are some eamples of parameter combinations you shoul try (initial conitions are always 9 an y ): Case I: Lower coupling of the proucts with κ 000 δ fie point in NW δ cycle with perio in NW δ perio oubling δ chaotic attractor ("two stripes" in NW) δ chaotic attractor ("prouct set" in NW) δ 6 chaotic attractor ("prouct set" in SW) δ 7 chaotic attractor (epansion of the "prouct set" with sparse points) δ 8 chaotic attractor (epane "prouct set" with sparse points only in NW) Case II: Meium coupling of the proucts with κ 0006 δ chaotic "Henon"-like attractor in NW δ 4 perioic cycles in SW δ chaotic attractor in SW δ 7 chaotic attractor from SW epaning to SE an NW Case III: Strong coupling of the proucts with κ 00 δ 7 strange attractor in NE δ 4 fie point in SW Literature: Puu, T: Nonlinear Economic Dynamics Berlin/Heielberg 997, - Robinson, J: The Economics of Imperfect Competition n e, Lonon 969 A Thiemer, 00 monopolymc,60006
12 ORIGIN A Thiemer, 00 monopolymc,60006
A Simple Exchange Economy with Complex Dynamics
FH-Kiel Universitf Alie Sciences Prof. Dr. Anreas Thiemer, 00 e-mail: anreas.thiemer@fh-kiel.e A Simle Exchange Economy with Comlex Dynamics (Revision: Aril 00) Summary: Mukherji (999) shows that a stanar
More informationLecture 6: Calculus. In Song Kim. September 7, 2011
Lecture 6: Calculus In Song Kim September 7, 20 Introuction to Differential Calculus In our previous lecture we came up with several ways to analyze functions. We saw previously that the slope of a linear
More informationSYNCHRONOUS SEQUENTIAL CIRCUITS
CHAPTER SYNCHRONOUS SEUENTIAL CIRCUITS Registers an counters, two very common synchronous sequential circuits, are introuce in this chapter. Register is a igital circuit for storing information. Contents
More information6 General properties of an autonomous system of two first order ODE
6 General properties of an autonomous system of two first orer ODE Here we embark on stuying the autonomous system of two first orer ifferential equations of the form ẋ 1 = f 1 (, x 2 ), ẋ 2 = f 2 (, x
More informationCalculus in the AP Physics C Course The Derivative
Limits an Derivatives Calculus in the AP Physics C Course The Derivative In physics, the ieas of the rate change of a quantity (along with the slope of a tangent line) an the area uner a curve are essential.
More informationIERCU. Institute of Economic Research, Chuo University 50th Anniversary Special Issues. Discussion Paper No.210
IERCU Institute of Economic Research, Chuo University 50th Anniversary Special Issues Discussion Paper No.210 Discrete an Continuous Dynamics in Nonlinear Monopolies Akio Matsumoto Chuo University Ferenc
More informationUnit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule
Unit # - Families of Functions, Taylor Polynomials, l Hopital s Rule Some problems an solutions selecte or aapte from Hughes-Hallett Calculus. Critical Points. Consier the function f) = 54 +. b) a) Fin
More informationLearning in Monopolies with Delayed Price Information
Learning in Monopolies with Delaye Price Information Akio Matsumoto y Chuo University Ferenc Sziarovszky z University of Pécs February 28, 2013 Abstract We call the intercept of the price function with
More informationby using the derivative rules. o Building blocks: d
Calculus for Business an Social Sciences - Prof D Yuen Eam Review version /9/01 Check website for any poste typos an upates Eam is on Sections, 5, 6,, 1,, Derivatives Rules Know how to fin the formula
More informationA. Incorrect! The letter t does not appear in the expression of the given integral
AP Physics C - Problem Drill 1: The Funamental Theorem of Calculus Question No. 1 of 1 Instruction: (1) Rea the problem statement an answer choices carefully () Work the problems on paper as neee (3) Question
More informationTable of Common Derivatives By David Abraham
Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec
More informationOnline Appendix for Trade Policy under Monopolistic Competition with Firm Selection
Online Appenix for Trae Policy uner Monopolistic Competition with Firm Selection Kyle Bagwell Stanfor University an NBER Seung Hoon Lee Georgia Institute of Technology September 6, 2018 In this Online
More informationChapter 2 Lagrangian Modeling
Chapter 2 Lagrangian Moeling The basic laws of physics are use to moel every system whether it is electrical, mechanical, hyraulic, or any other energy omain. In mechanics, Newton s laws of motion provie
More information18 EVEN MORE CALCULUS
8 EVEN MORE CALCULUS Chapter 8 Even More Calculus Objectives After stuing this chapter you shoul be able to ifferentiate an integrate basic trigonometric functions; unerstan how to calculate rates of change;
More informationSummary: Differentiation
Techniques of Differentiation. Inverse Trigonometric functions The basic formulas (available in MF5 are: Summary: Differentiation ( sin ( cos The basic formula can be generalize as follows: Note: ( sin
More informationBasic Differentiation Rules and Rates of Change. The Constant Rule
460_00.q //04 4:04 PM Page 07 SECTION. Basic Differentiation Rules an Rates of Change 07 Section. The slope of a horizontal line is 0. Basic Differentiation Rules an Rates of Change Fin the erivative of
More informationCalculus BC Section II PART A A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS
Calculus BC Section II PART A A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS. An isosceles triangle, whose base is the interval from (0, 0) to (c, 0), has its verte on the graph
More informationx = c of N if the limit of f (x) = L and the right-handed limit lim f ( x)
Limit We say the limit of f () as approaches c equals L an write, lim L. One-Sie Limits (Left an Right-Hane Limits) Suppose a function f is efine near but not necessarily at We say that f has a left-hane
More informationMath Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors
Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+
More informationChapter 1. Functions, Graphs, and Limits
Review for Final Exam Lecturer: Sangwook Kim Office : Science & Tech I, 226D math.gmu.eu/ skim22 Chapter 1. Functions, Graphs, an Limits A function is a rule that assigns to each objects in a set A exactly
More informationCalculus and optimization
Calculus an optimization These notes essentially correspon to mathematical appenix 2 in the text. 1 Functions of a single variable Now that we have e ne functions we turn our attention to calculus. A function
More informationTMA 4195 Matematisk modellering Exam Tuesday December 16, :00 13:00 Problems and solution with additional comments
Problem F U L W D g m 3 2 s 2 0 0 0 0 2 kg 0 0 0 0 0 0 Table : Dimension matrix TMA 495 Matematisk moellering Exam Tuesay December 6, 2008 09:00 3:00 Problems an solution with aitional comments The necessary
More information05 The Continuum Limit and the Wave Equation
Utah State University DigitalCommons@USU Founations of Wave Phenomena Physics, Department of 1-1-2004 05 The Continuum Limit an the Wave Equation Charles G. Torre Department of Physics, Utah State University,
More informationu t v t v t c a u t b a v t u t v t b a
Nonlinear Dynamical Systems In orer to iscuss nonlinear ynamical systems, we must first consier linear ynamical systems. Linear ynamical systems are just systems of linear equations like we have been stuying
More informationθ x = f ( x,t) could be written as
9. Higher orer PDEs as systems of first-orer PDEs. Hyperbolic systems. For PDEs, as for ODEs, we may reuce the orer by efining new epenent variables. For example, in the case of the wave equation, (1)
More informationWeb-Based Technical Appendix: Multi-Product Firms and Trade Liberalization
Web-Base Technical Appeni: Multi-Prouct Firms an Trae Liberalization Anrew B. Bernar Tuck School of Business at Dartmouth & NBER Stephen J. Reing LSE, Yale School of Management & CEPR Peter K. Schott Yale
More informationChaos in adaptive expectation Cournot Puu duopoly model
Chaos in adaptive expectation Cournot Puu duopoly model Varun Pandit 1, Dr. Brahmadeo 2, and Dr. Praveen Kulshreshtha 3 1 Economics, HSS, IIT Kanpur(India) 2 Ex-Professor, Materials Science and Engineering,
More informationIntegration Review. May 11, 2013
Integration Review May 11, 2013 Goals: Review the funamental theorem of calculus. Review u-substitution. Review integration by parts. Do lots of integration eamples. 1 Funamental Theorem of Calculus In
More informationOptimization of Geometries by Energy Minimization
Optimization of Geometries by Energy Minimization by Tracy P. Hamilton Department of Chemistry University of Alabama at Birmingham Birmingham, AL 3594-140 hamilton@uab.eu Copyright Tracy P. Hamilton, 1997.
More informationELEC3114 Control Systems 1
ELEC34 Control Systems Linear Systems - Moelling - Some Issues Session 2, 2007 Introuction Linear systems may be represente in a number of ifferent ways. Figure shows the relationship between various representations.
More informationMath 1271 Solutions for Fall 2005 Final Exam
Math 7 Solutions for Fall 5 Final Eam ) Since the equation + y = e y cannot be rearrange algebraically in orer to write y as an eplicit function of, we must instea ifferentiate this relation implicitly
More informationOrdinary Differential Equations: Homework 1
Orinary Differential Equations: Homework 1 M. Gameiro, J.-P. Lessar, J.D. Mireles James, K. Mischaikow January 12, 2017 2 Chapter 1 Motivation 1.1 Exercises Exercise 1.1.1. (Frictionless spring) Consier
More informationEC5555 Economics Masters Refresher Course in Mathematics September 2013
EC5555 Economics Masters Reresher Course in Mathematics September 3 Lecture 5 Unconstraine Optimization an Quaratic Forms Francesco Feri We consier the unconstraine optimization or the case o unctions
More informationThe Principle of Least Action
Chapter 7. The Principle of Least Action 7.1 Force Methos vs. Energy Methos We have so far stuie two istinct ways of analyzing physics problems: force methos, basically consisting of the application of
More informationSlide10 Haykin Chapter 14: Neurodynamics (3rd Ed. Chapter 13)
Slie10 Haykin Chapter 14: Neuroynamics (3r E. Chapter 13) CPSC 636-600 Instructor: Yoonsuck Choe Spring 2012 Neural Networks with Temporal Behavior Inclusion of feeback gives temporal characteristics to
More informationSchrödinger s equation.
Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of
More informationLecture 1b. Differential operators and orthogonal coordinates. Partial derivatives. Divergence and divergence theorem. Gradient. A y. + A y y dy. 1b.
b. Partial erivatives Lecture b Differential operators an orthogonal coorinates Recall from our calculus courses that the erivative of a function can be efine as f ()=lim 0 or using the central ifference
More informationWeb Appendix to Firm Heterogeneity and Aggregate Welfare (Not for Publication)
Web ppeni to Firm Heterogeneity an ggregate Welfare Not for Publication Marc J. Melitz Harvar University, NBER, an CEPR Stephen J. Reing Princeton University, NBER, an CEPR March 6, 203 Introuction his
More informationMath 115 Section 018 Course Note
Course Note 1 General Functions Definition 1.1. A function is a rule that takes certain numbers as inputs an assigns to each a efinite output number. The set of all input numbers is calle the omain of
More information2-7. Fitting a Model to Data I. A Model of Direct Variation. Lesson. Mental Math
Lesson 2-7 Fitting a Moel to Data I BIG IDEA If you etermine from a particular set of ata that y varies irectly or inversely as, you can graph the ata to see what relationship is reasonable. Using that
More informationMath 342 Partial Differential Equations «Viktor Grigoryan
Math 342 Partial Differential Equations «Viktor Grigoryan 6 Wave equation: solution In this lecture we will solve the wave equation on the entire real line x R. This correspons to a string of infinite
More informationExperiment I Electric Force
Experiment I Electric Force Twenty-five hunre years ago, the Greek philosopher Thales foun that amber, the harene sap from a tree, attracte light objects when rubbe. Only twenty-four hunre years later,
More informationII. First variation of functionals
II. First variation of functionals The erivative of a function being zero is a necessary conition for the etremum of that function in orinary calculus. Let us now tackle the question of the equivalent
More informationAPPPHYS 217 Thursday 8 April 2010
APPPHYS 7 Thursay 8 April A&M example 6: The ouble integrator Consier the motion of a point particle in D with the applie force as a control input This is simply Newton s equation F ma with F u : t q q
More informationThe Ehrenfest Theorems
The Ehrenfest Theorems Robert Gilmore Classical Preliminaries A classical system with n egrees of freeom is escribe by n secon orer orinary ifferential equations on the configuration space (n inepenent
More informationSAMPLE FINAL EXAM MATH 16A WINTER 2017
SAMPLE FINAL EXAM MATH 16A WINTER 2017 The final eam consists of 5 parts, worth a total of 40 points. You are not allowe to use books, calculators, mobile phones or anything else besies your writing utensils.
More informationChapter 3 Notes, Applied Calculus, Tan
Contents 3.1 Basic Rules of Differentiation.............................. 2 3.2 The Prouct an Quotient Rules............................ 6 3.3 The Chain Rule...................................... 9 3.4
More informationDesigning Information Devices and Systems I Spring 2018 Lecture Notes Note 16
EECS 16A Designing Information Devices an Systems I Spring 218 Lecture Notes Note 16 16.1 Touchscreen Revisite We ve seen how a resistive touchscreen works by using the concept of voltage iviers. Essentially,
More informationDiagonalization of Matrices Dr. E. Jacobs
Diagonalization of Matrices Dr. E. Jacobs One of the very interesting lessons in this course is how certain algebraic techniques can be use to solve ifferential equations. The purpose of these notes is
More informationEuler equations for multiple integrals
Euler equations for multiple integrals January 22, 2013 Contents 1 Reminer of multivariable calculus 2 1.1 Vector ifferentiation......................... 2 1.2 Matrix ifferentiation........................
More informationIntroduction to the Vlasov-Poisson system
Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its
More informationMATH 205 Practice Final Exam Name:
MATH 205 Practice Final Eam Name:. (2 points) Consier the function g() = e. (a) (5 points) Ientify the zeroes, vertical asymptotes, an long-term behavior on both sies of this function. Clearly label which
More informationThermal conductivity of graded composites: Numerical simulations and an effective medium approximation
JOURNAL OF MATERIALS SCIENCE 34 (999)5497 5503 Thermal conuctivity of grae composites: Numerical simulations an an effective meium approximation P. M. HUI Department of Physics, The Chinese University
More informationVectors in two dimensions
Vectors in two imensions Until now, we have been working in one imension only The main reason for this is to become familiar with the main physical ieas like Newton s secon law, without the aitional complication
More informationIMPLICIT DIFFERENTIATION
IMPLICIT DIFFERENTIATION CALCULUS 3 INU0115/515 (MATHS 2) Dr Arian Jannetta MIMA CMath FRAS Implicit Differentiation 1/ 11 Arian Jannetta Explicit an implicit functions Explicit functions An explicit function
More informationCalculus of Variations
Calculus of Variations Lagrangian formalism is the main tool of theoretical classical mechanics. Calculus of Variations is a part of Mathematics which Lagrangian formalism is base on. In this section,
More informationQuantum mechanical approaches to the virial
Quantum mechanical approaches to the virial S.LeBohec Department of Physics an Astronomy, University of Utah, Salt Lae City, UT 84112, USA Date: June 30 th 2015 In this note, we approach the virial from
More informationLinear and quadratic approximation
Linear an quaratic approximation November 11, 2013 Definition: Suppose f is a function that is ifferentiable on an interval I containing the point a. The linear approximation to f at a is the linear function
More informationMATH2231-Differentiation (2)
-Differentiation () The Beginnings of Calculus The prime occasion from which arose my iscovery of the metho of the Characteristic Triangle, an other things of the same sort, happene at a time when I ha
More informationConvective heat transfer
CHAPTER VIII Convective heat transfer The previous two chapters on issipative fluis were evote to flows ominate either by viscous effects (Chap. VI) or by convective motion (Chap. VII). In either case,
More informationThe Hamiltonian particle-mesh method for the spherical shallow water equations
ATMOSPHERIC SCIENCE LETTERS Atmos. Sci. Let. 5: 89 95 (004) Publishe online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.100/asl.70 The Hamiltonian particle-mesh metho for the spherical
More informationMathcad Lecture #5 In-class Worksheet Plotting and Calculus
Mathca Lecture #5 In-class Worksheet Plotting an Calculus At the en of this lecture, you shoul be able to: graph expressions, functions, an matrices of ata format graphs with titles, legens, log scales,
More informationBohr Model of the Hydrogen Atom
Class 2 page 1 Bohr Moel of the Hyrogen Atom The Bohr Moel of the hyrogen atom assumes that the atom consists of one electron orbiting a positively charge nucleus. Although it oes NOT o a goo job of escribing
More informationMany problems in physics, engineering, and chemistry fall in a general class of equations of the form. d dx. d dx
Math 53 Notes on turm-liouville equations Many problems in physics, engineering, an chemistry fall in a general class of equations of the form w(x)p(x) u ] + (q(x) λ) u = w(x) on an interval a, b], plus
More informationChiang/Wainwright: Fundamental Methods of Mathematical Economics
Chiang/Wainwright: Fundamental Methods of Mathematical Economics CHAPTER 9 EXERCISE 9.. Find the stationary values of the following (check whether they are relative maima or minima or inflection points),
More informationUNDERSTANDING INTEGRATION
UNDERSTANDING INTEGRATION Dear Reaer The concept of Integration, mathematically speaking, is the "Inverse" of the concept of result, the integration of, woul give us back the function f(). This, in a way,
More informationMake graph of g by adding c to the y-values. on the graph of f by c. multiplying the y-values. even-degree polynomial. graph goes up on both sides
Reference 1: Transformations of Graphs an En Behavior of Polynomial Graphs Transformations of graphs aitive constant constant on the outsie g(x) = + c Make graph of g by aing c to the y-values on the graph
More informationProblem Set 2: Solutions
UNIVERSITY OF ALABAMA Department of Physics an Astronomy PH 102 / LeClair Summer II 2010 Problem Set 2: Solutions 1. The en of a charge rubber ro will attract small pellets of Styrofoam that, having mae
More informationlim Prime notation can either be directly applied to a function as previously seen with f x 4.1 Basic Techniques for Finding Derivatives
MATH 040 Notes: Unit Page 4. Basic Techniques for Fining Derivatives In the previous unit we introuce the mathematical concept of the erivative: f f ( h) f ( ) lim h0 h (assuming the limit eists) In this
More informationOne Dimensional Convection: Interpolation Models for CFD
One Dimensional Convection: Interpolation Moels for CFD ME 448/548 Notes Geral Recktenwal Portlan State University Department of Mechanical Engineering gerry@p.eu ME 448/548: D Convection-Di usion Equation
More informationChapter 3 Definitions and Theorems
Chapter 3 Definitions an Theorems (from 3.1) Definition of Tangent Line with slope of m If f is efine on an open interval containing c an the limit Δy lim Δx 0 Δx = lim f (c + Δx) f (c) = m Δx 0 Δx exists,
More informationChapter Primer on Differentiation
Capter 0.01 Primer on Differentiation After reaing tis capter, you soul be able to: 1. unerstan te basics of ifferentiation,. relate te slopes of te secant line an tangent line to te erivative of a function,.
More informationcosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) 2t 1 + t 2 cos x = 1 t2 sin x =
6.4 Integration using tan/ We will revisit the ouble angle ientities: sin = sin/ cos/ = tan/ sec / = tan/ + tan / cos = cos / sin / tan = = tan / sec / tan/ tan /. = tan / + tan / So writing t = tan/ we
More informationThe Press-Schechter mass function
The Press-Schechter mass function To state the obvious: It is important to relate our theories to what we can observe. We have looke at linear perturbation theory, an we have consiere a simple moel for
More informationChapter 2 Derivatives
Chapter Derivatives Section. An Intuitive Introuction to Derivatives Consier a function: Slope function: Derivative, f ' For each, the slope of f is the height of f ' Where f has a horizontal tangent line,
More informationLinear First-Order Equations
5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
More informationThe Derivative and the Tangent Line Problem. The Tangent Line Problem
96 CHAPTER Differentiation Section ISAAC NEWTON (6 77) In aition to his work in calculus, Newton mae revolutionar contributions to phsics, incluing the Law of Universal Gravitation an his three laws of
More informationStable Poiseuille Flow Transfer for a Navier-Stokes System
Proceeings of the 26 American Control Conference Minneapolis, Minnesota, USA, June 14-16, 26 WeB2.2 Stable Poiseuille Flow Transfer for a Navier-Stokes System Rafael Vázquez, Emmanuel Trélat an Jean-Michel
More informationDynamic Equations and Nonlinear Dynamics of Cascade Two-Photon Laser
Commun. Theor. Phys. (Beiing, China) 45 (6) pp. 4 48 c International Acaemic Publishers Vol. 45, No. 6, June 5, 6 Dynamic Equations an Nonlinear Dynamics of Cascae Two-Photon Laser XIE Xia,,, HUANG Hong-Bin,
More informationThe derivative of a function f(x) is another function, defined in terms of a limiting expression: f(x + δx) f(x)
Y. D. Chong (2016) MH2801: Complex Methos for the Sciences 1. Derivatives The erivative of a function f(x) is another function, efine in terms of a limiting expression: f (x) f (x) lim x δx 0 f(x + δx)
More informationDynamical Systems and a Brief Introduction to Ergodic Theory
Dynamical Systems an a Brief Introuction to Ergoic Theory Leo Baran Spring 2014 Abstract This paper explores ynamical systems of ifferent types an orers, culminating in an examination of the properties
More informationCable holds system BUT at t=0 it breaks!! θ=20. Copyright Luis San Andrés (2010) 1
EAMPLE # for MEEN 363 SPRING 6 Objectives: a) To erive EOMS of a DOF system b) To unerstan concept of static equilibrium c) To learn the correct usage of physical units (US system) ) To calculate natural
More informationIntroduction to variational calculus: Lecture notes 1
October 10, 2006 Introuction to variational calculus: Lecture notes 1 Ewin Langmann Mathematical Physics, KTH Physics, AlbaNova, SE-106 91 Stockholm, Sween Abstract I give an informal summary of variational
More informationSection 2.1 The Derivative and the Tangent Line Problem
Chapter 2 Differentiation Course Number Section 2.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan
More informationNonlinear Dielectric Response of Periodic Composite Materials
onlinear Dielectric Response of Perioic Composite aterials A.G. KOLPAKOV 3, Bl.95, 9 th ovember str., ovosibirsk, 639 Russia the corresponing author e-mail: agk@neic.nsk.su, algk@ngs.ru A. K.TAGATSEV Ceramics
More informationAdjoint Transient Sensitivity Analysis in Circuit Simulation
Ajoint Transient Sensitivity Analysis in Circuit Simulation Z. Ilievski 1, H. Xu 1, A. Verhoeven 1, E.J.W. ter Maten 1,2, W.H.A. Schilers 1,2 an R.M.M. Mattheij 1 1 Technische Universiteit Einhoven; e-mail:
More informationDynamical Systems. 1.0 Ordinary Differential Equations. 2.0 Dynamical Systems
. Ordinary Differential Equations. An ordinary differential equation (ODE, for short) is an equation involves a dependent variable and its derivatives with respect to an independent variable. When we speak
More informationTechniques of Differentiation. Chapter 2 Section 2 Techniques of Differentiation 109. The Constant Rule For any constant c,
Chapter 2 Section 2 Techniques of Differentiation 109 UNEMPLOYMENT 34. In economics, the graph in Figure 2.3 is calle the Phillips curve, after A. W. Phillips, a New Zealaner associate with the Lonon School
More informationSHORT-CUTS TO DIFFERENTIATION
Chapter Three SHORT-CUTS TO DIFFERENTIATION In Chapter, we efine the erivative function f () = lim h 0 f( + h) f() h an saw how the erivative represents a slope an a rate of change. We learne how to approimate
More informationcosh x sinh x So writing t = tan(x/2) we have 6.4 Integration using tan(x/2) = 2 2t 1 + t 2 cos x = 1 t2 We will revisit the double angle identities:
6.4 Integration using tanx/) We will revisit the ouble angle ientities: sin x = sinx/) cosx/) = tanx/) sec x/) = tanx/) + tan x/) cos x = cos x/) sin x/) tan x = = tan x/) sec x/) tanx/) tan x/). = tan
More informationA SIMPLE ENGINEERING MODEL FOR SPRINKLER SPRAY INTERACTION WITH FIRE PRODUCTS
International Journal on Engineering Performance-Base Fire Coes, Volume 4, Number 3, p.95-3, A SIMPLE ENGINEERING MOEL FOR SPRINKLER SPRAY INTERACTION WITH FIRE PROCTS V. Novozhilov School of Mechanical
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 20 Quantum Mechanics in Three Dimensions Lecture 20 Physics 342 Quantum Mechanics I Monay, March 24th, 2008 We begin our spherical solutions with the simplest possible case zero potential.
More informationCMSC 313 Preview Slides
CMSC 33 Preview Slies These are raft slies. The actual slies presente in lecture may be ifferent ue to last minute changes, scheule slippage,... UMBC, CMSC33, Richar Chang CMSC 33 Lecture
More informationAnswer Key: Problem Set 3
Answer Key: Problem Set Econ 409 018 Fall Question 1 a This is a standard monopoly problem; using MR = a 4Q, let MR = MC and solve: Q M = a c 4, P M = a + c, πm = (a c) 8 The Lerner index is then L M P
More informationLATTICE-BASED D-OPTIMUM DESIGN FOR FOURIER REGRESSION
The Annals of Statistics 1997, Vol. 25, No. 6, 2313 2327 LATTICE-BASED D-OPTIMUM DESIGN FOR FOURIER REGRESSION By Eva Riccomagno, 1 Rainer Schwabe 2 an Henry P. Wynn 1 University of Warwick, Technische
More informationConstraint Reformulation and a Lagrangian Relaxation based Solution Algorithm for a Least Expected Time Path Problem Abstract 1.
Constraint Reformulation an a Lagrangian Relaation base Solution Algorithm for a Least Epecte Time Path Problem Liing Yang State Key Laboratory of Rail Traffic Control an Safety, Being Jiaotong University,
More informationQ(s, t) = S M = S M [ G 1 (t) G 2 (t) G 3 1(t) G 4 (t) ] T
Curves an Surfaces: Parametric Bicubic Surfaces - Intro Surfaces are generalizations of curves Use s in place of t in parametric equation: Q(s) = S M G where S equivalent to T in Q(t) = T M G If G is parameterize
More information3.7 Implicit Differentiation -- A Brief Introduction -- Student Notes
Fin these erivatives of these functions: y.7 Implicit Differentiation -- A Brief Introuction -- Stuent Notes tan y sin tan = sin y e = e = Write the inverses of these functions: y tan y sin How woul we
More informationarxiv:hep-th/ v1 3 Feb 1993
NBI-HE-9-89 PAR LPTHE 9-49 FTUAM 9-44 November 99 Matrix moel calculations beyon the spherical limit arxiv:hep-th/93004v 3 Feb 993 J. Ambjørn The Niels Bohr Institute Blegamsvej 7, DK-00 Copenhagen Ø,
More informationOptimization of a point-mass walking model using direct collocation and sequential quadratic programming
Optimization of a point-mass walking moel using irect collocation an sequential quaratic programming Chris Dembia June 5, 5 Telescoping actuator y Stance leg Point-mass boy m (x,y) Swing leg x Leg uring
More information