1.4 Phase Line and Bifurcation Diag

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "1.4 Phase Line and Bifurcation Diag"

Transcription

1 Dynamical Systems: Pat 2 2 Bifucation Theoy In pactical applications that involve diffeential equations it vey often happens that the diffeential equation contains paametes and the value of these paametes ae often only known appoimately. In paticula they ae geneally detemined by measuements which 42 ae not eact. Fo that eason it is impotant to study the behavio of solutions and eamine thei dependence on the paametes. This study leads to the aea efeed to as bifucation 1.4 Phase Line and Bifucation Diag theoy. It can happen that a slight vaiation in a paamete can have significant impact on the solution. Bifucation theoy is a vey deep and complicated aea involving lots of cuent Technical publications may use special diagams to display qu eseach. A complete eaminationinfomation of of the field would aboutbe the impossible. equilibium points of the diffeential e A fied point (o equilibium point) of a diffeential equation y = f(y) is a oot of the (1) y = f(y). equation f(y) = 0. As we have aleady seenfo autonomous poblems fied points can be vey useful in detemining the long This timequation behavio is of solutions. independent of, hence thee ae no etena tems that depend on. Due to the lack of Qualitative infomation about the equilibium points of the diffeential equation y etenal contols, = f(y) tion is said to be self-govening o autonomous. can be obtained fom special diagams called phase diagams. A phase line diagam fo the autonomous equation y = f(y A phase line diagam fo the autonomous segment with equation labels y sink, = f(y) souce is a line segment o node, with one labels fo each oot of fo so-called sinks, souces o nodes, i.e. one each foequilibium; each oot of f(y) see= Figue 0, i.e. each 11. equilibium. souce sink Figue 11. A phase line diagam y 0 y 1 autonomous equation y = f(y). The names ae boowed fom The the theoy labels of ae fluids boowed and they fom ae the theoy of fluids, and they following special definitions: 6 defined as follows: 1. Sink An equilibium y 0 which attacts neaby solutions at t =, i.e., thee eists Sink y = y 0 The equilibium y = y 0 attacts neaby solu M > 0 so that if y(0) y 0 < M, then y() y 0 t 0 = : fo some H > 0, y(0) y 0 < H y() y 0 deceases to 0 as. 2. Souce An equilibium y 1 which epels neaby solutions at t =, i.e., hee eists Souce y = y 1 The equilibium y = y 1 epels neaby solut M > 0 so that if y(0) y 1 < M, then y() y 1 inceases as t. = : fo some H > 0, y(0) y 1 < H that y() y 1 inceases as. 3. Node An equilibium y 2 which is neithe a sink o a souce. In fluids, sink means fluid Node y = y 2 The equilibium y = y 2 is neithe a sink no a is lost and souce means fluid is ceated. In fluids, sink 1 means fluid is lost and souce means fluid is c memoy device fo these concepts is the kitchen sink, wheein t is the souce and the dain is the sink. The stability test

2 Stability Test: The tem stable means that solutions that stat nea the equilibium will stay neaby as t. The tem unstable means not stable. Theefoe, a sink is stable and a souce is unstable. Pecisely, an equilibium y 0 is stable povided fo given ɛ > 0 thee eists some δ > 0 such that y(0) y 0 < δ implies y(t) eists fo t 0 and y(t)?y 0 < ɛ. Theoem 2.1 (Stability Conditions). Let f and f be continuous. The equation y = f(y) has a sink at y = y 0 povided f(y 0 ) = 0 and f (y0) < 0. An equilibium y = y 1 is a souce povided f(y 1 ) = 0 and f (y 1 ) > 0. Thee is no test when f is zeo at an equilibium. Ou objective in this section (fo fist ode equations) is to biefly eamine the thee simplest types of bifucations: 1) Saddle Node; 2) Tanscitical; 3) Pitchfok. 2.1 Saddle Bode Bifucation We begin with the Saddle Node bifucation (also called the blue sky bifucation) coesponding to the ceation and destuction of fied points. The nomal fom fo this type of bifucation is given by the eample = + 2 (1) The thee cases of < 0, = 0 and > 0 give vey diffeent stuctue fo the solutions. < 0 = 0 > 0 We obseve that thee is a bifucation at = 0. Fo < 0 thee ae two fied points given by = ±. The equilibium = is stable, i.e., solutions beginning nea this equilibium convege to it as time inceases. Futhe, initial conditions nea divege fom it. 2

3 At = 0 thee is a single fied point at = 0 and initial conditions less than zeo give solutions that convege to zeo while positive initial conditions give solutions that incease without bound. Finally if > 0 thee ae no fied points at all. Fo any initial condition solutions incease without bound. Thee ae seveal ways we depict this type of bifucation one of which is the so called bifucation diagam. Note that if instead we conside = 2 the the so-called phase line can be dawn as < 0 = 0 > 0 Eecise: Analyze the bifucation popeties of the following following poblems. 1. = = cosh() 3. = + ln(1 + ) 3

4 2.2 Tanscitical Bifucation Net we conside the tanscitical bifucation coesponding to the echange of stability of fied points. The nomal fom fo this type of bifucation is given by the eample = 2 (2) In this case thee is eithe one ( = 0) o two ( 0) fied points. When = 0 the only fied point is = 0 which is semi-stable (i.e., stable fom the ight and unstable fom the left). Fo 0 thee ae two fied points given by = 0 and =. So we note in this case = 0 is a fied point fo all. Fo < 0 the nonzeo fied point is unstable but fo > 0 the nonzeo fied point becomes stable. Thus we say that the stability of this fied point has switched fom unstable to stable. < 0 = 0 > 0 Bifucation diagam fo a tanscitical bifucation. Eecise: Analyze the bifucation popeties of the following following poblems. 1. = = ln(1 + ) 4

5 3. = (1 ) 2.3 Pitchfok Bifucation Finally we conside the pitchfok bifucation. The nomal fom fo this type of bifucation is given by the eample = 3 (3) The cases of 0 and > 0, once again, give vey diffeent stuctue fo the solutions. < 0 = 0 > 0 Supe Citical Pitchfok Bifucation Diagam Now conside the eample = + 3. (4) Fo this eample we obtain the so-called sub-citical pitchfok bifucation. Notice that solutions blow-up in finite time, i.e., satisfy (t) ± as t a <. 5

6 Sub Citical Pitchfok Bifucation Diagam Eecise: Analyze the bifucation popeties of the following following poblems. 1. = + β tanh() 2. = = sin() 4. = = sinh() 6. = = Hysteesis: a moe complicated bifucation In this subsection we conside an even moe complicated eample which contains pitchfokand saddle node bifucations. Conside the eample = (5) 1. Fo small initial conditions the bifucation diagam looks just like the sub-citical bifucation diagam. The oigin is locally stable fo < 0 and the two banches ae unstable. The two backwad unstable banches bifucated fom = 0. The tem 5 6

7 has now ceated a new phenomenon: at a value of < 0, denoted by, the unstable banches tun aound aound and become stable. These new banches eist fo all > 2. Note that fo < < 0 thee ae thee stable solutions. The initial condition detemines which of these thee fied points the solution conveges to as time inceases. 3. This eample demonstates an impotant physically obseved phenomenon known as Hysteesis. If we stat the system with an initial condition close to = 0 Bifucation Diagam showing Hysteesis 7

(3) Bipolar Transistor Current Sources

(3) Bipolar Transistor Current Sources B73 lectonics Analysis & Design (3) Bipola Tansisto Cuent Souces Leaning utcome Able to descibe and: Analyze and design a simple twotansisto BJT cuent-souce cicuit to poduce a given bias cuent. Analyze

More information

UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

More information

Chapter 6. Gradually-Varied Flow in Open Channels

Chapter 6. Gradually-Varied Flow in Open Channels Chapte 6 Gadually-Vaied Flow in Open Channels 6.. Intoduction A stea non-unifom flow in a pismatic channel with gadual changes in its watesuface elevation is named as gadually-vaied flow (GVF). The backwate

More information

Questions for Review. By buying bonds This period you save s, next period you get s(1+r)

Questions for Review. By buying bonds This period you save s, next period you get s(1+r) MACROECONOMICS 2006 Week 5 Semina Questions Questions fo Review 1. How do consumes save in the two-peiod model? By buying bonds This peiod you save s, next peiod you get s() 2. What is the slope of a consume

More information

Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapter 3 Savings, Present Value and Ricardian Equivalence Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

More information

Semipartial (Part) and Partial Correlation

Semipartial (Part) and Partial Correlation Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated

More information

Chapter For the deep-groove 02-series ball bearing with R = 0.90, the design life x D, in multiples of rating life, is ( ) 1.

Chapter For the deep-groove 02-series ball bearing with R = 0.90, the design life x D, in multiples of rating life, is ( ) 1. hapte 11 11-1 Fo the deep-goove 02-seies ball beaing with = 0.90, the design life, in multiples of ating life, is L 0 0( 25000) 350 n = 525 Ans. L = L L = = The design adial load is F = 1..5 = 3.0 kn Eq.

More information

Financing Terms in the EOQ Model

Financing Terms in the EOQ Model Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad

More information

EAS Groundwater Hydrology Lecture 13: Well Hydraulics 2 Dr. Pengfei Zhang

EAS Groundwater Hydrology Lecture 13: Well Hydraulics 2 Dr. Pengfei Zhang EAS 44600 Goundwate Hydology Lectue 3: Well Hydaulics D. Pengfei Zhang Detemining Aquife Paametes fom Time-Dawdown Data In the past lectue we discussed how to calculate dawdown if we know the hydologic

More information

Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Work, Power and Kinetic Energy Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

More information

2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90

2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90 . Tigonometic Ratios of An Angle Focus on... detemining the distance fom the oigin to a point (, ) on the teminal am of an angle detemining the value of sin, cos, o tan given an point (, ) on the teminal

More information

Physics: Electromagnetism Spring PROBLEM SET 6 Solutions

Physics: Electromagnetism Spring PROBLEM SET 6 Solutions Physics: Electomagnetism Sping 7 Physics: Electomagnetism Sping 7 PROBEM SET 6 Solutions Electostatic Enegy Basics: Wolfson and Pasachoff h 6 Poblem 7 p 679 Thee ae si diffeent pais of equal chages and

More information

1. How is the IS curve derived and what factors determine its slope? What happens to the slope of the IS curve if consumption is interest elastic?

1. How is the IS curve derived and what factors determine its slope? What happens to the slope of the IS curve if consumption is interest elastic? Chapte 7 Review Questions 1. How is the IS cuve deived and what factos detemine its slope? What happens to the slope of the IS cuve if consumption is inteest elastic? The IS cuve epesents equilibium in

More information

Trigonometric Functions of Any Angle

Trigonometric Functions of Any Angle Tigonomet Module T2 Tigonometic Functions of An Angle Copight This publication The Nothen Albeta Institute of Technolog 2002. All Rights Reseved. LAST REVISED Decembe, 2008 Tigonometic Functions of An

More information

Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E

Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known

More information

Life Insurance Purchasing to Reach a Bequest. Erhan Bayraktar Department of Mathematics, University of Michigan Ann Arbor, Michigan, USA, 48109

Life Insurance Purchasing to Reach a Bequest. Erhan Bayraktar Department of Mathematics, University of Michigan Ann Arbor, Michigan, USA, 48109 Life Insuance Puchasing to Reach a Bequest Ehan Bayakta Depatment of Mathematics, Univesity of Michigan Ann Abo, Michigan, USA, 48109 S. David Pomislow Depatment of Mathematics, Yok Univesity Toonto, Ontaio,

More information

Lecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3

Lecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3 Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each

More information

TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS 14. NEWTON'S RINGS

TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS 14. NEWTON'S RINGS 4. NEWTON'S RINGS. Obective Detemining adius of cuvatue of a long focal length plano-convex lens (lage adius of cuvatue).. Equipment needed Measuing micoscope, plano-convex long focal length lens, monochomatic

More information

Carter-Penrose diagrams and black holes

Carter-Penrose diagrams and black holes Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

More information

1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2

1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2 Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the

More information

On Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables.

On Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables. C.Candan EE3/53-METU On Coelation Coefficient The coelation coefficient indicates the degee of linea dependence of two andom vaiables. It is defined as ( )( )} σ σ Popeties: 1. 1. (See appendi fo the poof

More information

Forces & Magnetic Dipoles. r r τ = μ B r

Forces & Magnetic Dipoles. r r τ = μ B r Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent

More information

An Introduction to Omega

An Introduction to Omega An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei isk-ewad chaacteistics? The Finance Development Cente 2002 1 Fom

More information

9. Mathematics Practice Paper for Class XII (CBSE) Available Online Tutoring for students of classes 4 to 12 in Physics, Chemistry, Mathematics

9. Mathematics Practice Paper for Class XII (CBSE) Available Online Tutoring for students of classes 4 to 12 in Physics, Chemistry, Mathematics Available Online Tutoing fo students of classes 4 to 1 in Physics, 9. Mathematics Class 1 Pactice Pape 1 3 1. Wite the pincipal value of cos.. Wite the ange of the pincipal banch of sec 1 defined on the

More information

Trajectory Following Method on Output Regulation of Affine Nonlinear Control Systems with Relative Degree not Well Defined

Trajectory Following Method on Output Regulation of Affine Nonlinear Control Systems with Relative Degree not Well Defined ITB J. Sci., Vol. 43 A, No., 011, 73-86 73 Taectoy Following Method on Output Regulation of Affine Nonlinea Contol Systems with Relative Degee not Well Defined Janson Naibohu Industial Financial Mathematics

More information

Coordinate Systems L. M. Kalnins, March 2009

Coordinate Systems L. M. Kalnins, March 2009 Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

More information

Experiment MF Magnetic Force

Experiment MF Magnetic Force Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuent-caying conducto is basic to evey electic moto -- tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating

More information

2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES . TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

More information

The Supply of Loanable Funds: A Comment on the Misconception and Its Implications

The Supply of Loanable Funds: A Comment on the Misconception and Its Implications JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently Fields-Hat

More information

8-1 Newton s Law of Universal Gravitation

8-1 Newton s Law of Universal Gravitation 8-1 Newton s Law of Univesal Gavitation One of the most famous stoies of all time is the stoy of Isaac Newton sitting unde an apple tee and being hit on the head by a falling apple. It was this event,

More information

Gauss Law in dielectrics

Gauss Law in dielectrics Gauss Law in dielectics We fist deive the diffeential fom of Gauss s law in the pesence of a dielectic. Recall, the diffeential fom of Gauss Law is This law is always tue. E In the pesence of dielectics,

More information

Economics 326: Input Demands. Ethan Kaplan

Economics 326: Input Demands. Ethan Kaplan Economics 326: Input Demands Ethan Kaplan Octobe 24, 202 Outline. Tems 2. Input Demands Tems Labo Poductivity: Output pe unit of labo. Y (K; L) L What is the labo poductivity of the US? Output is ouhgly

More information

Learning Objectives. Decreasing size. ~10 3 m. ~10 6 m. ~10 10 m 1/22/2013. Describe ionic, covalent, and metallic, hydrogen, and van der Waals bonds.

Learning Objectives. Decreasing size. ~10 3 m. ~10 6 m. ~10 10 m 1/22/2013. Describe ionic, covalent, and metallic, hydrogen, and van der Waals bonds. Lectue #0 Chapte Atomic Bonding Leaning Objectives Descibe ionic, covalent, and metallic, hydogen, and van de Waals bonds. Which mateials exhibit each of these bonding types? What is coulombic foce of

More information

LINES AND TANGENTS IN POLAR COORDINATES

LINES AND TANGENTS IN POLAR COORDINATES LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Pola-coodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and

More information

YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH

YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH nd INTERNATIONAL TEXTILE, CLOTHING & ESIGN CONFERENCE Magic Wold of Textiles Octobe 03 d to 06 th 004, UBROVNIK, CROATIA YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH Jana VOBOROVA; Ashish GARG; Bohuslav

More information

PAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII - SPETO - 1995. pod patronatem. Summary

PAN STABILITY TESTING OF DC CIRCUITS USING VARIATIONAL METHODS XVIII - SPETO - 1995. pod patronatem. Summary PCE SEMINIUM Z PODSTW ELEKTOTECHNIKI I TEOII OBWODÓW 8 - TH SEMIN ON FUNDMENTLS OF ELECTOTECHNICS ND CICUIT THEOY ZDENĚK BIOLEK SPŠE OŽNO P.., CZECH EPUBLIC DLIBO BIOLEK MILITY CDEMY, BNO, CZECH EPUBLIC

More information

MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION

MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION MULTIPLE SOLUTIONS OF THE PRESCRIBED MEAN CURVATURE EQUATION K.C. CHANG AND TAN ZHANG In memoy of Pofesso S.S. Chen Abstact. We combine heat flow method with Mose theoy, supe- and subsolution method with

More information

Chris J. Skinner The probability of identification: applying ideas from forensic statistics to disclosure risk assessment

Chris J. Skinner The probability of identification: applying ideas from forensic statistics to disclosure risk assessment Chis J. Skinne The pobability of identification: applying ideas fom foensic statistics to disclosue isk assessment Aticle (Accepted vesion) (Refeeed) Oiginal citation: Skinne, Chis J. (2007) The pobability

More information

In the lecture on double integrals over non-rectangular domains we used to demonstrate the basic idea

In the lecture on double integrals over non-rectangular domains we used to demonstrate the basic idea Double Integals in Pola Coodinates In the lectue on double integals ove non-ectangula domains we used to demonstate the basic idea with gaphics and animations the following: Howeve this paticula example

More information

Converting knowledge Into Practice

Converting knowledge Into Practice Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading

More information

Criterion Specification Simul. Result Supply. 50MHz +- 10% 53.83MHz j1.52

Criterion Specification Simul. Result Supply. 50MHz +- 10% 53.83MHz j1.52 Low Noise Amplifie 6.776 Lab 1 Taeg Sang Cho, Tao Pan Massachusetts Institute of Technology Cambidge, MA Intoduction Low-noise amplifie plays a citical ole in the design of Radio Fequency (RF) system because

More information

Seshadri constants and surfaces of minimal degree

Seshadri constants and surfaces of minimal degree Seshadi constants and sufaces of minimal degee Wioletta Syzdek and Tomasz Szembeg Septembe 29, 2007 Abstact In [] we showed that if the multiple point Seshadi constants of an ample line bundle on a smooth

More information

The Critical Angle and Percent Efficiency of Parabolic Solar Cookers

The Critical Angle and Percent Efficiency of Parabolic Solar Cookers The Citical Angle and Pecent Eiciency o Paabolic Sola Cookes Aiel Chen Abstact: The paabola is commonly used as the cuve o sola cookes because o its ability to elect incoming light with an incoming angle

More information

NUCLEAR MAGNETIC RESONANCE

NUCLEAR MAGNETIC RESONANCE 19 Jul 04 NMR.1 NUCLEAR MAGNETIC RESONANCE In this expeiment the phenomenon of nuclea magnetic esonance will be used as the basis fo a method to accuately measue magnetic field stength, and to study magnetic

More information

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it. Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing

More information

Personal Saving Rate (S Households /Y) SAVING AND INVESTMENT. Federal Surplus or Deficit (-) Total Private Saving Rate (S Private /Y) 12/18/2009

Personal Saving Rate (S Households /Y) SAVING AND INVESTMENT. Federal Surplus or Deficit (-) Total Private Saving Rate (S Private /Y) 12/18/2009 1 Pesonal Saving Rate (S Households /Y) 2 SAVING AND INVESTMENT 16.0 14.0 12.0 10.0 80 8.0 6.0 4.0 2.0 0.0-2.0-4.0 1959 1961 1967 1969 1975 1977 1983 1985 1991 1993 1999 2001 2007 2009 Pivate Saving Rate

More information

Originally TRIGONOMETRY was that branch of mathematics concerned with solving triangles using trigonometric ratios which were seen as properties of

Originally TRIGONOMETRY was that branch of mathematics concerned with solving triangles using trigonometric ratios which were seen as properties of Oiginall TRIGONOMETRY was that banch of mathematics concened with solving tiangles using tigonometic atios which wee seen as popeties of tiangles athe than of angles. The wod Tigonomet comes fom the Geek

More information

UNIVERSITY OF CALIFORNIA BERKELEY Structural Engineering, Professor: S. Govindjee. Elastic-Perfectly Plastic Thick Walled Sphere

UNIVERSITY OF CALIFORNIA BERKELEY Structural Engineering, Professor: S. Govindjee. Elastic-Perfectly Plastic Thick Walled Sphere UNIVERSITY OF CALIFORNIA BERKELEY Stuctual Engineeing, Depatment of Civil Engineeing Mechanics and Mateials Fall 00 Pofesso: S Govindjee Elastic-Pefectly Plastic Thick Walled Sphee Conside a thick walled

More information

Universal Cycles. Yu She. Wirral Grammar School for Girls. Department of Mathematical Sciences. University of Liverpool

Universal Cycles. Yu She. Wirral Grammar School for Girls. Department of Mathematical Sciences. University of Liverpool Univesal Cycles 2011 Yu She Wial Gamma School fo Gils Depatment of Mathematical Sciences Univesity of Livepool Supeviso: Pofesso P. J. Giblin Contents 1 Intoduction 2 2 De Buijn sequences and Euleian Gaphs

More information

The Role of Gravity in Orbital Motion

The Role of Gravity in Orbital Motion ! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State

More information

Chapter F. Magnetism. Blinn College - Physics Terry Honan

Chapter F. Magnetism. Blinn College - Physics Terry Honan Chapte F Magnetism Blinn College - Physics 46 - Tey Honan F. - Magnetic Dipoles and Magnetic Fields Electomagnetic Duality Thee ae two types of "magnetic chage" o poles, Noth poles N and South poles S.

More information

So we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1)

So we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1) Lectue 17 Cicula Motion (Chapte 7) Angula Measue Angula Speed and Velocity Angula Acceleation We ve aleady dealt with cicula motion somewhat. Recall we leaned about centipetal acceleation: when you swing

More information

Problem Set # 9 Solutions

Problem Set # 9 Solutions Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new high-speed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease

More information

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360! 1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

More information

BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR. John R. Graham Adapted from S. Viswanathan FUQUA SCHOOL OF BUSINESS

BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR. John R. Graham Adapted from S. Viswanathan FUQUA SCHOOL OF BUSINESS BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR John R. Gaham Adapted fom S. Viswanathan FUQUA SCHOOL OF BUSINESS DUKE UNIVERSITY 1 In this lectue we conside the effect of govenment

More information

Introduction to Stock Valuation. Background

Introduction to Stock Valuation. Background Intoduction to Stock Valuation (Text efeence: Chapte 5 (Sections 5.4-5.9)) Topics backgound dividend discount models paamete estimation gowth oppotunities pice-eanings atios some final points AFM 271 -

More information

International Monetary Economics Note 1

International Monetary Economics Note 1 36-632 Intenational Monetay Economics Note Let me biefly ecap on the dynamics of cuent accounts in small open economies. Conside the poblem of a epesentative consume in a county that is pefectly integated

More information

In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.

In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. Radians At school we usually lean to measue an angle in degees. Howeve, thee ae othe ways of measuing an angle. One that we ae going to have a look at hee is measuing angles in units called adians. In

More information

Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project

Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.

More information

Gravitational Mechanics of the Mars-Phobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning

Gravitational Mechanics of the Mars-Phobos System: Comparing Methods of Orbital Dynamics Modeling for Exploratory Mission Planning Gavitational Mechanics of the Mas-Phobos System: Compaing Methods of Obital Dynamics Modeling fo Exploatoy Mission Planning Alfedo C. Itualde The Pennsylvania State Univesity, Univesity Pak, PA, 6802 This

More information

Infinite-dimensional Bäcklund transformations between isotropic and anisotropic plasma equilibria.

Infinite-dimensional Bäcklund transformations between isotropic and anisotropic plasma equilibria. Infinite-dimensional äcklund tansfomations between isotopic and anisotopic plasma equilibia. Infinite symmeties of anisotopic plasma equilibia. Alexei F. Cheviakov Queen s Univesity at Kingston, 00. Reseach

More information

Nontrivial lower bounds for the least common multiple of some finite sequences of integers

Nontrivial lower bounds for the least common multiple of some finite sequences of integers J. Numbe Theoy, 15 (007), p. 393-411. Nontivial lowe bounds fo the least common multiple of some finite sequences of integes Bai FARHI bai.fahi@gmail.com Abstact We pesent hee a method which allows to

More information

Concept and Experiences on using a Wiki-based System for Software-related Seminar Papers

Concept and Experiences on using a Wiki-based System for Software-related Seminar Papers Concept and Expeiences on using a Wiki-based System fo Softwae-elated Semina Papes Dominik Fanke and Stefan Kowalewski RWTH Aachen Univesity, 52074 Aachen, Gemany, {fanke, kowalewski}@embedded.wth-aachen.de,

More information

PY1052 Problem Set 8 Autumn 2004 Solutions

PY1052 Problem Set 8 Autumn 2004 Solutions PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

More information

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

More information

Promised Lead-Time Contracts Under Asymmetric Information

Promised Lead-Time Contracts Under Asymmetric Information OPERATIONS RESEARCH Vol. 56, No. 4, July August 28, pp. 898 915 issn 3-364X eissn 1526-5463 8 564 898 infoms doi 1.1287/ope.18.514 28 INFORMS Pomised Lead-Time Contacts Unde Asymmetic Infomation Holly

More information

Unit Vectors. the unit vector rˆ. Thus, in the case at hand, 5.00 rˆ, means 5.00 m/s at 36.0.

Unit Vectors. the unit vector rˆ. Thus, in the case at hand, 5.00 rˆ, means 5.00 m/s at 36.0. Unit Vectos What is pobabl the most common mistake involving unit vectos is simpl leaving thei hats off. While leaving the hat off a unit vecto is a nast communication eo in its own ight, it also leads

More information

Simple Harmonic Motion

Simple Harmonic Motion Simple Hamonic Motion Intoduction Simple hamonic motion occus when the net foce acting on an object is popotional to the object s displacement fom an equilibium position. When the object is at an equilibium

More information

NURBS Drawing Week 5, Lecture 10

NURBS Drawing Week 5, Lecture 10 CS 43/585 Compute Gaphics I NURBS Dawing Week 5, Lectue 1 David Been, William Regli and Maim Pesakhov Geometic and Intelligent Computing Laboato Depatment of Compute Science Deel Univesit http://gicl.cs.deel.edu

More information

STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts

More information

The Detection of Obstacles Using Features by the Horizon View Camera

The Detection of Obstacles Using Features by the Horizon View Camera The Detection of Obstacles Using Featues b the Hoizon View Camea Aami Iwata, Kunihito Kato, Kazuhiko Yamamoto Depatment of Infomation Science, Facult of Engineeing, Gifu Univesit aa@am.info.gifu-u.ac.jp

More information

Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.

Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities. Gaphs of Equations CHAT Pe-Calculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such

More information

Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27 Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field

More information

Open Economies. Chapter 32. A Macroeconomic Theory of the Open Economy. Basic Assumptions of a Macroeconomic Model of an Open Economy

Open Economies. Chapter 32. A Macroeconomic Theory of the Open Economy. Basic Assumptions of a Macroeconomic Model of an Open Economy Chapte 32. A Macoeconomic Theoy of the Open Economy Open Economies An open economy is one that inteacts feely with othe economies aound the wold. slide 0 slide 1 Key Macoeconomic Vaiables in an Open Economy

More information

AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM

AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,

More information

Database Management Systems

Database Management Systems Contents Database Management Systems (COP 5725) D. Makus Schneide Depatment of Compute & Infomation Science & Engineeing (CISE) Database Systems Reseach & Development Cente Couse Syllabus 1 Sping 2012

More information

Charges, Coulomb s Law, and Electric Fields

Charges, Coulomb s Law, and Electric Fields Q&E -1 Chages, Coulomb s Law, and Electic ields Some expeimental facts: Expeimental fact 1: Electic chage comes in two types, which we call (+) and ( ). An atom consists of a heavy (+) chaged nucleus suounded

More information

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses, 3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

More information

Gauss Law. Physics 231 Lecture 2-1

Gauss Law. Physics 231 Lecture 2-1 Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

More information

Geostrophic balance. John Marshall, Alan Plumb and Lodovica Illari. March 4, 2003

Geostrophic balance. John Marshall, Alan Plumb and Lodovica Illari. March 4, 2003 Geostophic balance John Mashall, Alan Plumb and Lodovica Illai Mach 4, 2003 Abstact We descibe the theoy of Geostophic Balance, deive key equations and discuss associated physical balances. 1 1 Geostophic

More information

Section 5-3 Angles and Their Measure

Section 5-3 Angles and Their Measure 5 5 TRIGONOMETRIC FUNCTIONS Section 5- Angles and Thei Measue Angles Degees and Radian Measue Fom Degees to Radians and Vice Vesa In this section, we intoduce the idea of angle and two measues of angles,

More information

92.131 Calculus 1 Optimization Problems

92.131 Calculus 1 Optimization Problems 9 Calculus Optimization Poblems ) A Noman window has the outline of a semicicle on top of a ectangle as shown in the figue Suppose thee is 8 + π feet of wood tim available fo all 4 sides of the ectangle

More information

Experiment 6: Centripetal Force

Experiment 6: Centripetal Force Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee

More information

Chapter 3: Vectors and Coordinate Systems

Chapter 3: Vectors and Coordinate Systems Coodinate Systems Chapte 3: Vectos and Coodinate Systems Used to descibe the position of a point in space Coodinate system consists of a fied efeence point called the oigin specific aes with scales and

More information

CHAPTER 10 Aggregate Demand I

CHAPTER 10 Aggregate Demand I CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income

More information

Butterfly Network Analysis and The Beneˇ s Network

Butterfly Network Analysis and The Beneˇ s Network 6.895 Theoy of Paallel Systems Lectue 17 Buttefly Netwok Analysis and The Beneˇ s Netwok Lectue: Chales Leiseson Lectue Summay 1. Netwok with N Nodes This section poves pat of the lowe bound on expected

More information

Magnetism. The Magnetic Force. B x x x x x x x x x x x x v x x x x x x. F = q

Magnetism. The Magnetic Force. B x x x x x x x x x x x x v x x x x x x. F = q Magnetism The Magnetic Foce F = qe + qv x x x x x x x x x x x x v x x x x x x F q v q F v F = q 0 IM intoduced the fist had disk in 1957, when data usually was stoed on tapes. It consisted of 50 plattes,

More information

Problem Set 6: Solutions

Problem Set 6: Solutions UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 16-4 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente

More information

Performance Analysis of an Inverse Notch Filter and Its Application to F 0 Estimation

Performance Analysis of an Inverse Notch Filter and Its Application to F 0 Estimation Cicuits and Systems, 013, 4, 117-1 http://dx.doi.og/10.436/cs.013.41017 Published Online Januay 013 (http://www.scip.og/jounal/cs) Pefomance Analysis of an Invese Notch Filte and Its Application to F 0

More information

Physics 235 Chapter 5. Chapter 5 Gravitation

Physics 235 Chapter 5. Chapter 5 Gravitation Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

More information

Economics 212 Microeconomic Theory I Final Exam. June Faculty of Arts and Sciences Queen s University Answer Key

Economics 212 Microeconomic Theory I Final Exam. June Faculty of Arts and Sciences Queen s University Answer Key Instuctions Economics 1 Micoeconomic Theoy I Final Exam June 008 Faculty of Ats and Sciences ueen s Univesity Anse Key The exam is thee hous in length. The exam consists of to sections: Section A has five

More information

875 Grocery Products Purchase Order

875 Grocery Products Purchase Order 875 Gocey Poducts Puchase Ode Functional Goup ID=OG Intoduction: This X12 Tansaction Set contains the fomat establishes the data contents of the Gocey Poducts Puchase Ode Tansaction Set (875) fo use within

More information

PHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013

PHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013 PHYSICS 111 HOMEWORK SOLUTION #5 Mach 3, 2013 0.1 You 3.80-kg physics book is placed next to you on the hoizontal seat of you ca. The coefficient of static fiction between the book and the seat is 0.650,

More information

12. Rolling, Torque, and Angular Momentum

12. Rolling, Torque, and Angular Momentum 12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.

More information

VISCOSITY OF BIO-DIESEL FUELS

VISCOSITY OF BIO-DIESEL FUELS VISCOSITY OF BIO-DIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use

More information

Lesson 9 Dipoles and Magnets

Lesson 9 Dipoles and Magnets Lesson 9 Dipoles and Magnets Lawence B. Rees 007. You may make a single copy of this document fo pesonal use without witten pemission. 9.0 Intoduction In this chapte we will lean about an assotment of

More information

Uncertainties in Fault Tree Analysis

Uncertainties in Fault Tree Analysis ncetainties in Fault Tee nalysis Yue-Lung Cheng Depatment of Infomation Management Husan Chuang College 48 Husan-Chuang Rd. HsinChu Taiwan R.O.C bstact Fault tee analysis is one kind of the pobilistic

More information

Notes on Electric Fields of Continuous Charge Distributions

Notes on Electric Fields of Continuous Charge Distributions Notes on Electic Fields of Continuous Chage Distibutions Fo discete point-like electic chages, the net electic field is a vecto sum of the fields due to individual chages. Fo a continuous chage distibution

More information

Physics 505 Homework No. 5 Solutions S5-1. 1. Angular momentum uncertainty relations. A system is in the lm eigenstate of L 2, L z.

Physics 505 Homework No. 5 Solutions S5-1. 1. Angular momentum uncertainty relations. A system is in the lm eigenstate of L 2, L z. Physics 55 Homewok No. 5 s S5-. Angula momentum uncetainty elations. A system is in the lm eigenstate of L 2, L z. a Show that the expectation values of L ± = L x ± il y, L x, and L y all vanish. ψ lm

More information

Episode 401: Newton s law of universal gravitation

Episode 401: Newton s law of universal gravitation Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce

More information