The Multiplicative Derivative as a Measure of Elasticity in Economics
|
|
- Kellie Harris
- 7 years ago
- Views:
Transcription
1 The Multiplicative Deivative as a Measue o Elasticity i Ecoomics - Itoductio Feado Códova-Lepe Istituto de Ciecias Básicas Uivesidad Católica del Maule codova@ucmcl I [] it was itoduced a kid o deivative o multiplicative type This deivative pemits to compae elative chages betwee a depedet positive vaiable ad the idepedet espective vaiable that it is also positive Thee we deied a ew opeatio o the positive eal umbes that makes commutative the aise to a powe opeatio Give a ad b i l(b) ] itoducig a b a as a secod opeatio this is ate the usual multiplicatio we do o the positive eal umbes a odeed ad complete ield ie a model o the eal umbes ield that lives i its iteio Fo a uctio :] ] we deie the multiplicative deivative o i ] [ as the limit i it eists Q ( ( h) l( h) h ) lim ( ) () Some o the picipal popeties that chaacteize the calculus iduced with this ew deivative ae listed i the ollowig theoem Theoem : Let g :] ] be uctios such that the deivatives Q ( ) ad Qg ( ) eist o some ] [ the: ( i) I is a costat uctio the Q ( ) ( ii) I ( ) with a eal umbe the Q ( ) e (iii) We have β β Q ( g )( ) ( Q ( )) ( Qg( )) o all ad β eal umbes (iv) We have Q( g ) ( Qg) ( Q ) ( g ) So i a > the Q ( a ) a ( v) We have Q( So i a > the + g) ( Q ) Q( + a) ( Q ) + g + a ( Qg) g + g
2 Othe impotat esult that pemits to calculate the multiplicative deivative o compouded uctios is the aalogous o the Chai Rule; it is the theoem that ollows Theoem : Let g :] ] be uctios such that the deivatives Q g( )) ad Qg ( ) eist o some ] [ the: l(qg ) ( Q ( g) ) Qg ( Q ( g)) Q ( o g) ( This deivative geeates a calculus that we called Popotioal Calculus PC Thee is a isomophism betwee PC ad the Dieetial Calculus DC This idetiicatio puts the PC ito de Newto s Calculus It is hoped the that the ew deivative that coespod to a multiplicative poit o view o the vaiatio's epesetatios (i cotast to the additive oe which is the usual pespective) will help i the solutio o poblems o the taditioal deivative Sice what it was said we ca ie that i the PC is a image that keeps the stuctue o the DC the ew calculus also is a whole that cotais aothe calculus i its iteio geeatig i this mae a iiite chai o deivatives ad its espective calculus We cite the ollowig esult that is vey useul techically because it ties both deivatives Theoem 3: Let :] ] ad some positive eal umbes The Q ) eists i oly i D ( ) eists i that case ( ) D ( ) l( Q ( )) () The multiplicative deivative has i some cases techical advatages o calculus picipally i the teatmet o uctios costucted by mi o powes multiplicatios ad thei ivese opeatios I this wok we peted to show some possible ad eplicit applicatios o the PC i Ecoomics Moe pecisely i sectio we itoduce a elatio that appeas betwee the multiplicative deivative ad the kow cocept o the elasticity The elasticity cocept is too much eteded ad useul a shot glossay cosides: Coss-pice elasticity o demad Coss-pice elasticity o supply Elastic (elasticity > ) Icome elasticity o demad Ielastic (elasticity < ) Ieio goods (icome elasticity < ) Luuies (icome elasticity > ) Necessity (icome elasticity < ) Nomal goods (icome elasticity > ) Peectly elastic cuves Peectly ielastic cuves Pice elasticity o demad Pice elasticity o supply Pice elasticity ad Uit elastic (elasticity ) We thik that the cocept o elasticity deseves to have i popety a associate calculus ad the PC is a good cadidate I sectio 3 we coect the ew deivative with the homogeeous uctios essetially we pesets a ew epesetatio o the Eule Theoem Moeove we do some special emaks o two impotat classes o these uctios we epeset the Cobb Douglas Fuctios as liea tasomatios ad o Costat Elasticity o Substitutio Fuctios (o to calculate the substitutio elasticity) we get a omula that is easy to emembe (
3 - The Elasticity Elasticity ca be thought as the cocept that measues the esposiveess o oe vaiable i espose to aothe vaiable I ecoomics the best measue o this esposiveess is the popotioal o the pecet chage i oe vaiable elative to the popotioal chage i aothe vaiable hee we ee to positive vaiables I :] ] is a uctio that elates two vaiables ad y by y () so that > ad y > ae give such that y ( ) the the popotioal chages i idepedet ad depedet vaiables ae / ad y / y espectively whee ad y y y The compaisos betwee these popotioal chages depedet vesus idepedet ae usually got by the quotiet opeatio y / y (3) / Fiig > this umbe is a uctio o so that it detemies a aveage elasticity o o the iteval + ] i > o + ] i < Fo to obtai a [ [ istataeous elasticity (poit elasticity) o at it is ecessay to do Epessio (3) i this case covets to: dy '( ) ( ) (4) y d ( ) Let be positive i has bee calculated the the lieaizatio o at this is y / '( ) i allows us to estimate the pecet chage y / y give a pecet chage / with the omula y (5) y Fo eample suppose tha i a couty is the uctio that elates the quatity demaded o meat ( y ) with the icome ( ) i i some > the icome elasticity o demad is 8 ad the pecetage chage i icome is 5% What ca we hope i the pecetage chage i demad? The aswe is y 8 5% 9% y We ca coect epessio (4) with () to wite the istataeous elasticity o at this is l( Q ( )) (6) So the elasticity is the atual logaithm o the multiplicative deivative o at A diect applicatio o (6) ad the ive items o Theoem is that they pemit to get quickly
4 the ollowig elatios amog the elasticity o a uctio ad the elasticity o its compoets Theoem 4: Let g :] ] be uctios such that ad g eist o some ] the: ( i) I is a costat uctio the l( Q ( )) l() ( ii) I ( ) with a eal umbe the l( Q ( )) l( e ) (iii) We have β β l[( Q ( )) ( Qg( )) ] g + β g o all ad β eal umbes (iv) We have l[( Qg) ( Q ) ( g )] ( ) [ g + g( )] g So i a > the l( Q( a )) l( a ) l( ) ( v) We have So i a > the a a g + g + + g + g g l[( Q ) ( Qg) ] + + g g g + a + a l[( Q ) ] + a The elatio o the chai ule o to calculate elasticity has the om l( Qg ) l[( Q ( g)) ] o g g ( ) g All these omulas the Theoem 4 ad the chai ule ae well kow; hee we do ot wat to show a uoldig oigiality oly to emak the acility i mae o to do the calculatios the immediateess o to obtai them kowig well the calculus ules o the PC which is the cost that we must pay I the PC i y () whee :] ] the the elative chage i the idepedet vaiable om to ca be measued by similaly o the idepedet vaiable this is y with y y These magitudes ca be compaed usig the ivese o the secod ( ) opeatio o the ield (] ) so we get y y l( / )
5 Epessio that i the limit whe is deoted by Q ( ) Clealy it also measue the esposiveess o vaiable y i espose to the vaiable ; so we pemit us to call it PCelasticity Let y () be the demad uctio o meat at a icome i i some > the icome PC-elasticity o demad is 65 ad the icome gowth is 5% the o to estimate the elative chage o the demad we use y y Q ( ) 65 l(5) 5 l(65) 9 So we coclude that the demad icease i 9% Theeoe ow the elasticity is a kid o deivative we do ot eed to lea a omula to calculate it A classic eecise is to id what uctios :] ] have costat elasticity With the additive deivative this poblem passes o solvig the odiay dieetial equatio '( ) + ( ) (7) With ou deivative the espective poblem is to solve Q ( ) e (8) It is vey easy to get the solutio o (7) by the method o sepaated vaiables but o solvig (8) it is eough to thik i uctios with multiplicative costat deivative seeig the ist thee popeties i Theoem we have ( ) > 3- Homogeeous Fuctios A uctio elatio: :] ] is a homogeeous uctio o ode (degee) i it satisies the ( t) t ( ) o all t > (9) It is easy to pove tha the ( ) k moeove this is equivalet to Q ( ) Q( k) Q( ) e Whe we wok with uctios o seveal vaiables this is :] ] the the coespodig deiitio is the same epessio (9) but with a vecto i ] The Eule s Homogeeous Fuctio Theoem that elates the degee o homogeeity with the patial deivatives aims that: ' ( ) + ' ( ) + L + ' ( ) i ' i ( ) () I tems o the multiplicative deivative the elatio () takes the om: Q ) Q ( ) Q ( ) e ( K ()
6 Whee () with i K is the patial multiplicative deivative o at this is Q i / l( h) [ ( K ih K ) / ( )] as h We ca ote some simple advatages o () cooted with () o istace: the autoomy o this epessio with espect to the idepedet vaiable the possibility o vebalizig as a homogeeous uctio has costat the poduct o all its patial elasticities This setece is ot diicult to memoize Thee ae two classical amilies with high utility i ecoomic theoy o homogeeous uctios they ae: a) The Cobb-Douglas Fuctios ad b) The CES (Costat Elasticity o Substitutio) Fuctios A uctio :] ] is a Cobb Douglas Fuctio (CD-uctio) i ( L L ) k whee k > o all i L The elatio () o a CD-uctio takes the om i e K e e so it is immediate that + L+ The way o to thik ad chaacteize a cocept i its elatio with othes has diect coectio with the cotet whee it is thought Fo istace we ca aim that a CDuctio is a liea tasomatio betwee vecto spaces deied o the ield (] ) ( R + ) It is hee the behid easo o to speak sometimes o the logaithmic- liea model Fo that we deie ove the set ] the poduct ( L ) ( y L y) ( y L y) It is tivial to pove that we have a commutative goup ad i this goup the eutal elemet take the om ( L) ad the ivese o a elemet ( L ) is ( L ) It is also possible to deie the scala poduct o a elemet L ) by a positive eal umbe ] by the equality ( ( L ) ( l( ) L l( ) ) This ew scala poduct satisies: ( β ) ( β ) ( y) ( ) ( y) ( β ) ( ) ( β ) ad that e o all β > ad y ] All these popeties pemit us to claim (] ) is a vecto space ove the itoduced ield o the positive eal umbes With the sets ] [ ad ] [ thought as vecto spaces o (] [ ) each liea tasomatio T :] ] is such that T[( ) ( β y)] ( T ( )) ( β T ( y)) this is the image o a liea combiatio o vectos is the liea combiatio o the images o such vectos Sice [ ( e L)] [ ( e L)] L[ ( L e)] to deie T i a uique mae it is suiciet to deie the images o the base omed by ( e L) ( e L) ( L e) i thei images ae espectively L the we have T ) ( ) ( ) L( ) (
7 β β β ie T ( ) L whee β i l( i) i K So that a CD-uctio ca be see i this cotet as a liea tasomatio Pehaps could be iteestig the study i the ecoomic ield o the mati epesetatio o a liea tasomatio betwee spaces o the om ] picipally i elatio with the tascedetal logaithm uctioal 5- CES Fuctios β β Fo a CD-uctio o two vaiables ( β > > it is deied o ) i i ( y) ( ) ] : ( ) y each > the cotou o at this value as the set C y { } The gaph o this cotou is a cocave level cuve whe we put the vaiable i uctio o the vaiable Fo to obtai the slopes we calculates implicit additive deivatives i the elatio that deie the cotou as a uctio o gettig the equality I Ecoomics the people is iteestig i to kow how this slope vaies alog o the cuve o that it is studied its itesectio with the staight lie C ad it is studied the vaiatio o such slope whe C vaies Fo a bette measue o the sesibility to the chages ad obviatig the sig it is cosideed the elasticity this is ' ' ' ( ) ' The elasticity o substitutio is deied asσ A eample o a uctio with costat elasticity o substitutio is k ( ) () v ρ δ δ + ρ ρ It ca be poved that this uctio has costat elasticity o substitutio σ (3) + ρ I i the give deiitio o substitutio elasticity we epess the Newtoias deivatives i tems o the aalogous multiplicative deivatives we have: σ + l Q l( Q l( Q ( / ) ) )
8 Note the similaity o this last epessio which is satisied o ay homogeeous uctio with the epessio (3) that was obtaied o a paticula case o CES uctio the uctio deied i () It is also possible to combie both deivatives to measue some dieet kids o esposiveess 6- Reeeces Códova-Lepe F Fom quotiet opeatio towad a popotioal calculus Joual o Mathematics Game Theoy ad Algeba (4) Accepted Madde P Cocavidad y optimizació e micoecoomía Aliaza Editoial S A Madid (987)
Understanding Financial Management: A Practical Guide Guideline Answers to the Concept Check Questions
Udestadig Fiacial Maagemet: A Pactical Guide Guidelie Aswes to the Cocept Check Questios Chapte 4 The Time Value of Moey Cocept Check 4.. What is the meaig of the tems isk-etu tadeoff ad time value of
More informationTwo degree of freedom systems. Equations of motion for forced vibration Free vibration analysis of an undamped system
wo degee of feedom systems Equatios of motio fo foced vibatio Fee vibatio aalysis of a udamped system Itoductio Systems that equie two idepedet d coodiates to descibe thei motio ae called two degee of
More informationPeriodic Review Probabilistic Multi-Item Inventory System with Zero Lead Time under Constraints and Varying Order Cost
Ameica Joual of Applied Scieces (8: 3-7, 005 ISS 546-939 005 Sciece Publicatios Peiodic Review Pobabilistic Multi-Item Ivetoy System with Zeo Lead Time ude Costaits ad Vayig Ode Cost Hala A. Fegay Lectue
More informationFinance Practice Problems
Iteest Fiace Pactice Poblems Iteest is the cost of boowig moey. A iteest ate is the cost stated as a pecet of the amout boowed pe peiod of time, usually oe yea. The pevailig maket ate is composed of: 1.
More information580.439 Course Notes: Nonlinear Dynamics and Hodgkin-Huxley Equations
58.439 Couse Notes: Noliea Dyamics ad Hodgki-Huxley Equatios Readig: Hille (3 d ed.), chapts 2,3; Koch ad Segev (2 d ed.), chapt 7 (by Rizel ad Emetout). Fo uthe eadig, S.H. Stogatz, Noliea Dyamics ad
More informationLearning Objectives. Chapter 2 Pricing of Bonds. Future Value (FV)
Leaig Objectives Chapte 2 Picig of Bods time value of moey Calculate the pice of a bod estimate the expected cash flows detemie the yield to discout Bod pice chages evesely with the yield 2-1 2-2 Leaig
More informationMoney Math for Teens. Introduction to Earning Interest: 11th and 12th Grades Version
Moey Math fo Tees Itoductio to Eaig Iteest: 11th ad 12th Gades Vesio This Moey Math fo Tees lesso is pat of a seies ceated by Geeatio Moey, a multimedia fiacial liteacy iitiative of the FINRA Ivesto Educatio
More informationDerivation of Annuity and Perpetuity Formulae. A. Present Value of an Annuity (Deferred Payment or Ordinary Annuity)
Aity Deivatios 4/4/ Deivatio of Aity ad Pepetity Fomlae A. Peset Vale of a Aity (Defeed Paymet o Odiay Aity 3 4 We have i the show i the lecte otes ad i ompodi ad Discoti that the peset vale of a set of
More informationThe dinner table problem: the rectangular case
The ie table poblem: the ectagula case axiv:math/009v [mathco] Jul 00 Itouctio Robeto Tauaso Dipatimeto i Matematica Uivesità i Roma To Vegata 00 Roma, Italy tauaso@matuiomait Decembe, 0 Assume that people
More informationAnnuities and loan. repayments. Syllabus reference Financial mathematics 5 Annuities and loan. repayments
8 8A Futue value of a auity 8B Peset value of a auity 8C Futue ad peset value tables 8D Loa epaymets Auities ad loa epaymets Syllabus efeece Fiacial mathematics 5 Auities ad loa epaymets Supeauatio (othewise
More informationBreakeven Holding Periods for Tax Advantaged Savings Accounts with Early Withdrawal Penalties
Beakeve Holdig Peiods fo Tax Advataged Savigs Accouts with Ealy Withdawal Pealties Stephe M. Hoa Depatmet of Fiace St. Boavetue Uivesity St. Boavetue, New Yok 4778 Phoe: 76-375-209 Fax: 76-375-29 e-mail:
More informationTHE PRINCIPLE OF THE ACTIVE JMC SCATTERER. Seppo Uosukainen
THE PRINCIPLE OF THE ACTIVE JC SCATTERER Seppo Uoukaie VTT Buildig ad Tapot Ai Hadlig Techology ad Acoutic P. O. Bo 1803, FIN 02044 VTT, Filad Seppo.Uoukaie@vtt.fi ABSTRACT The piciple of fomulatig the
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationSolving Logarithms and Exponential Equations
Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:
More informationMechanics 1: Motion in a Central Force Field
Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.
More informationSkills Needed for Success in Calculus 1
Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell
More informationChapter 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 informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationOn the Optimality and Interconnection of Valiant Load-Balancing Networks
O the Optimality ad Itecoectio of Valiat Load-Balacig Netwoks Moshe Babaioff ad Joh Chuag School of Ifomatio Uivesity of Califoia at Bekeley Bekeley, Califoia 94720 4600 {moshe,chuag}@sims.bekeley.edu
More information2 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 information3. Greatest Common Divisor - Least Common Multiple
3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More informationCoordinate 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 informationUNIT 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 informationNegotiation Programs
Negotiatio Pogams Javie Espaza 1 ad Jög Desel 2 1 Fakultät fü Ifomatik, Techische Uivesität Müche, Gemay espaza@tum.de 2 Fakultät fü Mathematik ud Ifomatik, FeUivesität i Hage, Gemay joeg.desel@feui-hage.de
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least
More informationLong-Term Trend Analysis of Online Trading --A Stochastic Order Switching Model
Asia Pacific Maagemet Review (24) 9(5), 893-924 Log-Tem Ted Aalysis of Olie Tadig --A Stochastic Ode Switchig Model Shalig Li * ad Zili Ouyag ** Abstact Olie bokeages ae eplacig bokes ad telephoes with
More informationI. Chi-squared Distributions
1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.
More informationMechanics 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 informationCHAPTER 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 informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationbetween Modern Degree Model Logistics Industry in Gansu Province 2. Measurement Model 1. Introduction 2.1 Synergetic Degree
www.ijcsi.og 385 Calculatio adaalysis alysis of the Syegetic Degee Model betwee Mode Logistics ad Taspotatio Idusty i Gasu Povice Ya Ya 1, Yogsheg Qia, Yogzhog Yag 3,Juwei Zeg 4 ad Mi Wag 5 1 School of
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationNontrivial 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 informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationTime Value of Money, NPV and IRR equation solving with the TI-86
Time Value of Moey NPV ad IRR Equatio Solvig with the TI-86 (may work with TI-85) (similar process works with TI-83, TI-83 Plus ad may work with TI-82) Time Value of Moey, NPV ad IRR equatio solvig with
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationSymmetric polynomials and partitions Eugene Mukhin
Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation
More informationDisplacement, Velocity And Acceleration
Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,
More informationPaper SD-07. Key words: upper tolerance limit, macros, order statistics, sample size, confidence, coverage, binomial
SESUG 212 Pae SD-7 Samle Size Detemiatio fo a Noaametic Ue Toleace Limit fo ay Ode Statistic D. Deis Beal, Sciece Alicatios Iteatioal Cooatio, Oak Ridge, Teessee ABSTRACT A oaametic ue toleace limit (UTL)
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationBENEFIT-COST ANALYSIS Financial and Economic Appraisal using Spreadsheets
BENEIT-CST ANALYSIS iacial ad Ecoomic Appraisal usig Spreadsheets Ch. 2: Ivestmet Appraisal - Priciples Harry Campbell & Richard Brow School of Ecoomics The Uiversity of Queeslad Review of basic cocepts
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More information2. 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 informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory
More informationEuler, Goldbach and Exact Values of Trigonometric Functions. Hieu D. Nguyen and Elizabeth Volz Rowan University Glassboro, NJ 08028 nguyen@rowan.
I. Itoductio Eule, Goldbach ad Exact Values of Tigoometic Fuctios Hieu D. Nguye ad Elizabeth Volz Rowa Uivesity Glassboo, NJ 0808 guye@owa.edu Jauay 9, 04 I a potio of a lette set to Chistia Goldbach o
More informationMulticomponent Systems
CE 6333, Levicky 1 Multicompoet Systems MSS TRNSFER. Mass tasfe deals with situatios i which thee is moe tha oe compoet peset i a system; fo istace, situatios ivolvig chemical eactios, dissolutio, o mixig
More informationNotes on Power System Load Flow Analysis using an Excel Workbook
Notes o owe System Load Flow Aalysis usig a Excel Woboo Abstact These otes descibe the featues of a MS-Excel Woboo which illustates fou methods of powe system load flow aalysis. Iteative techiques ae epeseted
More informationLogistic Regression, AdaBoost and Bregman Distances
A exteded abstact of this joual submissio appeaed ipoceedigs of the Thiteeth Aual Cofeece o ComputatioalLeaig Theoy, 2000 Logistic Regessio, Adaoost ad egma istaces Michael Collis AT&T Labs Reseach Shao
More informationComplex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have
Comple Numbers I spite of Calvi s discomfiture, imagiar umbers (a subset of the set of comple umbers) eist ad are ivaluable i mathematics, egieerig, ad sciece. I fact, i certai fields, such as electrical
More informationANNUITIES SOFTWARE ASSIGNMENT TABLE OF CONTENTS... 1 ANNUITIES SOFTWARE ASSIGNMENT... 2 WHAT IS AN ANNUITY?... 2 EXAMPLE 1... 2 QUESTIONS...
ANNUITIES SOFTWARE ASSIGNMENT TABLE OF CONTENTS ANNUITIES SOFTWARE ASSIGNMENT TABLE OF CONTENTS... 1 ANNUITIES SOFTWARE ASSIGNMENT... WHAT IS AN ANNUITY?... EXAMPLE 1... QUESTIONS... EXAMPLE BRANDON S
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationCHAPTER 4: NET PRESENT VALUE
EMBA 807 Corporate Fiace Dr. Rodey Boehe CHAPTER 4: NET PRESENT VALUE (Assiged probles are, 2, 7, 8,, 6, 23, 25, 28, 29, 3, 33, 36, 4, 42, 46, 50, ad 52) The title of this chapter ay be Net Preset Value,
More informationEkkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the
More informationAn 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 informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More information7.1 Finding Rational Solutions of Polynomial Equations
4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?
More informationDomain 1: Designing a SQL Server Instance and a Database Solution
Maual SQL Server 2008 Desig, Optimize ad Maitai (70-450) 1-800-418-6789 Domai 1: Desigig a SQL Server Istace ad a Database Solutio Desigig for CPU, Memory ad Storage Capacity Requiremets Whe desigig a
More informationVector 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 informationAsian Development Bank Institute. ADBI Working Paper Series
DI Wokig Pape Seies Estimatig Dual Deposit Isuace Pemium Rates ad oecastig No-pefomig Loas: Two New Models Naoyuki Yoshio, ahad Taghizadeh-Hesay, ad ahad Nili No. 5 Jauay 5 sia Developmet ak Istitute Naoyuki
More information16. Mean Square Estimation
6 Me Sque stmto Gve some fomto tht s elted to uow qutty of teest the poblem s to obt good estmte fo the uow tems of the obseved dt Suppose epeset sequece of dom vbles bout whom oe set of obsevtos e vlble
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More informationThe analysis of the Cournot oligopoly model considering the subjective motive in the strategy selection
The aalysis of the Courot oligopoly model cosiderig the subjective motive i the strategy selectio Shigehito Furuyama Teruhisa Nakai Departmet of Systems Maagemet Egieerig Faculty of Egieerig Kasai Uiversity
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationThe force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges
The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee
More informationSaturated and weakly saturated hypergraphs
Satuated and weakly satuated hypegaphs Algebaic Methods in Combinatoics, Lectues 6-7 Satuated hypegaphs Recall the following Definition. A family A P([n]) is said to be an antichain if we neve have A B
More informationSimple Annuities Present Value.
Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX-9850GB PLUS to efficietly compute values associated with preset value auities.
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationHeat (or Diffusion) equation in 1D*
Heat (or Diffusio) equatio i D* Derivatio of the D heat equatio Separatio of variables (refresher) Worked eamples *Kreysig, 8 th Ed, Sectios.4b Physical assumptios We cosider temperature i a log thi wire
More informationOMG! Excessive Texting Tied to Risky Teen Behaviors
BUSIESS WEEK: EXECUTIVE EALT ovember 09, 2010 OMG! Excessive Textig Tied to Risky Tee Behaviors Kids who sed more tha 120 a day more likely to try drugs, alcohol ad sex, researchers fid TUESDAY, ov. 9
More informationEstimating Surface Normals in Noisy Point Cloud Data
Estiatig Suface Noals i Noisy Poit Cloud Data Niloy J. Mita Stafod Gaphics Laboatoy Stafod Uivesity CA, 94305 iloy@stafod.edu A Nguye Stafod Gaphics Laboatoy Stafod Uivesity CA, 94305 aguye@cs.stafod.edu
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationFXA 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 informationBuilding Blocks Problem Related to Harmonic Series
TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite
More informationHypergeometric Distributions
7.4 Hypergeometric Distributios Whe choosig the startig lie-up for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationOn Some Functions Involving the lcm and gcd of Integer Tuples
SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 6, 2 (2014), 91-100. On Some Functions Involving the lcm and gcd of Intege Tuples O. Bagdasa Abstact:
More informationVoltage ( = Electric Potential )
V-1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is
More informationTHE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationOPTIMALLY EFFICIENT MULTI AUTHORITY SECRET BALLOT E-ELECTION SCHEME
OPTIMALLY EFFICIENT MULTI AUTHORITY SECRET BALLOT E-ELECTION SCHEME G. Aja Babu, 2 D. M. Padmavathamma Lectue i Compute Sciece, S.V. Ats College fo Me, Tiupati, Idia 2 Head, Depatmet of Compute Applicatio.
More informationNetwork Theorems - J. R. Lucas. Z(jω) = jω L
Netwo Theoems - J.. Lucas The fudametal laws that gove electic cicuits ae the Ohm s Law ad the Kichoff s Laws. Ohm s Law Ohm s Law states that the voltage vt acoss a esisto is diectly ootioal to the cuet
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationElementary Theory of Russian Roulette
Elemetary Theory of Russia Roulette -iterestig patters of fractios- Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some
More informationTHE HEIGHT OF q-binary SEARCH TREES
THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average
More informationQuestions 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 informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationLab #7: Energy Conservation
Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 1-4 Intoduction: Pehaps one of the most unusual
More informationStrategic Remanufacturing Decision in a Supply Chain with an External Local Remanufacturer
Assoiatio fo Ifomatio Systems AIS Eletoi Libay (AISeL) WHICEB 013 Poeedigs Wuha Iteatioal Cofeee o e-busiess 5-5-013 Stategi Remaufatuig Deisio i a Supply Chai with a Exteal Loal Remaufatue Xu Tiatia Shool
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More informationProblem 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 informationThe Binomial Distribution
The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More information