Important result on the first passage time and its integral functional for a certain diffusion process


 Miles Bates
 1 years ago
 Views:
Transcription
1 Lcturs Mtmátics Volumn 22 (21), págins 5 9 Importnt rsult on th first pssg tim nd its intgrl functionl for crtin diffusion procss Yousf ALZlzlh nd Bsl M. ALEidh Kuwit Univrsity, Kuwit Abstrct. In this ppr w considr th gnrl birth nd dth procss nd study som importnt rsults on th first pssg tim nd its intgrl functionl, in prticulr w driv th symptotic distribution of th intgrl functionl. Finlly w driv th symptotic rgrssion qution of first pssg tim on its intgrl functionl. Ky words nd phrss. Birth & dth procss, rgrssion qutions Mthmtics Subjct Clssifiction. Primry 62J99. Scondry 62N99. Rsumn. En st rtículo s considrn l procso gnrl d nciminto y murt y s studin lgunos rsultdos importnts ntr l primr timpo d pso y su funcionl intgrl. En prticulr, s driv l distribución sintótic dl funcionl intgrl. Finlmnt, s driv l corrspondint cución d rgrsión sintótic dl primr timpo d cruc sobr su funcionl intgrl.
2 6 YOUSEF ALZALZALAH AND BASEL M. ALEIDEH 1. Introduction Mny importnt rsults rltd to rndom vribl nd thir intgrls hv bn studid from point of viws of diffrnt uthors. For xmpl, Puri (1966), (1968) hs invstigtd th joint distribution of th numbr of survivors X(t) in th procss nd th ssocitd intgrl t Y (t) = X(z)dz. In prticulr h hs obtind limiting rsults s t. McNil (197) hs drivd th distribution of th intgrl τx functionl W x g(x(t))dt, whrτ x is th first pssg tim to th origin in gnrl birthdth procss with X() = x nd g is n rbitrry function. Also Gni nd McNil (1971) hv studid th joint distribution of both {X(t),Y(t)} nd {T x,w x } for gnrl birthdth procsss. Also in prticulr, thy hv studid th joint distribution of {T x,w x } for diffusion procss whos bckwrd qution is givn by dt = βxα d2 f dx 2, whr f(y; x, t) =P (y <X(t) y + dy X() = x). Functionls of th form ris nturlly in trffic nd storg thory. In th prsnt ppr, w considr th most gnrl diffusion procss in which th drift condition is lso xists. W driv th symptotic W x distribution of nd som symptotic rsults on th joint distribution of {T x,w x }. Also, w driv th symptotic rgrssion qution (prdiction qution) of T x on W x. W follow th nlysis of Gni nd McNil (1971). 2. Joint Distribution of {T x,w x } Considr th birthdth diffusion procss whos bckwrd Kolmogorov qution is
3 IMPORTANT RESULT ON THE FIRST PASSAGE TIME... 7 whr dt = x dx + βxα d2 f dx 2, (1) f(y; x, t) =P (y <X(t) y + dy X() = x). In ordr to insur tht th first pssg tim is finit, w tk α<2, β is th diffusion condition nd is th drift condition. Now lt M x (θ, φ) =E( θt x φw x ) (2) whr R (θ), R (φ) >. Not tht M x is th joint Lplc trnsform of {T x,w x }. This stisfis th qution (θ + φx)m x (θ, φ) =xm x(θ, φ)+βx α M x (θ, φ), (3) whr M x nd M x indict first nd scond drivtiv with rspct to x rspctivly. This cn b usd to obtin th symptotic rgrssion qution of T x on W x, whn g(x) =x. Dfin K x (φ) := d dθ M x(θ, φ) = φu E[T x W x = u]dp (W x u). (4) Ltting θ =, thn K x (φ) stisfis th diffrntil qution ( ) ( ) φ K x(φ)+ x 1 α K x x 1 α K x (φ) = 1 β β β x α M x (,φ). (5) Also, using θ = nd th boundry conditions M (,φ) = 1 nd M (,φ) = in qution (3), w gt ( M x (,φ)= βφ βφ ) + 2 4β 2 + φβ β 2 + φβ +
4 8 YOUSEF ALZALZALAH AND BASEL M. ALEIDEH Also th solution of th diffrntil qution in (5), is givn by 1 K x (φ) = 1 x1 M x 2 4β 2 + 4φβ + 1 x1 M x 2 4β 2 + 4φ β 2 + 4φ 4β 2 β 2 4β 2 + 4φβ, whr M x is th solution in qution (6). Th invrsion of qution (6), using th bhvior of th Lplc trnsforms lim φm x(,φ) = lim f W φ x x (u), whr f Wx (u) is th probbility dnsity function of W x,forlrgx. Now ftr som mnipultion w gt, 2 x 2 x f Wx = 4β 4β u if u (6) if u< This implis tht W x hs n symptotic xponntil distribution with prmtr 2 x 4β. Similrly, th invrsion of qution (7), using th symptotic bhvior of th Lplc trnsforms lim φk x(φ) = lim E [T x W x = u] f Wx (u), φ x nd ftr som mnipultion for lrg vlus of x, wgt E [T x W x = u] f Wx (u) 2 x 4β u 2 16 u u. (7) Now dividing qution (9) into qution (8), w finlly obtin th symptotic rgrssion qution of T x on W x.i.. ( ) 2 4β E [T x W x = u] u 2 16 u, u, (8) for lrg vlu of x nd α =.
5 IMPORTANT RESULT ON THE FIRST PASSAGE TIME... 9 Rfrncs [1] Gni,J. & McNil, D.R. Joint Distributions of Rndom Vribls nd thir Intgrls for Crtin BirthDth nd Diffusion Procss. Adv. Appl. Prob. 3, (1971), págs [2] McNil, D.R. Intgrl functionls of birth nd dth procsss nd rltd limiting distributions. Ann. Mth. Sttist. 41, (197), págs [3] Puri, Prms S. On th homognous birth nd dth procss nd its intgrl. Biomtrik 53,(1966), págs [4] Puri, Prm S. Som furthr rsults on th birth nd dth procss nd its intgrl. Proc. Cmb. Phil. Soc. 64, (1968), págs (Rcibido n fbrro d 21) Yousf ALZlzlh Bsl M. ALEidh Dprtmnt Of Quntittiv Mthods And Informtion Systms Kuwit Univrsity, Kuwit P.O.Box 5486, Sft 1355, Kuwit
Quality and Pricing for Outsourcing Service: Optimal Contract Design
Qulity nd Pricing for Outsourcing Srvic: Optiml Contrct Dsign Smr K. Mukhopdhyy Univrsity of WisconsinMilwuk Couthor: Xiowi Zhu, Wst Chstr Univrsity of PA Third nnul confrnc, POMS Collg of Srvic Oprtions
More informationHigher. Exponentials and Logarithms 160
hsn uknt Highr Mthmtics UNIT UTCME Eponntils nd Logrithms Contnts Eponntils nd Logrithms 6 Eponntils 6 Logrithms 6 Lws of Logrithms 6 Eponntils nd Logrithms to th Bs 65 5 Eponntil nd Logrithmic Equtions
More informationNonHomogeneous Systems, Euler s Method, and Exponential Matrix
NonHomognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous firstordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach
More information(Analytic Formula for the European Normal Black Scholes Formula)
(Analytic Formula for th Europan Normal Black Schols Formula) by Kazuhiro Iwasawa Dcmbr 2, 2001 In this short summary papr, a brif summary of Black Schols typ formula for Normal modl will b givn. Usually
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) 92.222  Linar Algbra II  Spring 2006 by D. Klain prliminary vrsion Corrctions and commnts ar wlcom! Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial
More informationThe example is taken from Sect. 1.2 of Vol. 1 of the CPN book.
Rsourc Allocation Abstract This is a small toy xampl which is wllsuitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of Cnts. Hnc, it can b rad by popl
More informationOutline. Binary Tree
Outlin Similrity Srh Th Nikolus Augstn Fr Univrsity of BoznBolzno Fulty of Computr Sin DIS 1 Binry Rprsnttion of Tr Binry Brnhs Lowr Boun for th Eit Distn Unit 10 My 17, 2012 Nikolus Augstn (DIS) Similrity
More informationChapter 3 Chemical Equations and Stoichiometry
Chptr Chmicl Equtions nd Stoichiomtry Homwork (This is VERY importnt chptr) Chptr 27, 29, 1, 9, 5, 7, 9, 55, 57, 65, 71, 75, 77, 81, 87, 91, 95, 99, 101, 111, 117, 121 1 2 Introduction Up until now w hv
More informationLast time Interprocedural analysis Dimensions of precision (flow and contextsensitivity) FlowSensitive Pointer Analysis
FlowInsnsitiv Pointr Anlysis Lst tim Intrprocurl nlysis Dimnsions of prcision (flow n contxtsnsitivity) FlowSnsitiv Pointr Anlysis Toy FlowInsnsitiv Pointr Anlysis CIS 570 Lctur 12 FlowInsnsitiv
More informationMODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
More informationLecture 3: Diffusion: Fick s first law
Lctur 3: Diffusion: Fick s first law Today s topics What is diffusion? What drivs diffusion to occur? Undrstand why diffusion can surprisingly occur against th concntration gradint? Larn how to dduc th
More informationAC Circuits ThreePhase Circuits
AC Circuits ThrPhs Circuits Contnts Wht is ThrPhs Circuit? Blnc ThrPhs oltgs Blnc ThrPhs Connction Powr in Blncd Systm Unblncd ThrPhs Systms Aliction Rsidntil Wiring Sinusoidl voltg sourcs A siml
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More informationM5A42 APPLIED STOCHASTIC PROCESSES PROBLEM SHEET 1 SOLUTIONS Term 1 20102011
M5A42 APPLIED STOCHASTIC PROCESSES PROBLEM SHEET 1 SOLUTIONS Term 1 21211 1. Clculte the men, vrince nd chrcteristic function of the following probbility density functions. ) The exponentil distribution
More informationManaging Supply Chain Backorders under Vendor Managed Inventory: A PrincipalAgent Approach and Empirical Analysis
nging Supply Chin Bckordrs undr Vndor ngd Invntory: A PrinciplAgnt Approch nd Empiricl Anlysis Yuling Yo Collg o Businss & Economics, high Univrsity Emil: yuy3@lhigh.du, Tl: 6 758676 Yn ong Crlson School
More informationTraffic Flow Analysis (2)
Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. GangLn Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,
More informationBusiness Process Simulation for Operational Decision Support
Businss Procss Simultion for Oprtionl Dcision Support M. T. Wynn 1, M. Dums 1, C. J. Fidg 1, A. H. M. tr Hofstd 1, nd W. M. P. vn dr Alst 1,2 1 Fculty of Informtion Tchnology, Qunslnd Univrsity of Tchnology,
More informationA Note on Approximating. the Normal Distribution Function
Applid Mathmatical Scincs, Vol, 00, no 9, 4549 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and
More informationChange Your History How Can Soccer Knowledge Improve Your Business Processes?
Symposium Inuurl Lctur o Hjo Rijrs, VU, 2662015 Chn Your History How Cn Soccr Knowl Improv Your Businss Procsss? Wil vn r Alst TU/ n DSC/ 1970 born Oostrbk 19881992 CS TU/ 19921994 TS TU/ 19941996
More informationg(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany
Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More informationQUANTITATIVE METHODS CLASSES WEEK SEVEN
QUANTITATIVE METHODS CLASSES WEEK SEVEN Th rgrssion modls studid in prvious classs assum that th rspons variabl is quantitativ. Oftn, howvr, w wish to study social procsss that lad to two diffrnt outcoms.
More informationEcon 371: Answer Key for Problem Set 1 (Chapter 1213)
con 37: Answr Ky for Problm St (Chaptr 23) Instructor: Kanda Naknoi Sptmbr 4, 2005. (2 points) Is it possibl for a country to hav a currnt account dficit at th sam tim and has a surplus in its balanc
More informationSharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means
Qian t al. Journal of Inqualitis and Applications (015) 015:1 DOI 10.1186/s166001507411 R E S E A R C H Opn Accss Sharp bounds for Sándor man in trms of arithmtic, gomtric and harmonic mans WiMao Qian
More informationTaxation Search in Boolean Games
Txtion Srch in Booln Gms Vdim Lvit Dpt. of Computr Scinc BnGurion Univrsity BrShv, Isrl lvitv@cs.bgu.c.il Tl Grinshpoun Dpt. of Computr Scinc BnGurion Univrsity BrShv, Isrl grinshpo@cs.bgu.c.il An
More informationSUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT. Eduard N. Klenov* RostovonDon. Russia
SUBATOMIC PARTICLES AND ANTIPARTICLES AS DIFFERENT STATES OF THE SAME MICROCOSM OBJECT Eduard N. Klnov* RostovonDon. Russia Th distribution law for th valus of pairs of th consrvd additiv quantum numbrs
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More informationOstrowski Type Inequalities and Applications in Numerical Integration. Edited By: Sever S. Dragomir. and. Themistocles M. Rassias
Ostrowski Type Inequlities nd Applictions in Numericl Integrtion Edited By: Sever S Drgomir nd Themistocles M Rssis SS Drgomir) School nd Communictions nd Informtics, Victori University of Technology,
More informationFundamentals of Tensor Analysis
MCEN 503/ASEN 50 Chptr Fundmntls of Tnsor Anlysis Fll, 006 Fundmntls of Tnsor Anlysis Concpts of Sclr, Vctor, nd Tnsor Sclr α Vctor A physicl quntity tht cn compltly dscrid y rl numr. Exmpl: Tmprtur; Mss;
More informationROMAI J., 2, 1(2006), Consider three system in fig.1.
ROMAI J.,, 1(6), 19 5 AN INVESTIGATION OF MATHEMATICAL MODELS OF A PIPELINE  PRESSURE SENSOR MECHANICAL SYSTEM Petr A. Velmisov, Yuliy V. Pokldov Ulynovsk Stte Technicl University, Russi velmisov@ulstu.ru
More informationAP Calculus AB 2008 Scoring Guidelines
AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a notforprofit mmbrship association whos mission is to connct studnts to collg succss and opportunity.
More informationOn the moments of the aggregate discounted claims with dependence introduced by a FGM copula
On th momnts of th aggrgat discountd claims with dpndnc introducd by a FGM copula  Mathiu BARGES Univrsité Lyon, Laboratoir SAF, Univrsité Laval  Hélèn COSSETTE Ecol Actuariat, Univrsité Laval, Québc,
More informationLecture note on Solid State Physics de Haasvan Alphen effect
Ltur not on Solid Stt Phsis d Hsvn Alphn fft Mstsugu Suzuki nd Itsuko S. Suzuki Stt Univrsit of Nw York t Binghmton Binghmton Nw York 396 (April 6 6) ABSTRACT Hr th phsis on th d Hsvn Alphn (dhva) fft
More informationThe Normal Distribution: A derivation from basic principles
Th Normal Distribution: A drivation from basic principls Introduction Dan Tagu Th North Carolina School of Scinc and Mathmatics Studnts in lmntary calculus, statistics, and finit mathmatics classs oftn
More informationThe Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the singlevrible chin rule extends to n inner
More informationStock and Foreign Exchange Market Linkages in Emerging Economies
Working Ppr 01013 Stock nd Forign Exchng Mrkt Linkgs in Emrging Economis Eln Androu, Mri Mtsi nd Andrs Svvids Dprtmnt of Economics, Univrsity of Cyprus, P.O. Box 0537, 1678 Nicosi, Cyprus Tl.: +357893700,
More informationInstruction: Solving Exponential Equations without Logarithms. This lecture uses a fourstep process to solve exponential equations:
49 Instuction: Solving Eponntil Equtions without Logithms This lctu uss foustp pocss to solv ponntil qutions: Isolt th bs. Wit both sids of th qution s ponntil pssions with lik bss. St th ponnts qul to
More informationLectures 8 and 9 1 Rectangular waveguides
1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves
More informationLecture 20: Emitter Follower and Differential Amplifiers
Whits, EE 3 Lctur 0 Pag of 8 Lctur 0: Emittr Followr and Diffrntial Amplifirs Th nxt two amplifir circuits w will discuss ar ry important to lctrical nginring in gnral, and to th NorCal 40A spcifically.
More informationME 612 Metal Forming and Theory of Plasticity. 6. Strain
Mtal Forming and Thory of Plasticity mail: azsnalp@gyt.du.tr Makin Mühndisliği Bölümü Gbz Yüksk Tknoloji Enstitüsü 6.1. Uniaxial Strain Figur 6.1 Dfinition of th uniaxial strain (a) Tnsil and (b) Comprssiv.
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationSIMULATION OF THE PERFECT COMPETITION AND MONOPOLY MARKET STRUCTURE IN THE COMPANY THEORY
1 SIMULATION OF THE PERFECT COMPETITION AND MONOPOLY MARKET STRUCTURE IN THE COMPANY THEORY ALEXA Vasil ABSTRACT Th prsnt papr has as targt to crat a programm in th Matlab ara, in ordr to solv, didactically
More informationUNIVERSITY OF NOTTINGHAM. Discussion Papers in Economics STRATEGIC SECOND SOURCING IN A VERTICAL STRUCTURE
UNVERSTY OF NOTTNGHAM Discussion Ppers in Economics Discussion Pper No. 04/15 STRATEGC SECOND SOURCNG N A VERTCAL STRUCTURE By Arijit Mukherjee September 004 DP 04/15 SSN 10438 UNVERSTY OF NOTTNGHAM Discussion
More informationCPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions
CPS 22 Thory of Computation REGULAR LANGUAGES Rgular xprssions Lik mathmatical xprssion (5+3) * 4. Rgular xprssion ar built using rgular oprations. (By th way, rgular xprssions show up in various languags:
More informationDeer: Predation or Starvation
: Prdation or Starvation National Scinc Contnt Standards: Lif Scinc: s and cosystms Rgulation and Bhavior Scinc in Prsonal and Social Prspctiv s, rsourcs and nvironmnts Unifying Concpts and Procsss Systms,
More informationINTERCHANGING TWO LIMITS. Zoran Kadelburg and Milosav M. Marjanović
THE TEACHING OF MATHEMATICS 2005, Vol. VIII, 1, pp. 15 29 INTERCHANGING TWO LIMITS Zorn Kdelburg nd Milosv M. Mrjnović This pper is dedicted to the memory of our illustrious professor of nlysis Slobodn
More informationNew Basis Functions. Section 8. Complex Fourier Series
Nw Basis Functions Sction 8 Complx Fourir Sris Th complx Fourir sris is prsntd first with priod 2, thn with gnral priod. Th connction with th ralvalud Fourir sris is xplaind and formula ar givn for convrting
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationAdverse Selection and Moral Hazard in a Model With 2 States of the World
Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams,
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions Section 3 Integrl Equtions Integrl Opertors nd Liner Integrl Equtions As we sw in Section on opertor nottion, we work with functions
More informationSPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM
RESEARCH PAPERS IN MANAGEMENT STUDIES SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM M.A.H. Dmpstr & S.S.G. Hong WP 26/2000 Th Judg Institut of Managmnt Trumpington Strt Cambridg CB2 1AG Ths paprs
More informationFive Dimensional Spherical Symmetric Universe in Creation Field Cosmology
Th African Rviw of Physics (2014) 9:003 277 iv Dimnsional Sphrical Symmtric Univrs in Cration ild Cosmoloy U. K. Panirahi 1, R. Patra 2 and ita Sharma 3,* 1 Acadmy of Tchnocrats, Brhampur, Odisha, India
More informationA new generalized Jacobi Galerkin operational matrix of derivatives: two algorithms for solving fourthorder boundary value problems
AbdElhmeed et l. Advnces in Difference Equtions (2016) 2016:22 DOI 10.1186/s1366201607532 R E S E A R C H Open Access A new generlized Jcobi Glerkin opertionl mtrix of derivtives: two lgorithms for
More informationApr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa
Calculus with Algebra and Trigonometry II Lecture 23 Final Review: Curve sketching and parametric equations Apr 23, 2015 Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23,
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More informationQuasilog concavity conjecture and its applications in statistics
Wen et l. Journl of Inequlities nd Applictions 2014, 2014:339 http://www.journlofinequlitiesndpplictions.com/content/2014/1/339 R E S E A R C H Open Access Qusilog concvity conjecture nd its pplictions
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More information3. Building a Binary Search Tree. 5. Splay Trees: A SelfAdjusting Data Structure
Chptr 10 BINARY TREES 1. Gnrl Binry Trs 2. Binry Srch Trs 3. Builing Binry Srch Tr 4. Hight Blnc: AVL Trs 5. Sply Trs: A SlfAjusting Dt Structur Outlin Trnsp. 1, Chptr 10, Binry Trs 243 1999 PrnticHll,
More informationEFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS
25 Vol. 3 () JanuaryMarch, pp.375/tripathi EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS *Shilpa Tripathi Dpartmnt of Chmical Enginring, Indor Institut
More informationTHE EFFECT OF GROUND SETTLEMENTS ON THE AXIAL RESPONSE OF PILES: SOME CLOSED FORM SOLUTIONS CUED/DSOILS/TR 341 (Aug 2005) By A. Klar and K.
THE EFFECT OF GROUND SETTEMENTS ON THE AXIA RESPONSE OF PIES: SOME COSED FORM SOUTIONS CUED/DSOIS/TR 4 Aug 5 By A. Klr d K. Sog Klr d Sog "Th Effct of Groud Displcmt o Axil Rspos of Pils: Som Closd Form
More informationMathematics. Mathematics 3. hsn.uk.net. Higher HSN23000
hsn uknt Highr Mathmatics UNIT Mathmatics HSN000 This documnt was producd spcially for th HSNuknt wbsit, and w rquir that any copis or drivativ works attribut th work to Highr Still Nots For mor dtails
More informationPlanning and Managing Copper Cable Maintenance through Cost Benefit Modeling
Planning and Managing Coppr Cabl Maintnanc through Cost Bnfit Modling Jason W. Rup U S WEST Advancd Tchnologis Bouldr Ky Words: Maintnanc, Managmnt Stratgy, Rhabilitation, Costbnfit Analysis, Rliability
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationEconomics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999
Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551,
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nmwide region t x
More informationArchitecture of the proposed standard
Architctur of th proposd standard Introduction Th goal of th nw standardisation projct is th dvlopmnt of a standard dscribing building srvics (.g.hvac) product catalogus basd on th xprincs mad with th
More informationDistributions. (corresponding to the cumulative distribution function for the discrete case).
Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive
More informationVague Cosets. (2) aa Aa A is vague invariant. b G then. X [0,1], where as a vague set A of set X is a pair. (1) aa ba Aa Ab if A is vague symmetric.
World Journl of Rsrch nd Rviw (WJRR) ISSN:24553956, Volum3, Issu2, ugust 206 Pgs 283 Vgu Costs Ch Mllik, N Rmkrishn, nndh Ro strct In this ppr w study th vgu costs nd thir proprtisths concpts r usd
More information3 The Utility Maximization Problem
3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best
More informationDecember Homework Week 1
Dcmbr Hmwrk Wk 1 Mth Cmmn Cr Stndrds: K.CC.A.1  Cunt t 100 by ns nd by tns. K.CC.A.2  Cunt frwrd bginning frm givn numbr within th knwn squnc (instd f hving t bgin t 1). K.CC.B.4.A  Whn cunting bjcts,
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationConstrained Renewable Resource Allocation in Fuzzy Metagraphs via Min Slack
Intrntonl Journl of ppld Oprtonl Rsrch Vol 1, No 1, pp 717, Summr 011 Journl hompg: wwworlur Constrnd Rnwl Rsourc llocton n Fuzzy Mtgrphs v Mn Slck S S Hshmn* Rcvd: Jnury 31, 011 ; ccptd: My 1, 011 strct
More informationHomogenization of a parabolic equation in perforated domain with Neumann boundary condition
Proc. Indin Acd. Sci. Mth. Sci. Vol. 112, No. 1, Februry 22, pp. 195 27. Printed in Indi Homogeniztion of prbolic eqution in perforted domin with Neumnn boundry condition A K NANDAKUMARAN nd M RAJESH Deprtment
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More informationEuler Euler Everywhere Using the EulerLagrange Equation to Solve Calculus of Variation Problems
Euler Euler Everywhere Using the EulerLgrnge Eqution to Solve Clculus of Vrition Problems Jenine Smllwood Principles of Anlysis Professor Flschk My 12, 1998 1 1. Introduction Clculus of vritions is brnch
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationMATH 425, PRACTICE FINAL EXAM SOLUTIONS.
MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator
More informationHelicopter Theme and Variations
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
More information4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS
4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem
More informationDistributed Systems Principles and Paradigms. Chapter 11: Distributed File Systems. Distributed File Systems. Example: NFS Architecture
Distriut Systms Prinipls n Prigms Mrtn vn Stn VU mstrm, Dpt. Computr Sin stn@s.vu.nl Chptr 11: Vrsion: Dmr 10, 2012 1 / 14 Gnrl gol Try to mk fil systm trnsprntly vill to rmot lints. 1. Fil mov to lint
More informationAP STATISTICS SUMMER MATH PACKET
AP STATISTICS SUMMER MATH PACKET This pcket is review of Algebr I, Algebr II, nd bsic probbility/counting. The problems re designed to help you review topics tht re importnt to your success in the clss.
More informationISE 576 Industrial Ecology 1
ISE 576 Industril Ecology 1 Instructors: Dr. M. Rhii; Offic: GER 202B; Tl: 2137404016; Fx: 2137401120; Eil: rhii@usc.du COURSE DESCRIPTIO Approchs nd tools to vlut products, procsss, nd systs in
More informationMicroanalyses of Ferroelectric Properties of BaTiO 3
Micronly of Frrolctric Proprti of BTiO io i Liu Domin Grin boundry Micronly of Frrolctric Proprti of BTiO Im Fchbrich Elktrotchnik und Informtiontchnik dr Brgichn UnivritätGmthochchul Wupprtl Zur Erlngn
More informationC H A P T E R 1 Writing Reports with SAS
C H A P T E R 1 Writing Rports with SAS Prsnting information in a way that s undrstood by th audinc is fundamntally important to anyon s job. Onc you collct your data and undrstand its structur, you nd
More informationEffect of Interrupt Logic on Delay Balancing Circuit
Intrntionl Journl of Computr Applictions (0975 8887) Volum 27 No.4, Auu 2011 Effct of Intrrupt Loc on Dly Blncin Circuit N. Sursh Kumr GIT, GITAM Univrsity Viskhptnm, Indi Dr. D.V. m Koti ddy Coll of Ennrin,
More informationInternational Association of Scientific Innovation and Research (IASIR) (An Association Unifying the Sciences, Engineering, and Applied Research)
Intrnational Association of Scintific Innovation and Rsarch (IASIR) (An Association Unifing th Scincs, Enginring, and Applid Rsarch) ISSN (Print): 79000 ISSN (Onlin): 79009 Intrnational Journal of Enginring,
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationMath Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.
Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while
More information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More information11 + Nonverbal Reasoning
Prti Tst + Nonvrl Rsoning R th instrutions rfully. Do not gin th tst or opn th ooklt until tol to o so. Work s quikly n s rfully s you n. Cirl th orrt lttr from th options givn to nswr h qustion. You
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
More informationMath 1105: Calculus II (Math/Sci majors) MWF 11am / 12pm, Campion 235 Written homework 5
Mth 5: Clculus II Mth/Sci mjos) MWF m / pm, Cmpion 35 Witten homewok 5 6.6, p. 458 3,33), 6.7, p. 467 8,3), 6.875), 7.58,6,6), 7.44,48) Fo pctice not to tun in): 6.6, p. 458,8,,3,4), 6.7, p. 467 4,6,8),
More informationSystems of First Order Linear Differential Equations
Sysms of Firs Ordr Linr Diffrnil Equions W will now urn our nion o solving sysms of simulnous homognous firs ordr linr diffrnil quions Th soluions of such sysms rquir much linr lgbr (Mh Bu sinc i is no
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationQUADRATURE METHODS. July 19, 2011. Kenneth L. Judd. Hoover Institution
QUADRATURE METHODS Kenneth L. Judd Hoover Institution July 19, 2011 1 Integrtion Most integrls cnnot be evluted nlyticlly Integrls frequently rise in economics Expected utility Discounted utility nd profits
More information200506 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 256 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
More informationtrademark and symbol guidelines FOR CORPORATE STATIONARY APPLICATIONS reviewed 01.02.2007
trdemrk nd symbol guidelines trdemrk guidelines The trdemrk Cn be plced in either of the two usul configurtions but horizontl usge is preferble. Wherever possible the trdemrk should be plced on blck bckground.
More information