STOCHASTIC CALCULUS I STOCHASTIC DIFFERENTIAL EQUATION
|
|
- Alicia Gallagher
- 5 years ago
- Views:
Transcription
1 The Bk of Thld Fcl Isuos Polcy Group Que Models & Fcl Egeerg Tem Fcl Mhemcs Foudo Noe 8 STOCHASTIC CALCULUS I STOCHASTIC DIFFERENTIAL EQUATION. ก Through he use of ordry d/or prl deres, ODE/PDE c rele oupu rle o oe or more pu rles, w.r.. whom he former s deermsc fuco s well s smuleously soluo o sd ODE/PDE. Cosder PDE wo pu rles, oe of whom specfclly deoes he pssge of me [,, whle he oher s some quy whose lue depeds deermsclly o me, whece self fuco of me x x. The soluo s of he form y y x,, [,, whose ol dere dfferel oo or ol dfferel s ge he ch rule y: y y y dx y y dx y y y x, dy dx d d d d x x d x d A me-dom sochsc process { X, } s fmly of me-dexed rdom rles, d s such c lso e hough of eher s re deermsc fuco of me d relso, X X, ω, [,, ω Ω, or s ure rdom fuco of me, X X, [,, u eer o e equed wh ure deermsc fuco of me. Dffcules rse whe oe res o formule PDE wo pu rles, oe of whom specfclly deoes he pssge of me [,, whle he oher s some quy whose lue depeds o me d rdom chces, hece self sochsc process { X, }. [Poomj Ncskul, Ph.D.][BOT-QMFE.FclMhemcsFoudo.doc][p.78][Mr.55B.E.]
2 The soluo, for whom o ol dere my e defed, les o y wy of he rdol ch rule, s of he form: Y Y X, Y X, ω,, [,, ω Ω 3 STOCHASTIC DIFFERENTIAL oo for fesml chge sochsc process {, },.e. he sese of ryg o defe: X dx lm { X X } 4 BRONIAN INNOVATION TERM sochsc dfferel oo for sdrd Brow moo/eer process: d lm{ } 5 ~ Ν, d ~ Ν, d STOCHASTIC DIFFERENTIAL EQUATION SDE geerlsed dfferel equo h cos les oe sochsc process s pu rle, hece he presece somewhere of sochsc dfferel, prculr he Brow oo erm esselly wh mkes SDE much more h jus regulr PDE wh some rdom oses dded o. GEOMETRIC BRONIAN MOTION GBM sochsc process { S, } descrle y he followg SDE: ds S µ d d, d ~ Ν, d 6 σ ITO PROCESS sochsc process { X, } descrle y he followg SDE: s deermsc pr dx d sochsc pr µ d σ d, d ~ Ν, σ d 7 3 drf 443 dffuso { oo [Poomj Ncskul, Ph.D.][BOT-QMFE.FclMhemcsFoudo.doc][p.79][Mr.55B.E.]
3 DRIFT TERM deermsc s pr of SDE o s ow forms ODE/PDE. DIFFUSION TERM sochsc d pr of SDE cog sochsc dfferels. GENERALISED IENER PROCESS sochsc process { X, } descrle y he followg SDE: s deermsc pr dx d sochsc pr µ S, d σ S, d, d ~ Ν, σ d drf 443 dffuso { oo I y ge modellg pplcos, such geerlsed eer process s o prode he uderlyg rdom dymcs w.r.. whom some quy of eres s deermsc fuco hereof; y ee, SDE s formuled erms of such ques d her deres. STOCHASTIC CALCULUS I ITO INTEGRAL As wh deermsc clculus, where he rue meg of dy y' dx s fc y dx dy ', sochsc clculus SDE s smlrly megful sofr s s hough of s smply shorhd oo for he correspodg egrl equo ; he rems o defe wh s me excly y egrl or performg egro oer sochsc dfferels d egrds! h fxed me erl [, ], sdrd Brow moo/eer process, d some rdom rgh-couous fuco of me, [, ], [, ] F : Sep Defe PARTITIONING h ddes [, ] up o > coguous, ooerlppg, lef-closed, equl-legh suerls: [, [,,, K, [,, K, [, [,, ],, K, 9 [Poomj Ncskul, Ph.D.][BOT-QMFE.FclMhemcsFoudo.doc][p.8][Mr.55B.E.]
4 Sep Defe, w.r.., [, ], ogeher wh he suerls, he BRONIAN INCREMENTS { B,, K, }, whch s deed couous-lue, dscree-me sochsc process: B ~ Ν,,, K, Sep 3 Defe, w.r.. F, [, ], ogeher wh he suerls, pproxmg RANDOM STEP FUNCTION F, [, ], whch s deed dscree-lue, couous-me sochsc process: F < F,, K, F Noe how such pproxmo c e mde rrrly close sequece coergece erms of fuco orm: F F Sep 4 Sum-mulply he lues of he rdom sep fuco F, [, ] wh he correspodg Brow cremes { B,, K, }, herey defg hs rdom rle, wh relso y me, s he DISCRETE STOCHASTIC ITO INTEGRAL F : F F B F 3 Noe how he fuco rgume s F, s per Io s formulo, d delerely eher F, s per Mll s formulo, or F F, s per Srooch s formulo. [Poomj Ncskul, Ph.D.][BOT-QMFE.FclMhemcsFoudo.doc][p.8][Mr.55B.E.]
5 The choce reflecs s moed y he ure of modelled pplco wherey F represes some kd of decso fuco h depeds o hg ll he lle formo ll ucery resoled s of me. Techclly, F s dped o he { }, {, K, } flro, u eher F or F F s; hese ler wo re cully dped o slghly lrger flro { }, {, K,, }. Icdelly, he Rem-egro logue of he dscree sochsc Io egrl s smply clled he Rem sum,.e. ohg lke dscree deermsc Rem egrl! Sep 5 I he lm s, le hs rdom rle sequece coerges me-squre o ye oher rdom rle, deed oe defed s he ITO INTEGRAL: F F B F d me squre s dscree sochsc Io egrl Io egrl 4 I essece: F d 443 Io egrl } Θ rdom rle lm Ε Expeco [ F Θ ] F Θ dω Ω Lesesgue egrl I geerl, he Io egrl s defed w.r.. he dffuso erm, hece sochsc egro oer sochsc egrd d sochsc dfferel: 5 σ Su, u d 4 43 { u 6 sochsc egrd sochsc dfferel I prculr, cosder he followg kow resuls: [Poomj Ncskul, Ph.D.][BOT-QMFE.FclMhemcsFoudo.doc][p.8][Mr.55B.E.]
6 [Poomj Ncskul, Ph.D.][BOT-QMFE.FclMhemcsFoudo.doc][p.83][Mr.55B.E.] σ σ d dy d dy d d d Here s sruce o go hrough deled sep 4 clculo w.r.. oe cse prculr, he Io egrl of where he egrd s he sdrd Brow moo/eer process self T d : For hs, he dscree sochsc Io egrl s ge y he followg rdom rle: 8 Nog he dey: 9 I he follows h:, f ω 3 Usg he followg lm resul: lm d d 3
7 [Poomj Ncskul, Ph.D.][BOT-QMFE.FclMhemcsFoudo.doc][p.84][Mr.55B.E.] From whch he sochsc Io egrl he follows: { Io correco T d lm lm lm 3 Noe he cors wh he Rem egrl, wherey: x dx 33
Interval Estimation. Consider a random variable X with a mean of X. Let X be distributed as X X
ECON 37: Ecoomercs Hypohess Tesg Iervl Esmo Wh we hve doe so fr s o udersd how we c ob esmors of ecoomcs reloshp we wsh o sudy. The queso s how comforble re we wh our esmors? We frs exme how o produce
More informationLaplace Transform. Definition of Laplace Transform: f(t) that satisfies The Laplace transform of f(t) is defined as.
Lplce Trfor The Lplce Trfor oe of he hecl ool for olvg ordry ler dfferel equo. - The hoogeeou equo d he prculr Iegrl re olved oe opero. - The Lplce rfor cover he ODE o lgerc eq. σ j ple do. I he pole o
More informationIntegral Equations and their Relationship to Differential Equations with Initial Conditions
Scece Refleco SR Vol 6 wwwscecereflecocom Geerl Leers Mhemcs GLM 6 3-3 Geerl Leers Mhemcs GLM Wese: hp://wwwscecereflecocom/geerl-leers--mhemcs/ Geerl Leers Mhemcs Scece Refleco Iegrl Equos d her Reloshp
More informationChapter Simpson s 1/3 Rule of Integration. ( x)
Cper 7. Smpso s / Rule o Iegro Aer redg s per, you sould e le o. derve e ormul or Smpso s / rule o egro,. use Smpso s / rule o solve egrls,. develop e ormul or mulple-segme Smpso s / rule o egro,. use
More information14. Poisson Processes
4. Posso Processes I Lecure 4 we roduced Posso arrvals as he lmg behavor of Bomal radom varables. Refer o Posso approxmao of Bomal radom varables. From he dscusso here see 4-6-4-8 Lecure 4 " arrvals occur
More information4. Runge-Kutta Formula For Differential Equations
NCTU Deprme o Elecrcl d Compuer Egeerg 5 Sprg Course by Pro. Yo-Pg Ce. Ruge-Ku Formul For Derel Equos To solve e derel equos umerclly e mos useul ormul s clled Ruge-Ku ormul
More informationDecompression diagram sampler_src (source files and makefiles) bin (binary files) --- sh (sample shells) --- input (sample input files)
. Iroduco Probblsc oe-moh forecs gudce s mde b 50 esemble members mproved b Model Oupu scs (MO). scl equo s mde b usg hdcs d d observo d. We selec some prmeers for modfg forecs o use mulple regresso formul.
More informationContinuous Time Markov Chains
Couous me Markov chas have seay sae probably soluos f a oly f hey are ergoc, us lke scree me Markov chas. Fg he seay sae probably vecor for a couous me Markov cha s o more ffcul ha s he scree me case,
More informationThe Existence and Uniqueness of Random Solution to Itô Stochastic Integral Equation
Appled Mhemcs,, 3, 8-84 hp://dx.do.org/.436/m..379 Pulshed Ole July (hp://www.scrp.org/jourl/m) The Exsece d Uqueess of Rdom Soluo o Iô Sochsc Iegrl Equo Hmd Ahmed Alff, Csh Wg School of Mhemcs d Iformo
More information(1) Cov(, ) E[( E( ))( E( ))]
Impac of Auocorrelao o OLS Esmaes ECON 3033/Evas Cosder a smple bvarae me-seres model of he form: y 0 x The four key assumpos abou ε hs model are ) E(ε ) = E[ε x ]=0 ) Var(ε ) =Var(ε x ) = ) Cov(ε, ε )
More informationINTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY
[Mjuh, : Jury, 0] ISSN: -96 Scefc Jourl Impc Fcr: 9 ISRA, Impc Fcr: IJESRT INTERNATIONAL JOURNAL OF ENINEERIN SCIENCES & RESEARCH TECHNOLOY HAMILTONIAN LACEABILITY IN MIDDLE RAPHS Mjuh*, MurlR, B Shmukh
More information4. Runge-Kutta Formula For Differential Equations. A. Euler Formula B. Runge-Kutta Formula C. An Example for Fourth-Order Runge-Kutta Formula
NCTU Deprme o Elecrcl d Compuer Egeerg Seor Course By Pro. Yo-Pg Ce. Ruge-Ku Formul For Derel Equos A. Euler Formul B. Ruge-Ku Formul C. A Emple or Four-Order Ruge-Ku Formul
More informationKey words: Fractional difference equation, oscillatory solutions,
OSCILLATION PROPERTIES OF SOLUTIONS OF FRACTIONAL DIFFERENCE EQUATIONS Musafa BAYRAM * ad Ayd SECER * Deparme of Compuer Egeerg, Isabul Gelsm Uversy Deparme of Mahemacal Egeerg, Yldz Techcal Uversy * Correspodg
More informationWeek 8 Lecture 3: Problems 49, 50 Fourier analysis Courseware pp (don t look at French very confusing look in the Courseware instead)
Week 8 Lecure 3: Problems 49, 5 Fourier lysis Coursewre pp 6-7 (do look Frech very cofusig look i he Coursewre ised) Fourier lysis ivolves ddig wves d heir hrmoics, so i would hve urlly followed fer he
More informationThe Poisson Process Properties of the Poisson Process
Posso Processes Summary The Posso Process Properes of he Posso Process Ierarrval mes Memoryless propery ad he resdual lfeme paradox Superposo of Posso processes Radom seleco of Posso Pos Bulk Arrvals ad
More informationChapter Trapezoidal Rule of Integration
Cper 7 Trpezodl Rule o Iegro Aer redg s per, you sould e le o: derve e rpezodl rule o egro, use e rpezodl rule o egro o solve prolems, derve e mulple-segme rpezodl rule o egro, 4 use e mulple-segme rpezodl
More informationChapter 2. Review of Hydrodynamics and Vector Analysis
her. Ree o Hdrodmcs d Vecor Alss. Tlor seres L L L L ' ' L L " " " M L L! " ' L " ' I s o he c e romed he Tlor seres. O he oher hd ' " L . osero o mss -dreco: L L IN ] OUT [mss l [mss l] mss ccmled h me
More informationModeling and Predicting Sequences: HMM and (may be) CRF. Amr Ahmed Feb 25
Modelg d redcg Sequeces: HMM d m be CRF Amr Ahmed 070 Feb 25 Bg cure redcg Sgle Lbel Ipu : A se of feures: - Bg of words docume - Oupu : Clss lbel - Topc of he docume - redcg Sequece of Lbels Noo Noe:
More informationThe Products of Regularly Solvable Operators with Their Spectra in Direct Sum Spaces
Advces Pure Mhemcs 3 3 45-49 h://dxdoorg/436/m3346 Pulshed Ole July 3 (h://wwwscrorg/ourl/m) he Producs of Regulrly Solvle Oerors wh her Secr Drec Sum Sces Sohy El-Syed Irhm Derme of Mhemcs Fculy of Scece
More informationAsymptotic Behavior of Solutions of Nonlinear Delay Differential Equations With Impulse
P a g e Vol Issue7Ver,oveber Global Joural of Scece Froer Research Asypoc Behavor of Soluos of olear Delay Dffereal Equaos Wh Ipulse Zhag xog GJSFR Classfcao - F FOR 3 Absrac Ths paper sudes he asypoc
More informationStat 6863-Handout 5 Fundamentals of Interest July 2010, Maurice A. Geraghty
S 6863-Hou 5 Fuels of Ieres July 00, Murce A. Gerghy The pror hous resse beef cl occurreces, ous, ol cls e-ulero s ro rbles. The fl copoe of he curl oel oles he ecooc ssupos such s re of reur o sses flo.
More information8. Queueing systems lect08.ppt S Introduction to Teletraffic Theory - Fall
8. Queueg sysems lec8. S-38.45 - Iroduco o Teleraffc Theory - Fall 8. Queueg sysems Coes Refresher: Smle eleraffc model M/M/ server wag laces M/M/ servers wag laces 8. Queueg sysems Smle eleraffc model
More informationSolution of Impulsive Differential Equations with Boundary Conditions in Terms of Integral Equations
Joural of aheacs ad copuer Scece (4 39-38 Soluo of Ipulsve Dffereal Equaos wh Boudary Codos Ters of Iegral Equaos Arcle hsory: Receved Ocober 3 Acceped February 4 Avalable ole July 4 ohse Rabba Depare
More informationP-Convexity Property in Musielak-Orlicz Function Space of Bohner Type
J N Sce & Mh Res Vol 3 No (7) -7 Alble ole h://orlwlsogocd/deh/sr P-Coey Proery Msel-Orlcz Fco Sce o Boher ye Yl Rodsr Mhecs Edco Deree Fcly o Ss d echology Uerss sl Neger Wlsogo Cerl Jdoes Absrcs Corresodg
More information1. Consider an economy of identical individuals with preferences given by the utility function
CO 755 Problem Se e Cbrer. Cosder ecoomy o decl dduls wh reereces e by he uly uco U l l Pre- rces o ll hree oods re ormled o oe. Idduls suly ood lbor < d cosume oods d. The oerme c mose d lorem es o oods
More informationKINEMATICS OF RIGID BODIES RELATIVE VELOCITY RELATIVE ACCELERATION PROBLEMS
KINEMTICS F RIGID DIES RELTIVE VELCITY RELTIVE CCELERTIN PRLEMS 1. The crculr dsk rolls o he lef whou slppg. If.7 m s deerme he eloc d ccelero of he ceer of he dsk. (516) .7 m s?? . The eloc of roller
More informationIntroduction to Neural Networks Computing. CMSC491N/691N, Spring 2001
Iroduco o Neurl Neorks Compug CMSC49N/69N, Sprg 00 us: cvo/oupu: f eghs: X, Y j X Noos, j s pu u, for oher us, j pu sgl here f. s he cvo fuco for j from u o u j oher books use Y f _ j j j Y j X j Y j bs:
More informationThe ray paths and travel times for multiple layers can be computed using ray-tracing, as demonstrated in Lab 3.
C. Trael me cures for mulple reflecors The ray pahs ad rael mes for mulple layers ca be compued usg ray-racg, as demosraed Lab. MATLAB scrp reflec_layers_.m performs smple ray racg. (m) ref(ms) ref(ms)
More informationA Kalman filtering simulation
A Klmn filering simulion The performnce of Klmn filering hs been esed on he bsis of wo differen dynmicl models, ssuming eiher moion wih consn elociy or wih consn ccelerion. The former is epeced o beer
More informationMATRIX AND VECTOR NORMS
Numercl lyss for Egeers Germ Jord Uversty MTRIX ND VECTOR NORMS vector orm s mesure of the mgtude of vector. Smlrly, mtr orm s mesure of the mgtude of mtr. For sgle comoet etty such s ordry umers, the
More informationThe Cauchy Problem for the Heat Equation with a Random Right Part from the Space
Appled Mhemcs, 4, 5, 38-333 Publshed Ole Augus 4 ScRes hp://wwwscrporg/jourl/m hp://dxdoorg/436/m4556 The Cuch Problem for he He Equo wh Rdom Rgh Pr from he Spce Sub ϕ ( Ω Yur Kozcheo, A Slv-Tlshch Deprme
More informationThe Lucas congruence for Stirling numbers of the second kind
ACTA ARITHMETICA XCIV 2 The Luc cogruece for Srlg umber of he ecod kd by Robero Sáchez-Peregro Pdov Iroduco The umber roduced by Srlg 7 h Mehodu dfferel [], ubequely clled Srlg umber of he fr d ecod kd,
More informationThe Linear Regression Of Weighted Segments
The Lear Regresso Of Weghed Segmes George Dael Maeescu Absrac. We proposed a regresso model where he depede varable s made o up of pos bu segmes. Ths suao correspods o he markes hroughou he da are observed
More informationScience & Technologies GENERAL BIRTH-DEATH PROCESS AND SOME OF THEIR EM (EXPETATION- MAXIMATION) ALGORITHM
GEERAL BIRH-EAH ROCESS A SOME OF HEIR EM EXEAIO- MAXIMAIO) ALGORIHM Il Hl, Lz Ker, Ylldr Seer Se ery o eoo,, eoo Mcedo l.hl@e.ed.; lz.er@e.ed.; ylldr_@hol.co ABSRAC Brh d deh roce coo-e Mrco ch, h odel
More informationBianchi Type II Stiff Fluid Tilted Cosmological Model in General Relativity
Ieraoal Joural of Mahemacs esearch. IN 0976-50 Volume 6, Number (0), pp. 6-7 Ieraoal esearch Publcao House hp://www.rphouse.com Bach ype II ff Flud led Cosmologcal Model Geeral elay B. L. Meea Deparme
More informationBrownian Motion and Stochastic Calculus. Brownian Motion and Stochastic Calculus
Browa Moo Sochasc Calculus Xogzh Che Uversy of Hawa a Maoa earme of Mahemacs Seember, 8 Absrac Ths oe s abou oob decomoso he bascs of Suare egrable margales Coes oob-meyer ecomoso Suare Iegrable Margales
More informationModified Taylor's Method and Nonlinear Mixed Integral Equation
Uversl Jourl of Iegrl quos 4 (6), 9 wwwpperscecescom Modfed Tylor's Mehod d oler Mxed Iegrl quo R T Moog Fculy of Appled Scece, Umm Al Qurh Uversy Mkh, Kgdom of Sud Ar rmoog_777@yhoocom Asrc I hs pper,
More informationChapter 2 Intro to Math Techniques for Quantum Mechanics
Wter 3 Chem 356: Itroductory Qutum Mechcs Chpter Itro to Mth Techques for Qutum Mechcs... Itro to dfferetl equtos... Boudry Codtos... 5 Prtl dfferetl equtos d seprto of vrbles... 5 Itroducto to Sttstcs...
More informationAn Application of Linear Automata to Near Rings
ppled Mhemcs,, 3, 64-68 hp://dxdoorg/436/m33 Publshed Ole November (hp://wwwscrporg/ourl/m) pplco o Ler uom o Ner Rgs Sog You, Yu Feg, Mg Co, Ypg We School o Mhemcs d Compuer Scece, Hube Uversy, Wuh, Ch
More informationNONLINEAR SYSTEM OF SINGULAR PARTIAL DIFFERENTIAL EQUATIONS
Jourl of Mhemcl Sceces: dvces d pplcos Volume 43, 27, Pges 3-53 vlble hp://scefcdvces.co. DOI: hp://d.do.org/.8642/ms_72748 OLIER SYSTEM OF SIGULR PRTIL DIFFERETIL EQUTIOS PTRICE POGÉRRD Mhemcs Lborory
More informationASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DISCRETE EQUATIONS ON DISCRETE REAL TIME SCALES
ASYPTOTI BEHAVIOR OF SOLUTIONS OF DISRETE EQUATIONS ON DISRETE REAL TIE SALES J. Dlí B. Válvíová 2 Bro Uversy of Tehology Bro zeh Repul 2 Deprme of heml Alyss d Appled hems Fuly of See Uversy of Zl Žl
More informationLecture 3 summary. C4 Lecture 3 - Jim Libby 1
Lecue su Fes of efeece Ivce ude sfoos oo of H wve fuco: d-fucos Eple: e e - µ µ - Agul oeu s oo geeo Eule gles Geec slos cosevo lws d Noehe s heoe C4 Lecue - Lbb Fes of efeece Cosde fe of efeece O whch
More informationUnscented Transformation Unscented Kalman Filter
Usceed rsformo Usceed Klm Fler Usceed rcle Fler Flerg roblem Geerl roblem Seme where s he se d s he observo Flerg s he problem of sequell esmg he ses (prmeers or hdde vrbles) of ssem s se of observos become
More informationTEACHERS ASSESS STUDENT S MATHEMATICAL CREATIVITY COMPETENCE IN HIGH SCHOOL
Jourl o See d rs Yer 5, No., pp. 5-, 5 ORIGINL PPER TECHERS SSESS STUDENT S MTHEMTICL CRETIVITY COMPETENCE IN HIGH SCHOOL TRN TRUNG TINH Musrp reeved: 9..5; eped pper:..5; Pulsed ole:..5. sr. ssessme s
More informationKINEMATICS OF RIGID BODIES RELATIVE VELOCITY RELATIVE ACCELERATION PROBLEMS
KINEMTICS OF RIGID ODIES RELTIVE VELOCITY RELTIVE CCELERTION PROLEMS 1. The crculr dsk rolls o he lef whou slppg. If.7 m s deerme he eloc d ccelero of he ceer O of he dsk. (516) .7 m s O? O? . The ed rollers
More informationIsotropic Non-Heisenberg Magnet for Spin S=1
Ierol Jourl of Physcs d Applcos. IN 974- Volume, Number (, pp. 7-4 Ierol Reserch Publco House hp://www.rphouse.com Isoropc No-Heseberg Mge for p = Y. Yousef d Kh. Kh. Mumov.U. Umrov Physcl-Techcl Isue
More informationArea and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]
Are d the Defte Itegrl 1 Are uder Curve We wt to fd the re uder f (x) o [, ] y f (x) x The Prtto We eg y prttog the tervl [, ] to smller su-tervls x 0 x 1 x x - x -1 x 1 The Bsc Ide We the crete rectgles
More informationSolution set Stat 471/Spring 06. Homework 2
oluo se a 47/prg 06 Homework a Whe he upper ragular elemes are suppressed due o smmer b Le Y Y Y Y A weep o he frs colum o oba: A ˆ b chagg he oao eg ad ec YY weep o he secod colum o oba: Aˆ YY weep o
More informationVARIATIONAL ITERATION METHOD FOR DELAY DIFFERENTIAL-ALGEBRAIC EQUATIONS. Hunan , China,
Mahemacal ad Compuaoal Applcaos Vol. 5 No. 5 pp. 834-839. Assocao for Scefc Research VARIATIONAL ITERATION METHOD FOR DELAY DIFFERENTIAL-ALGEBRAIC EQUATIONS Hoglag Lu Aguo Xao Yogxag Zhao School of Mahemacs
More informationPartial Molar Properties of solutions
Paral Molar Properes of soluos A soluo s a homogeeous mxure; ha s, a soluo s a oephase sysem wh more ha oe compoe. A homogeeous mxures of wo or more compoes he gas, lqud or sold phase The properes of a
More informationSoo King Lim Figure 1: Figure 2: Figure 3: Figure 4: Figure 5: Figure 6: Figure 7: Figure 8: Figure 9: Figure 10: Figure 11:
Soo Kg Lm 1.0 Nested Fctorl Desg... 1.1 Two-Fctor Nested Desg... 1.1.1 Alss of Vrce... Exmple 1... 5 1.1. Stggered Nested Desg for Equlzg Degree of Freedom... 7 1.1. Three-Fctor Nested Desg... 8 1.1..1
More informationChapter 5 Transient Analysis
hpr 5 rs Alyss Jsug Jg ompl rspos rs rspos y-s rspos m os rs orr co orr Dffrl Equo. rs Alyss h ffrc of lyss of crcus wh rgy sorg lms (ucors or cpcors) & m-ryg sgls wh rss crcus s h h quos rsulg from r
More informationReinforcement Learning
Reiforceme Corol lerig Corol polices h choose opiml cios Q lerig Covergece Chper 13 Reiforceme 1 Corol Cosider lerig o choose cios, e.g., Robo lerig o dock o bery chrger o choose cios o opimize fcory oupu
More informationReal-Time Systems. Example: scheduling using EDF. Feasibility analysis for EDF. Example: scheduling using EDF
EDA/DIT6 Real-Tme Sysems, Chalmers/GU, 0/0 ecure # Updaed February, 0 Real-Tme Sysems Specfcao Problem: Assume a sysem wh asks accordg o he fgure below The mg properes of he asks are gve he able Ivesgae
More informationFALL HOMEWORK NO. 6 - SOLUTION Problem 1.: Use the Storage-Indication Method to route the Input hydrograph tabulated below.
Jorge A. Ramírez HOMEWORK NO. 6 - SOLUTION Problem 1.: Use he Sorage-Idcao Mehod o roue he Ipu hydrograph abulaed below. Tme (h) Ipu Hydrograph (m 3 /s) Tme (h) Ipu Hydrograph (m 3 /s) 0 0 90 450 6 50
More information10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n
0. Sere I th ecto, we wll troduce ere tht wll be dcug for the ret of th chpter. Wht ere? If we dd ll term of equece, we get whch clled fte ere ( or jut ere) d deoted, for hort, by the ymbol or Doe t mke
More informationUpper Bound For Matrix Operators On Some Sequence Spaces
Suama Uer Bou formar Oeraors Uer Bou For Mar Oeraors O Some Sequece Saces Suama Dearme of Mahemacs Gaah Maa Uersy Yogyaara 558 INDONESIA Emal: suama@ugmac masomo@yahoocom Isar D alam aer aa susa masalah
More informationAML710 CAD LECTURE 12 CUBIC SPLINE CURVES. Cubic Splines Matrix formulation Normalised cubic splines Alternate end conditions Parabolic blending
CUIC SLINE CURVES Cubc Sples Marx formulao Normalsed cubc sples Alerae ed codos arabolc bledg AML7 CAD LECTURE CUIC SLINE The ame sple comes from he physcal srume sple drafsme use o produce curves A geeral
More informationCyclone. Anti-cyclone
Adveco Cycloe A-cycloe Lorez (963) Low dmesoal aracors. Uclear f hey are a good aalogy o he rue clmae sysem, bu hey have some appealg characerscs. Dscusso Is he al codo balaced? Is here a al adjusme
More informationQuantum Mechanics II Lecture 11 Time-dependent perturbation theory. Time-dependent perturbation theory (degenerate or non-degenerate starting state)
Pro. O. B. Wrgh, Auum Quaum Mechacs II Lecure Tme-depede perurbao heory Tme-depede perurbao heory (degeerae or o-degeerae sarg sae) Cosder a sgle parcle whch, s uperurbed codo wh Hamloa H, ca exs a superposo
More informationExpectation and Moments
Her Sr d Joh W. Woods robbl Sscs d Rdom Vrbles or geers 4h ed. erso duco Ic.. ISB: 978----6 Cher 4 eco d omes Secos 4. eced Vlue o Rdom Vrble 5 O he Vld o quo 4.-8 8 4. Codol ecos Codol eco s Rdom Vrble
More information4.1 Schrödinger Equation in Spherical Coordinates
Phs 34 Quu Mehs D 9 9 Mo./ Wed./ Thus /3 F./4 Mo., /7 Tues. / Wed., /9 F., /3 4.. -. Shodge Sphe: Sepo & gu (Q9.) 4..-.3 Shodge Sphe: gu & d(q9.) Copuo: Sphe Shodge s 4. Hdoge o (Q9.) 4.3 gu Moeu 4.4.-.
More informationLeast Squares Fitting (LSQF) with a complicated function Theexampleswehavelookedatsofarhavebeenlinearintheparameters
Leas Squares Fg LSQF wh a complcaed fuco Theeampleswehavelookedasofarhavebeelearheparameers ha we have bee rg o deerme e.g. slope, ercep. For he case where he fuco s lear he parameers we ca fd a aalc soluo
More informationXidian University Liu Congfeng Page 1 of 49
dom Sgl Processg Cher4 dom Processes Cher 4 dom Processes Coes 4 dom Processes... 4. Deo o dom Process... 4. Chrcerzo o dom Process...4 4.. ol Chrcerzo o dom Process...4 4.. Frs-Order Deses o dom Process...5
More informationDetermination of Antoine Equation Parameters. December 4, 2012 PreFEED Corporation Yoshio Kumagae. Introduction
refeed Soluos for R&D o Desg Deermao of oe Equao arameers Soluos for R&D o Desg December 4, 0 refeed orporao Yosho Kumagae refeed Iroduco hyscal propery daa s exremely mpora for performg process desg ad
More informationSOME REMARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOMIAL ASYMPTOTE
D I D A C T I C S O F A T H E A T I C S No (4) 3 SOE REARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOIAL ASYPTOTE Tdeusz Jszk Abstct I the techg o clculus, we cosde hozotl d slt symptote I ths ppe the
More informationThe Properties of Probability of Normal Chain
I. J. Coep. Mah. Sceces Vol. 8 23 o. 9 433-439 HIKARI Ld www.-hkar.co The Properes of Proaly of Noral Cha L Che School of Maheacs ad Sascs Zheghou Noral Uversy Zheghou Cy Hea Provce 4544 Cha cluu6697@sa.co
More information4.8 Improper Integrals
4.8 Improper Inegrls Well you ve mde i hrough ll he inegrion echniques. Congrs! Unforunely for us, we sill need o cover one more inegrl. They re clled Improper Inegrls. A his poin, we ve only del wih inegrls
More informationThe z-transform. LTI System description. Prof. Siripong Potisuk
The -Trsform Prof. Srpog Potsuk LTI System descrpto Prevous bss fucto: ut smple or DT mpulse The put sequece s represeted s ler combto of shfted DT mpulses. The respose s gve by covoluto sum of the put
More informationMaximum likelihood estimate of phylogeny. BIOL 495S/ CS 490B/ MATH 490B/ STAT 490B Introduction to Bioinformatics April 24, 2002
Mmm lkelhood eme of phylogey BIO 9S/ S 90B/ MH 90B/ S 90B Iodco o Bofomc pl 00 Ovevew of he pobblc ppoch o phylogey o k ee ccodg o he lkelhood d ee whee d e e of eqece d ee by ee wh leve fo he eqece. he
More informationAvailable online Journal of Scientific and Engineering Research, 2014, 1(1): Research Article
Avalable ole wwwjsaercom Joural o Scec ad Egeerg Research, 0, ():0-9 Research Arcle ISSN: 39-630 CODEN(USA): JSERBR NEW INFORMATION INEUALITIES ON DIFFERENCE OF GENERALIZED DIVERGENCES AND ITS APPLICATION
More informationThe Signal, Variable System, and Transformation: A Personal Perspective
The Sgal Varable Syem ad Traformao: A Peroal Perpecve Sherv Erfa 35 Eex Hall Faculy of Egeerg Oule Of he Talk Iroduco Mahemacal Repreeao of yem Operaor Calculu Traformao Obervao O Laplace Traform SSB A
More informationMidterm Exam. Tuesday, September hour, 15 minutes
Ecoomcs of Growh, ECON560 Sa Fracsco Sae Uvers Mchael Bar Fall 203 Mderm Exam Tuesda, Sepember 24 hour, 5 mues Name: Isrucos. Ths s closed boo, closed oes exam. 2. No calculaors of a d are allowed. 3.
More informationTechnical Appendix for Inventory Management for an Assembly System with Product or Component Returns, DeCroix and Zipkin, Management Science 2005.
Techc Appedx fo Iveoy geme fo Assemy Sysem wh Poduc o Compoe eus ecox d Zp geme Scece 2005 Lemm µ µ s c Poof If J d µ > µ he ˆ 0 µ µ µ µ µ µ µ µ Sm gumes essh he esu f µ ˆ > µ > µ > µ o K ˆ If J he so
More informationChapter 7. Bounds for weighted sums of Random Variables
Chpter 7. Bouds for weghted sums of Rdom Vrbles 7. Itroducto Let d 2 be two depedet rdom vrbles hvg commo dstrbuto fucto. Htczeko (998 d Hu d L (2000 vestgted the Rylegh dstrbuto d obted some results bout
More informationQR factorization. Let P 1, P 2, P n-1, be matrices such that Pn 1Pn 2... PPA
QR facorzao Ay x real marx ca be wre as AQR, where Q s orhogoal ad R s upper ragular. To oba Q ad R, we use he Householder rasformao as follows: Le P, P, P -, be marces such ha P P... PPA ( R s upper ragular.
More informationAn improved Bennett s inequality
COMMUNICATIONS IN STATISTICS THEORY AND METHODS 017,VOL.0,NO.0,1 8 hps://do.org/10.1080/0361096.017.1367818 A mproved Bee s equly Sogfeg Zheg Deprme of Mhemcs, Mssour Se Uversy, Sprgfeld, MO, USA ABSTRACT
More informationChapter Lagrangian Interpolation
Chaper 5.4 agrangan Inerpolaon Afer readng hs chaper you should be able o:. dere agrangan mehod of nerpolaon. sole problems usng agrangan mehod of nerpolaon and. use agrangan nerpolans o fnd deraes and
More informationECE Microwave Engineering. Fall Prof. David R. Jackson Dept. of ECE. Notes 10. Waveguides Part 7: Transverse Equivalent Network (TEN)
EE 537-635 Microwve Engineering Fll 7 Prof. Dvid R. Jcson Dep. of EE Noes Wveguides Pr 7: Trnsverse Equivlen Newor (N) Wveguide Trnsmission Line Model Our gol is o come up wih rnsmission line model for
More informationRademacher Complexity. Examples
Algorthmc Foudatos of Learg Lecture 3 Rademacher Complexty. Examples Lecturer: Patrck Rebesch Verso: October 16th 018 3.1 Itroducto I the last lecture we troduced the oto of Rademacher complexty ad showed
More informationThrough the fractional Riemann Liouville integral x
Volue 7 Issue 5 M 7 ISSN: 77 8X Ierol ourl o Advced Reserch Copuer Scece d Sowre geerg Reserch Pper Avlle ole : wwwjrcsseco se he Soluo o Frcol erel quos wh Trscedel Fucos Mukesh Grover r Aru Kur Toer
More informationHow to explore replicator equations? G.P. Karev
How o explore replcor equos? GP Krev Locheed r SD Nol Isue of Helh Bldg 38 R 5N5N 86 Rocvlle Pe Behes D 2894 US E-l: rev@clhgov src Replcor equos RE) re og he sc ools hecl heory of seleco d evoluo We develop
More informationReview for the Midterm Exam.
Review for he iderm Exm Rememer! Gross re e re Vriles suh s,, /, p / p, r, d R re gross res 2 You should kow he disiio ewee he fesile se d he udge se, d kow how o derive hem The Fesile Se Wihou goverme
More informationIntegration by Parts for D K
Itertol OPEN ACCESS Jourl Of Moder Egeerg Reserc IJMER Itegrto y Prts for D K Itegrl T K Gr, S Ry 2 Deprtmet of Mtemtcs, Rgutpur College, Rgutpur-72333, Purul, West Begl, Id 2 Deprtmet of Mtemtcs, Ss Bv,
More information3. REVIEW OF PROPERTIES OF EIGENVALUES AND EIGENVECTORS
. REVIEW OF PROPERTIES OF EIGENVLUES ND EIGENVECTORS. EIGENVLUES ND EIGENVECTORS We hll ow revew ome bc fct from mtr theory. Let be mtr. clr clled egevlue of f there et ozero vector uch tht Emle: Let 9
More informationMathematics 805 Final Examination Answers
. 5 poins Se he Weiersrss M-es. Mhemics 85 Finl Eminion Answers Answer: Suppose h A R, nd f n : A R. Suppose furher h f n M n for ll A, nd h Mn converges. Then f n converges uniformly on A.. 5 poins Se
More informationPatterns of Continued Fractions with a Positive Integer as a Gap
IOSR Jourl of Mthemtcs (IOSR-JM) e-issn: 78-578, -ISSN: 39-765X Volume, Issue 3 Ver III (My - Ju 6), PP -5 wwwosrjourlsorg Ptters of Cotued Frctos wth Postve Iteger s G A Gm, S Krth (Mthemtcs, Govermet
More informationFundamentals of Speech Recognition Suggested Project The Hidden Markov Model
. Projec Iroduco Fudameals of Speech Recogo Suggesed Projec The Hdde Markov Model For hs projec, s proposed ha you desg ad mpleme a hdde Markov model (HMM) ha opmally maches he behavor of a se of rag sequeces
More informatione t dt e t dt = lim e t dt T (1 e T ) = 1
Improper Inegrls There re wo ypes of improper inegrls - hose wih infinie limis of inegrion, nd hose wih inegrnds h pproch some poin wihin he limis of inegrion. Firs we will consider inegrls wih infinie
More informationLeast squares and motion. Nuno Vasconcelos ECE Department, UCSD
Leas squares ad moo uo Vascocelos ECE Deparme UCSD Pla for oda oda we wll dscuss moo esmao hs s eresg wo was moo s ver useful as a cue for recogo segmeao compresso ec. s a grea eample of leas squares problem
More informationCalculation of Effective Resonance Integrals
Clculo of ffecve Resoce egrls S.B. Borzkov FLNP JNR Du Russ Clculo of e effecve oce egrl wc cludes e rel eerg deedece of euro flux des d correco o e euro cure e smle s eeded for ccure flux deermo d euro
More informationBEST PATTERN OF MULTIPLE LINEAR REGRESSION
ERI COADA GERMAY GEERAL M.R. SEFAIK AIR FORCE ACADEMY ARMED FORCES ACADEMY ROMAIA SLOVAK REPUBLIC IERAIOAL COFERECE of SCIEIFIC PAPER AFASES Brov 6-8 M BES PAER OF MULIPLE LIEAR REGRESSIO Corel GABER PEROLEUM-GAS
More informationONE RANDOM VARIABLE F ( ) [ ] x P X x x x 3
The Cumulive Disribuio Fucio (cd) ONE RANDOM VARIABLE cd is deied s he probbiliy o he eve { x}: F ( ) [ ] x P x x - Applies o discree s well s coiuous RV. Exmple: hree osses o coi x 8 3 x 8 8 F 3 3 7 x
More information0 for t < 0 1 for t > 0
8.0 Sep nd del funcions Auhor: Jeremy Orloff The uni Sep Funcion We define he uni sep funcion by u() = 0 for < 0 for > 0 I is clled he uni sep funcion becuse i kes uni sep = 0. I is someimes clled he Heviside
More informationFresnel Equations cont.
Lecure 12 Chaper 4 Fresel quaos co. Toal eral refleco ad evaesce waves Opcal properes of meals Laer: Famlar aspecs of he eraco of lgh ad maer Fresel quaos r 2 Usg Sell s law, we ca re-wre: r s s r a a
More informationThe sphere of radius a has the geographical form. r (,)=(acoscos,acossin,asin) T =(p(u)cos v, p(u)sin v,q(u) ) T.
Che 5. Dieeil Geome o Sces 5. Sce i meic om I 3D sce c be eeseed b. Elici om z =. Imlici om z = 3. Veco om = o moe geel =z deedig o wo mees. Emle. he shee o dis hs he geoghicl om =coscoscossisi Emle. he
More informationCyclically Interval Total Colorings of Cycles and Middle Graphs of Cycles
Ope Joural of Dsree Mahemas 2017 7 200-217 hp://wwwsrporg/joural/ojdm ISSN Ole: 2161-7643 ISSN Pr: 2161-7635 Cylally Ierval Toal Colorgs of Cyles Mddle Graphs of Cyles Yogqag Zhao 1 Shju Su 2 1 Shool of
More informationMTH 146 Class 11 Notes
8.- Are of Surfce of Revoluion MTH 6 Clss Noes Suppose we wish o revolve curve C round n is nd find he surfce re of he resuling solid. Suppose f( ) is nonnegive funcion wih coninuous firs derivive on he
More informationELEC 6041 LECTURE NOTES WEEK 3 Dr. Amir G. Aghdam Concordia University
ecre Noe Prepared b r G. ghda EE 64 ETUE NTE WEE r. r G. ghda ocorda Uer eceraled orol e - Whe corol heor appled o a e ha co of geographcall eparaed copoe or a e cog of a large ber of p-op ao ofe dered
More informationI I M O I S K J H G. b gb g. Chapter 8. Problem Solutions. Semiconductor Physics and Devices: Basic Principles, 3 rd edition Chapter 8
emcouc hyscs evces: Bsc rcles, r eo Cher 8 oluos ul rolem oluos Cher 8 rolem oluos 8. he fwr s e ex f The e ex f e e f ex () () f f f f l G e f f ex f 59.9 m 60 m 0 9. m m 8. e ex we c wre hs s e ex h
More information