A Modeling Method of SISO Discrete-Event Systems in Max Algebra

Size: px
Start display at page:

Download "A Modeling Method of SISO Discrete-Event Systems in Max Algebra"

Transcription

1 A Modelg Mehod of SISO Dscee-Eve Syses Max Algeba Jea-Lous Bood, Laue Hadou, P. Cho To ce hs veso: Jea-Lous Bood, Laue Hadou, P. Cho. A Modelg Mehod of SISO Dscee-Eve Syses Max Algeba. Euopea Cool Cofeece ECC 95, Sep 1995, Roe, Ialy. pp , <hal > HAL Id: hal hps://hal.achves-ouvees.f/hal Subed o 11 Jul 213 HAL s a ul-dscplay ope access achve fo he depos ad dsseao of scefc eseach docues, whehe hey ae publshed o o. The docues ay coe fo eachg ad eseach suos Face o aboad, o fo publc o pvae eseach cees. L achve ouvee pludscplae HAL, es desée au dépô e à la dffuso de docues scefues de veau echeche, publés ou o, éaa des éablssees d esegee e de echeche faças ou éages, des laboaoes publcs ou pvés.

2 A MODELIN METHOD OF SISO DISCRETE-EVENT SYSTEMS IN MAX-ALEBRA J.-L. Bood, L. Hadou, P. Cho L.I.S.A. 62, aveue Noe Dae du Lac 49 ANERS Face Keywods : Dscee-eve syses, (ax, +) algeba, defcao, asfe fuco, ARMA odel. Absac Ths pape deals wh he odelg of he e behavo of dscee-eve syses (ax, +) algeba. The syses we cosde ae lealy odeled hs algeba. The poposed ehod s sped by coveoal lea syse heoy : fo a ARMA fo of he odel ad he syse pulse espose, he odel paaees ae copued by usg a basc esul of Resduao Theoy ode o ze a eo ceo. 1 Ioduco I auoac cool we eed a aheacal odel o chaaceze syse popees (coadably, sably,...) o o desg he cool law of a syse. Theefoe, s poa o oba a odel of he syse whch should, as fa as possble, be boh ealsc ad sple. I coveoal lea syse heoy ay wos deal wh hs poble. These sudes have allowed o develop ue sple ad elavely effecve ehods ode o defy coveoal lea syses. The pupose of hs pape s o popose a odelg ehod of dscee-eve syse (DES). Ths ehod uses he ax-plus algebac oao ode o do aaloges wh he coveoal lea syse heoy. To have a lea epeseao of a syse (ax, +) algeba (fo shoess we wll oe ax-algeba), we cosde DES whch oly sychozao pheoea appea. Moeove, hey ae deesc ad we esc ouselves o sgle-pu sgle-oupu (SISO) syses. I he secod seco we befly ecall he axalgebac opeaos ad he a epeseaos of DES. A coplee oduco o ax-algeba ca be foud [2], [3]. The hd seco s coceed wh he defcao ehod ax-algeba whch s aly based o he Resduao Theoy [1], [2]. I he fouh seco, we popose a paccal odelg pocedue. 2 Noaos ad Descpos of DES Max- Algeba We cosde he seg ( { + } laws, ae defed by R,,, ) whee he a b= ax ( a, b), a b= a+ b The elee s eual fo he law ad absobg fo, he elee oed e s eual fo. We ca oe ha s depoe,.e., a a = a. Fsly we ecall he usual ecue lea euaos axalgeba x( + 1) = A x( ) B u( + 1) y( ) = C x( ) x = ( x1 x ) s he desoal odel sae. u, x ad y ae called cool, h odel sae ad odel oupu especvely. The odel descbed by E. (1) ca also be epeseed by a ed eve gaph whee u, ad y ae x (1)

3 asos ; u( ), x ( ) ad y( ) epese he daes whe u, x ad y ae especvely fed fo he h e. I he followg, he sg wll be oed. The use of he γ-asfo [2], [4], whee γ opeaes as he z 1 opeao of coveoal syse heoy, leads o he followg epeseao of E. (1) X ( γ ) = Aγ X ( γ ) BU ( γ ) Y ( γ ) = C X ( γ ) whee U( γ ), X ( γ ) ad Y( γ ) ae he γ-asfo of u, x ad y especvely. A basc heoe [2], [4] shows ha he leas soluo of E. (2) s gve by Y( γ) C( Aγ) BU( γ) = wh ( A γ ) = ( Aγ ) + = ad hus he pu-oupu behavo of E. (2) ca also be descbed by he asfe fuco h( γ) = C( Aγ) B whch ca also be expessed as a polyoal expesso of he fo whee p( γ) ν h( γ) = p( γ) γ ( γ)( sγ ) (3) ν 1 = = p γ 1 = =, ( γ) γ (2) ad s s a ooal [2]. Le us ed ha h(γ) ca be cosdeed as he γ-asfo of he pulse espose of he odel. Theefoe, he h(γ) expesso ca expess ha he pae epeseed by (γ) s defely epoduced because he ulplcao by s γ sybolzes a u abscssa shf ad a s u odae shf. Ths peodc behavo begs afe a ase behavo whch ay be epeseed by p(γ). We us oe ha such a ealzao s o ecessaly al [2, 6.5.4], howeve allows a sple epeao of h(γ). Fo exaple, he asfe fuco h( γ) = e 1γ 3γ γ ( 6 8γ 9γ )( 7γ ) leads o he pulse espose llusaed Fg. 1. The pupose s o defy paaees of he asfe fuco h( γ ) descbed by E. (3),.e., paaees of polyoals p( γ ) ad ( γ ) ad ooal s, by usg he syse pulse espose. The ( s γ ) e of he asfe fuco h( γ ) ples a fe epeo of he ( γ ) pae whch eas a fe ube of paaees o copue. Hece as coveoal syse heoy, we asfo he asfe fuco h( γ ) o a ARMA fo o oba a fe ube of paaees. A soluo s gve [2, 9.2.2] o oba hs ARMA fo, howeve we pefe a sple ehod whch s : Assug ha h( γ ) s defed by E. (3), he Y( γ ) ad U( γ ) sasfy he ARMA euao Y( γ) sγ p( γ) U( γ) = p( γ) γ ( γ) U( γ) sγ Y( γ) Poof s easly obaed by expessg E. (3) as Y( γ) = p( γ) γ ( γ)( e sγ ( sγ ) ) U( γ) whch ca be we as Y ( γ ) = 2 [ ] [ ] a1 p( γ ) γ ( γ ) U ( γ ) γ ( γ ) sγ ( sγ ) U ( γ ) a2 a3 O he ohe had he ulplcao of E. (3) by s γ leads o sγ Y( γ) = sγ p( γ) U( γ) sγ γ ( γ)( sγ ) U( γ) a4 Because a a5 a6 = a, hece we have 3 6 a1 a5 = a2 a3 a5 = a2 a5 a6 = a2 a4 whch leads o he esul h (γ) ase behavo ase ν pae s peodc behavo γ Fgue 1 : Ipulse espose coespodg o h( γ ). ( ν = 3, = 3, s = 7) (4)

4 3 Idefcao Mehod I hs seco we develop a defcao ehod of a ARMA odel ax-algeba whch wll be used o copue he paaees of he asfe fuco h( γ ). The poposed appoach offes a aalogy wh coveoal dscee e syse heoy. Le us ed ha a lea dscee e syse ca be expessed as he ARMA euao y ( ) + a y ( ) = b u( ) = 1 = To esae he paaees of hs ARMA euao,.e., a1,, a, b,, b, soe defcao ehods cosde he followg pedco eo ε( ) = y ( ) + a y ( ) b u( ) = 1 = whee y s he easued syse oupu ad a,, a, b,, b 1 ae he esaed paaees. Such a pedco eo allows o defe a ceo whch s zed fo he se of paaees seached. Fo exaple a uadac ceo s used he well ow leas suae ehod [5, chap. 7]. ε = Y ( M θ ) whee ε ε ε = ( ) ( N) s he pedco eo veco, Y = y ( ) y ( N) s he easued syse oupu veco ad M s he ax ϕ possble o defe a ceo J as N J( θ ) = ( ) = ε ϕ N. Veco ε aes To deee he esaed paaees veco θ whch zes hs ceo, we cosde a basc esul of Resduao Theoy [1], [2] whch saes ha ( ) Y θ = M (6) s he geaes subsoluo of Y = M θ whee efes o he ulplcao of wo aces whch he opeao s used ahe ha he ax-opeao [2]. Thus, by usg hs geaes subsoluo, each elee of he pedco eo veco ε s always posve ad J( θ ) s zed. 3.1 Idefcao Mehod of Max-Algeba ARMA Model Le us suppose ha odel ca be descbed axalgeba by he ARMA euao 1 Y( γ) = b b γ U( γ) a γ a γ Y( γ) The defcao ehod we popose s based o he syse pulse espose,.e., u ( ) = Slaly o ohewse he coveoal syse heoy, we defe he pedco eo ε( ) y ( = ) ( ϕ θ ) whee y s he easued syse oupu, ϕ = u ( ) u ( ) y ( 1 ) y ( ) s he egesso veco a eve ad θ= bb a a s he esaed paaees veco. 1 By usg he daa of u( ) ad y ( ) wh = o N ad N > +, oe ca oba he ax expesso (5) 3.2 Idefcao Mehod of Tasfe Fuco h(γ) The defcao ehod poposed he pevous seco eeds o have a pacula lea ARMA fo (see E. (5)). Because hs codo s o vefed by he ARMA euao (4) (owg o he sγ p( γ) U( γ) e), we popose o defy ase pa ad peodc pa sepaaely. Ths pocedue leads us o cosde he wo followg euaos ahe ha E. (4). Whe we cosde a pulse espose whou peodc behavo,.e., ( γ ) =, E. (3) s educed o he sple MA euao Y( γ ) = p( γ ) U( γ ) (7) Slaly, a pulse espose whou ase behavo,.e., p( γ ) =, leads by usg E. (4) o Y( γ) = γ ( γ) U( γ) sγ Y( γ) (8)

5 Le us ed ha ν 1 ad 1 ae he polyoal odes of p( γ ) ad ( γ ) especvely. Theefoe we oba a esao of he p( γ ) paaees by applyg he esul of seco 3.1 o E. (7) whee θ p p θ= ϕ ϕ Y ν 1 (9) = ν 1 s he esaed veco of he p( γ ) paaees ad ϕ = u ( ) u ( ν+ 1 ) Y = y ( ) y ( ν 1) Le us oe ha we aually eed he fs daa of he easued syse oupu o coecly defy he ase pa. Slaly, we oba a esao of he paaees of ( γ ) ad s by applyg he esul of seco 3.1 o E. (8) θ= ϕ ϕ wh N Y ν N ν + whee θ= 1 s (1) s he esaed veco of he paaees of ( γ ) ad s ad Y = y ( ν) y ( N) ϕ = u ( ν) u ( ν + 1 ) y ( ) Accodg o he assupo whch leads o E. (8) p( γ ) =, we cosde ha y ( ) = = y ( ν 1) = o copue ϕ. 4 Paccal Modelg of h ( γ ) As coveoal lea syse heoy [5, chap. 16], he defcao of he asfe fuco h( γ ) by usg Es. (9) ad (1) poses o ow he ube of coeffces of polyoals p( γ ) ad ( γ ),.e., ν ad especvely. I pacce he owledge of hese values s a poa poble of he odelg. Ideed he odel sucue should as fa as possble be boh coplex eough o coecly ualfy he syse ad sple eough o educe he ube of paaees o be defed. To oba hs copose we eed a pelay aalyss o have a esao of ν oed l p, ca be coase bu us be geae ha he exac ase legh. The we popose a odelg ehod based o he aalyss of he ceo J (see seco 3.1) as a fuco of l p ad l (defed as he esao of he pae legh ). I ode o esae : - we ae l o 1 ; - we copue ( =,, 1) ad s by usg E. (1) ; we epea he lae calculao wh l J s al. l = l +1 ul ceo To esae ν we fx o s pevous esao. The we decease l p ad we calculae ( =,, 1) ad s by usg E. (1) ul ceo J ceases. Ths ccal value of l p coespods o he esao of ase pa legh (ν). 5 Cocluso We have poposed a defcao ehod of DES ax-algeba. Ths ehod aes possble he defcao of SISO syses. I offes a aalogy wh he coveoal lea syse heoy : s based boh o he aalyss of he syse pulse espose ad o a lea ARMA fo of he odel. To oba hs ARMA fo we sepaae he ase behavo ad he peodc oe. Such popey aes possble he use of a basc esul of Resduao Theoy whch leads o a sple ehod of paaees odel esao. Howeve we cao guaaee he aly of he odel due o he cosdeed expesso of he asfe fuco, we eep wog o hs poble. O he ohe had wll be eesg o chaaceze he aco of dsubaces o he ehod. Refeeces [1] R. Cughae-ee, Max Algeba, Lecue Noes Ecoocs ad Maheacal Syses, Spge- Velag, 166, [2] F. Baccell,. Cohe,.J. Olsde, J.P. Quada, Sychozao ad Leay. A algeba fo Dscee Eve Syses, New Yo : Wley, [3]. Cohe, D. Dubos, J.P. Quada ad M. Vo, A Lea-Syse-Theoec Vew of Dscee-Eve Pocesses ad s Use fo Pefoace Evaluao Maufacug, IEEE Tasacos o Auoac Cool, 3, 3, ach 1985, pp [4]. Cohe, P. Molle, J.P. Quada ad M. Vo, Algebac Tools fo he Pefoace Evaluao of Dscee Eve Syses, IEEE Tasacos o Auoac Cool, 77, 1, auay 1989, pp [5] L. Lug, Syse Idefcao : Theoy fo he Use, Pece-Hall, Ic., Eglewood Clffs, New Jesey, 1987.

Fault-tolerant Output Feedback Control for a Class of Multiple Input Fuzzy Bilinear Systems

Fault-tolerant Output Feedback Control for a Class of Multiple Input Fuzzy Bilinear Systems Sesos & asduces Vol 7 Issue 6 Jue 04 pp 47-5 Sesos & asduces 04 by IFSA Publshg S L hp://wwwsesospoalco Faul-olea Oupu Feedbac Cool fo a Class of Mulple Ipu Fuzzy Blea Syses * YU Yag WAG We School of Eleccal

More information

The Solutions of Initial Value Problems for Nonlinear Fourth-Order Impulsive Integro-Differential Equations in Banach Spaces

The Solutions of Initial Value Problems for Nonlinear Fourth-Order Impulsive Integro-Differential Equations in Banach Spaces WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo The Soluos of Ial Value Pobles fo Nolea Fouh-Ode Ipulsve Iego-Dffeeal Equaos Baach Spaces Zhag Lglg Y Jgy Lu Juguo Depae of aheacs of Ta Yua Uvesy

More information

Lecture 3 summary. C4 Lecture 3 - Jim Libby 1

Lecture 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 information

( ) ( ) Weibull Distribution: k ti. u u. Suppose t 1, t 2, t n are times to failure of a group of n mechanisms. The likelihood function is

( ) ( ) Weibull Distribution: k ti. u u. Suppose t 1, t 2, t n are times to failure of a group of n mechanisms. The likelihood function is Webll Dsbo: Des Bce Dep of Mechacal & Idsal Egeeg The Uvesy of Iowa pdf: f () exp Sppose, 2, ae mes o fale of a gop of mechasms. The lelhood fco s L ( ;, ) exp exp MLE: Webll 3//2002 page MLE: Webll 3//2002

More information

Exponential Synchronization of the Hopfield Neural Networks with New Chaotic Strange Attractor

Exponential Synchronization of the Hopfield Neural Networks with New Chaotic Strange Attractor ITM Web of Cofeeces, 0509 (07) DOI: 0.05/ mcof/070509 ITA 07 Expoeal Sychozao of he Hopfeld Neual Newos wh New Chaoc Sage Aaco Zha-J GUI, Ka-Hua WANG* Depame of Sofwae Egeeg, Haa College of Sofwae Techology,qogha,

More information

DUALITY IN MULTIPLE CRITERIA AND MULTIPLE CONSTRAINT LEVELS LINEAR PROGRAMMING WITH FUZZY PARAMETERS

DUALITY IN MULTIPLE CRITERIA AND MULTIPLE CONSTRAINT LEVELS LINEAR PROGRAMMING WITH FUZZY PARAMETERS Ida Joual of Fudameal ad ppled Lfe Sceces ISSN: 223 6345 (Ole) Ope ccess, Ole Ieaoal Joual valable a www.cbech.o/sp.ed/ls/205/0/ls.hm 205 Vol.5 (S), pp. 447-454/Noua e al. Reseach cle DULITY IN MULTIPLE

More information

Suppose we have observed values t 1, t 2, t n of a random variable T.

Suppose we have observed values t 1, t 2, t n of a random variable T. Sppose we have obseved vales, 2, of a adom vaable T. The dsbo of T s ow o belog o a cea ype (e.g., expoeal, omal, ec.) b he veco θ ( θ, θ2, θp ) of ow paamees assocaed wh s ow (whee p s he mbe of ow paamees).

More information

EMA5001 Lecture 3 Steady State & Nonsteady State Diffusion - Fick s 2 nd Law & Solutions

EMA5001 Lecture 3 Steady State & Nonsteady State Diffusion - Fick s 2 nd Law & Solutions EMA5 Lecue 3 Seady Sae & Noseady Sae ffuso - Fck s d Law & Soluos EMA 5 Physcal Popees of Maeals Zhe heg (6) 3 Noseady Sae ff Fck s d Law Seady-Sae ffuso Seady Sae Seady Sae = Equlbum? No! Smlay: Sae fuco

More information

On Probability Density Function of the Quotient of Generalized Order Statistics from the Weibull Distribution

On Probability Density Function of the Quotient of Generalized Order Statistics from the Weibull Distribution ISSN 684-843 Joua of Sac Voue 5 8 pp. 7-5 O Pobaby Dey Fuco of he Quoe of Geeaed Ode Sac fo he Webu Dbuo Abac The pobaby dey fuco of Muhaad Aee X k Y k Z whee k X ad Y k ae h ad h geeaed ode ac fo Webu

More information

ANSWERS TO ODD NUMBERED EXERCISES IN CHAPTER 2

ANSWERS TO ODD NUMBERED EXERCISES IN CHAPTER 2 Joh Rley Novembe ANSWERS O ODD NUMBERED EXERCISES IN CHAPER Seo Eese -: asvy (a) Se y ad y z follows fom asvy ha z Ehe z o z We suppose he lae ad seek a oado he z Se y follows by asvy ha z y Bu hs oads

More information

Asymptotic Behavior of Solutions of Nonlinear Delay Differential Equations With Impulse

Asymptotic 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 information

CONTROL ROUTH ARRAY AND ITS APPLICATIONS

CONTROL ROUTH ARRAY AND ITS APPLICATIONS 3 Asa Joual of Cool, Vol 5, No, pp 3-4, Mach 3 CONTROL ROUTH ARRAY AND ITS APPLICATIONS Dazha Cheg ad TJTa Bef Pape ABSTRACT I hs pape he Rouh sably ceo [6] has bee developed o cool Rouh aay Soe foulas

More information

Fakultas Matematika dan Ilmu Pengetahuan Alam, Institut Teknologi Bandung, Bandung, 40132, Indonesia *

Fakultas Matematika dan Ilmu Pengetahuan Alam, Institut Teknologi Bandung, Bandung, 40132, Indonesia * MacWllams Equvalece Theoem fo he Lee Wegh ove Z 4 leams Baa * Fakulas Maemaka da Ilmu Pegeahua lam, Isu Tekolog Badug, Badug, 403, Idoesa * oesodg uho: baa@mahbacd BSTRT Fo codes ove felds, he MacWllams

More information

are positive, and the pair A, B is controllable. The uncertainty in is introduced to model control failures.

are positive, and the pair A, B is controllable. The uncertainty in is introduced to model control failures. Lectue 4 8. MRAC Desg fo Affe--Cotol MIMO Systes I ths secto, we cosde MRAC desg fo a class of ult-ut-ult-outut (MIMO) olea systes, whose lat dyacs ae lealy aaetezed, the ucetates satsfy the so-called

More information

Verification and Validation of LAD2D Hydrocodes for Multi-material Compressible Flows

Verification and Validation of LAD2D Hydrocodes for Multi-material Compressible Flows 8 Ieaoal Cofeece o Physcs Maheacs Sascs Modellg ad Sulao (PMSMS 8) ISBN: 978--6595-558- Vefcao ad Valdao of ADD Hydocodes fo Mul-aeal Copessle Flows Ru-l WANG * ad Xao IANG Isue of Appled Physcs ad Copuaoal

More information

Continuous Time Markov Chains

Continuous 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 information

Determination of Antoine Equation Parameters. December 4, 2012 PreFEED Corporation Yoshio Kumagae. Introduction

Determination 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 information

( m is the length of columns of A ) spanned by the columns of A : . Select those columns of B that contain a pivot; say those are Bi

( m is the length of columns of A ) spanned by the columns of A : . Select those columns of B that contain a pivot; say those are Bi Assgmet /MATH 47/Wte Due: Thusday Jauay The poblems to solve ae umbeed [] to [] below Fst some explaatoy otes Fdg a bass of the colum-space of a max ad povg that the colum ak (dmeso of the colum space)

More information

Technical Appendix for Inventory Management for an Assembly System with Product or Component Returns, DeCroix and Zipkin, Management Science 2005.

Technical 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 information

Shrinkage Estimators for Reliability Function. Mohammad Qabaha

Shrinkage Estimators for Reliability Function. Mohammad Qabaha A - Najah Uv. J. es. (N. Sc.) Vol. 7 3 Shkage Esmaos fo elably Fuco مقدرات التقلص لدالة الفاعلية ohammad Qabaha محمد قبها Depame of ahemacs Faculy of Scece A-Najah Naoal Uvesy alese E-mal: mohqabha@mal.ajah.edu

More information

ON TOTAL TIME ON TEST TRANSFORM ORDER ABSTRACT

ON TOTAL TIME ON TEST TRANSFORM ORDER ABSTRACT V M Chacko E CONVE AND INCREASIN CONVE OAL IME ON ES RANSORM ORDER R&A # 4 9 Vol. Decembe ON OAL IME ON ES RANSORM ORDER V. M. Chacko Depame of Sascs S. homas Collee hss eala-68 Emal: chackovm@mal.com

More information

Anouncements. Conjugate Gradients. Steepest Descent. Outline. Steepest Descent. Steepest Descent

Anouncements. Conjugate Gradients. Steepest Descent. Outline. Steepest Descent. Steepest Descent oucms Couga Gas Mchal Kazha (6.657) Ifomao abou h Sma (6.757) hav b pos ol: hp://www.cs.hu.u/~msha Tch Spcs: o M o Tusay afoo. o Two paps scuss ach w. o Vos fo w s caa paps u by Thusay vg. Oul Rvw of Sps

More information

Thermal Properties of Functionally Graded Fibre Material

Thermal Properties of Functionally Graded Fibre Material Ja Tua, Elżbea Radaszewska Depae of Techcal Mechacs ad Ifoacs, Lodz Uvesy of Techology Żeoskego 6, 9-94 Łódź, Polad E-al: ja.ua@p.lodz.pl, elzbea.adaszewska@p.lodz.pl Theal Popees of Fucoally Gaded Fbe

More information

Minimizing spherical aberrations Exploiting the existence of conjugate points in spherical lenses

Minimizing spherical aberrations Exploiting the existence of conjugate points in spherical lenses Mmzg sphecal abeatos Explotg the exstece of cojugate pots sphecal leses Let s ecall that whe usg asphecal leses, abeato fee magg occus oly fo a couple of, so called, cojugate pots ( ad the fgue below)

More information

The Properties of Probability of Normal Chain

The 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 information

STABILITY CRITERIA FOR A CLASS OF NEUTRAL SYSTEMS VIA THE LMI APPROACH

STABILITY CRITERIA FOR A CLASS OF NEUTRAL SYSTEMS VIA THE LMI APPROACH Asan Jounal of Conol, Vol. 6, No., pp. 3-9, Mach 00 3 Bef Pape SABILIY CRIERIA FOR A CLASS OF NEURAL SYSEMS VIA HE LMI APPROACH Chang-Hua Len and Jen-De Chen ABSRAC In hs pape, he asypoc sably fo a class

More information

Generalisation on the Zeros of a Family of Complex Polynomials

Generalisation on the Zeros of a Family of Complex Polynomials Ieol Joul of hemcs esech. ISSN 976-584 Volume 6 Numbe 4. 93-97 Ieol esech Publco House h://www.house.com Geelso o he Zeos of Fmly of Comlex Polyomls Aee sgh Neh d S.K.Shu Deme of hemcs Lgys Uvesy Fdbd-

More information

(1) Cov(, ) E[( E( ))( E( ))]

(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 information

Spectrum of The Direct Sum of Operators. 1. Introduction

Spectrum of The Direct Sum of Operators. 1. Introduction Specu of The Diec Su of Opeaos by E.OTKUN ÇEVİK ad Z.I.ISMILOV Kaadeiz Techical Uivesiy, Faculy of Scieces, Depae of Maheaics 6080 Tabzo, TURKEY e-ail adess : zaeddi@yahoo.co bsac: I his wok, a coecio

More information

A Second Kind Chebyshev Polynomial Approach for the Wave Equation Subject to an Integral Conservation Condition

A Second Kind Chebyshev Polynomial Approach for the Wave Equation Subject to an Integral Conservation Condition SSN 76-7659 Eglad K Joural of forao ad Copug Scece Vol 7 No 3 pp 63-7 A Secod Kd Chebyshev olyoal Approach for he Wave Equao Subec o a egral Coservao Codo Soayeh Nea ad Yadollah rdokha Depare of aheacs

More information

Solution of Impulsive Differential Equations with Boundary Conditions in Terms of Integral Equations

Solution 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 information

Cyclone. Anti-cyclone

Cyclone. 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 information

Chapter 3: Vectors and Two-Dimensional Motion

Chapter 3: Vectors and Two-Dimensional Motion Chape 3: Vecos and Two-Dmensonal Moon Vecos: magnude and decon Negae o a eco: eese s decon Mulplng o ddng a eco b a scala Vecos n he same decon (eaed lke numbes) Geneal Veco Addon: Tangle mehod o addon

More information

Fairing of Parametric Quintic Splines

Fairing of Parametric Quintic Splines ISSN 46-69 Eglad UK Joual of Ifomato ad omputg Scece Vol No 6 pp -8 Fag of Paametc Qutc Sples Yuau Wag Shagha Isttute of Spots Shagha 48 ha School of Mathematcal Scece Fuda Uvesty Shagha 4 ha { P t )}

More information

A New Approach to Probabilistic Load Flow

A New Approach to Probabilistic Load Flow INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 73, DECEMBER 79, 837 A New Appoach o Pobablsc Load Flow T K Basu, R B Msa ad Puob Paoway Absac: Ths pape descbes a ew appoach o modellg of asmsso le uceaes usg

More information

Applying Eyring s Model to Times to Breakdown of Insulating Fluid

Applying Eyring s Model to Times to Breakdown of Insulating Fluid Ieaoal Joual of Pefomably Egeeg, Vol. 8, No. 3, May 22, pp. 279-288. RAMS Cosulas Ped Ida Applyg Eyg s Model o Tmes o Beakdow of Isulag Flud DANIEL I. DE SOUZA JR. ad R. ROCHA Flumese Fed. Uvesy, Cvl Egeeg

More information

XII. Addition of many identical spins

XII. Addition of many identical spins XII. Addto of may detcal sps XII.. ymmetc goup ymmetc goup s the goup of all possble pemutatos of obects. I total! elemets cludg detty opeato. Each pemutato s a poduct of a ceta fte umbe of pawse taspostos.

More information

The Non-Truncated Bulk Arrival Queue M x /M/1 with Reneging, Balking, State-Dependent and an Additional Server for Longer Queues

The Non-Truncated Bulk Arrival Queue M x /M/1 with Reneging, Balking, State-Dependent and an Additional Server for Longer Queues Alied Maheaical Sciece Vol. 8 o. 5 747-75 The No-Tucaed Bul Aival Queue M x /M/ wih Reei Bali Sae-Deede ad a Addiioal Seve fo Loe Queue A. A. EL Shebiy aculy of Sciece Meofia Uiveiy Ey elhebiy@yahoo.co

More information

Delay-Dependent Robust Asymptotically Stable for Linear Time Variant Systems

Delay-Dependent Robust Asymptotically Stable for Linear Time Variant Systems Delay-Depede Robus Asypocally Sable for Lear e Vara Syses D. Behard, Y. Ordoha, S. Sedagha ABSRAC I hs paper, he proble of delay depede robus asypocally sable for ucera lear e-vara syse wh ulple delays

More information

A Type-2 Fuzzy Rule-Based Expert System Model for Portfolio Selection

A Type-2 Fuzzy Rule-Based Expert System Model for Portfolio Selection Type- Fuzzy Rule-Based Epe Syse Model fo Pofolo Seleco M.H. Fazel Zaad, E. Hagol Yazd Eal: zaad@au.ac., elahehagol@gal.co Depae of Idusal Egeeg, kab Uvesy of Techology, P.O. Bo 5875-344, Teha, Ia bsac:

More information

dm dt = 1 V The number of moles in any volume is M = CV, where C = concentration in M/L V = liters. dcv v

dm dt = 1 V The number of moles in any volume is M = CV, where C = concentration in M/L V = liters. dcv v Mg: Pcess Aalyss: Reac ae s defed as whee eac ae elcy lue M les ( ccea) e. dm he ube f les ay lue s M, whee ccea M/L les. he he eac ae beces f a hgeeus eac, ( ) d Usually s csa aqueus eeal pcesses eac,

More information

Chapter 5. Long Waves

Chapter 5. Long Waves ape 5. Lo Waes Wae e s o compaed ae dep: < < L π Fom ea ae eo o s s ; amos ozoa moo z p s ; dosac pesse Dep-aeaed coseao o mass

More information

- 1 - Processing An Opinion Poll Using Fuzzy Techniques

- 1 - Processing An Opinion Poll Using Fuzzy Techniques - - Pocessg A Oo Poll Usg Fuzzy Techues by Da Peu Vaslu ABSTRACT: I hs ae we deal wh a mul cea akg oblem, based o fuzzy u daa : he uose s o comae he effec of dffee mecs defed o he sace of fuzzy umbes o

More information

The ray paths and travel times for multiple layers can be computed using ray-tracing, as demonstrated in Lab 3.

The 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 information

Lagrangian & Hamiltonian Mechanics:

Lagrangian & Hamiltonian Mechanics: XII AGRANGIAN & HAMITONIAN DYNAMICS Iouco Hamlo aaoal Pcple Geealze Cooaes Geealze Foces agaga s Euao Geealze Momea Foces of Cosa, agage Mulples Hamloa Fucos, Cosevao aws Hamloa Dyamcs: Hamlo s Euaos agaga

More information

MULTI-OBJECTIVE GEOMETRIC PROGRAMMING PROBLEM AND ITS APPLICATIONS

MULTI-OBJECTIVE GEOMETRIC PROGRAMMING PROBLEM AND ITS APPLICATIONS Yugoslav Joual of Opeaos Reseach Volume (), Numbe, -7 DOI:.98/YJORI MULTI-OBJECTIVE GEOMETRIC PROGRAMMING PROBLEM AND ITS APPLICATIONS Sahdul ISLAM Depame of Mahemacs, Guskaa Mahavdyalaya, Guskaa, Budwa

More information

SUBSEQUENCE CHARACTERIZAT ION OF UNIFORM STATISTICAL CONVERGENCE OF DOUBLE SEQUENCE

SUBSEQUENCE CHARACTERIZAT ION OF UNIFORM STATISTICAL CONVERGENCE OF DOUBLE SEQUENCE Reseach ad Coucatos atheatcs ad atheatcal ceces Vol 9 Issue 7 Pages 37-5 IN 39-6939 Publshed Ole o Novebe 9 7 7 Jyot cadec Pess htt//yotacadecessog UBEQUENCE CHRCTERIZT ION OF UNIFOR TTITIC CONVERGENCE

More information

Some Different Perspectives on Linear Least Squares

Some Different Perspectives on Linear Least Squares Soe Dfferet Perspectves o Lear Least Squares A stadard proble statstcs s to easure a respose or depedet varable, y, at fed values of oe or ore depedet varables. Soetes there ests a deterstc odel y f (,,

More information

Stress Analysis of Infinite Plate with Elliptical Hole

Stress Analysis of Infinite Plate with Elliptical Hole Sess Analysis of Infinie Plae ih Ellipical Hole Mohansing R Padeshi*, D. P. K. Shaa* * ( P.G.Suden, Depaen of Mechanical Engg, NRI s Insiue of Infoaion Science & Technology, Bhopal, India) * ( Head of,

More information

AML710 CAD LECTURE 12 CUBIC SPLINE CURVES. Cubic Splines Matrix formulation Normalised cubic splines Alternate end conditions Parabolic blending

AML710 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 information

On EPr Bimatrices II. ON EP BIMATRICES A1 A Hence x. is said to be EP if it satisfies the condition ABx

On EPr Bimatrices II. ON EP BIMATRICES A1 A Hence x. is said to be EP if it satisfies the condition ABx Iteatoal Joual of Mathematcs ad Statstcs Iveto (IJMSI) E-ISSN: 3 4767 P-ISSN: 3-4759 www.jms.og Volume Issue 5 May. 4 PP-44-5 O EP matces.ramesh, N.baas ssocate Pofesso of Mathematcs, ovt. ts College(utoomous),Kumbakoam.

More information

The Linear Regression Of Weighted Segments

The 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 information

ABSOLUTE INDEXED SUMMABILITY FACTOR OF AN INFINITE SERIES USING QUASI-F-POWER INCREASING SEQUENCES

ABSOLUTE INDEXED SUMMABILITY FACTOR OF AN INFINITE SERIES USING QUASI-F-POWER INCREASING SEQUENCES Available olie a h://sciog Egieeig Maheaics Lees 2 (23) No 56-66 ISSN 249-9337 ABSLUE INDEED SUMMABILIY FACR F AN INFINIE SERIES USING QUASI-F-WER INCREASING SEQUENCES SKAIKRAY * RKJAI 2 UKMISRA 3 NCSAH

More information

Least Squares Fitting (LSQF) with a complicated function Theexampleswehavelookedatsofarhavebeenlinearintheparameters

Least 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 information

The Stability of High Order Max-Type Difference Equation

The Stability of High Order Max-Type Difference Equation Aled ad Comuaoal Maemacs 6; 5(): 5-55 ://wwwsceceulsggoucom/j/acm do: 648/jacm653 ISSN: 38-565 (P); ISSN: 38-563 (Ole) Te Saly of g Ode Ma-Tye Dffeece Equao a Ca-og * L Lue Ta Xue Scool of Maemacs ad Sascs

More information

Two kinds of B-basis of the algebraic hyperbolic space *

Two kinds of B-basis of the algebraic hyperbolic space * 75 L e al. / J Zhejag Uv SCI 25 6A(7):75-759 Joual of Zhejag Uvesy SCIECE ISS 9-395 h://www.zju.edu.c/jzus E-al: jzus@zju.edu.c Two ds of B-bass of he algebac hyebolc sace * LI Ya-jua ( 李亚娟 ) WAG Guo-zhao

More information

Professor Wei Zhu. 1. Sampling from the Normal Population

Professor Wei Zhu. 1. Sampling from the Normal Population AMS570 Pofesso We Zhu. Samplg fom the Nomal Populato *Example: We wsh to estmate the dstbuto of heghts of adult US male. It s beleved that the heght of adult US male follows a omal dstbuto N(, ) Def. Smple

More information

ÖRNEK 1: THE LINEAR IMPULSE-MOMENTUM RELATION Calculate the linear momentum of a particle of mass m=10 kg which has a. kg m s

ÖRNEK 1: THE LINEAR IMPULSE-MOMENTUM RELATION Calculate the linear momentum of a particle of mass m=10 kg which has a. kg m s MÜHENDİSLİK MEKANİĞİ. HAFTA İMPULS- MMENTUM-ÇARPIŞMA Linea oenu of a paicle: The sybol L denoes he linea oenu and is defined as he ass ies he elociy of a paicle. L ÖRNEK : THE LINEAR IMPULSE-MMENTUM RELATIN

More information

Computation of an Over-Approximation of the Backward Reachable Set using Subsystem Level Set Functions

Computation of an Over-Approximation of the Backward Reachable Set using Subsystem Level Set Functions Compuao o a Ove-Appomao o he Backwad Reachable Se usg Subsysem Level Se Fucos Duša M Spaov, Iseok Hwag, ad Clae J oml Depame o Aeoaucs ad Asoaucs Saod Uvesy Saod, CA 94305-4035, USA E-mal: {dusko, shwag,

More information

χ be any function of X and Y then

χ be any function of X and Y then We have show that whe we ae gve Y g(), the [ ] [ g() ] g() f () Y o all g ()() f d fo dscete case Ths ca be eteded to clude fuctos of ay ube of ado vaables. Fo eaple, suppose ad Y ae.v. wth jot desty fucto,

More information

Chapter Linear Regression

Chapter Linear Regression Chpte 6.3 Le Regesso Afte edg ths chpte, ou should be ble to. defe egesso,. use sevel mmzg of esdul cte to choose the ght cteo, 3. deve the costts of le egesso model bsed o lest sques method cteo,. use

More information

On the Quasi-Hyperbolic Kac-Moody Algebra QHA7 (2)

On the Quasi-Hyperbolic Kac-Moody Algebra QHA7 (2) Ieaoal Reeach Joual of Egeeg ad Techology (IRJET) e-issn: 9 - Volume: Iue: May- www.e.e -ISSN: 9-7 O he Qua-Hyebolc Kac-Moody lgeba QH7 () Uma Mahewa., Khave. S Deame of Mahemac Quad-E-Mllah Goveme College

More information

k of the incident wave) will be greater t is too small to satisfy the required kinematics boundary condition, (19)

k of the incident wave) will be greater t is too small to satisfy the required kinematics boundary condition, (19) TOTAL INTRNAL RFLTION Kmacs pops Sc h vcos a coplaa, l s cosd h cd pla cocds wh h X pla; hc 0. y y y osd h cas whch h lgh s cd fom h mdum of hgh dx of faco >. Fo cd agls ga ha h ccal agl s 1 ( /, h hooal

More information

8. Queueing systems lect08.ppt S Introduction to Teletraffic Theory - Fall

8. 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 information

= y and Normed Linear Spaces

= y and Normed Linear Spaces 304-50 LINER SYSTEMS Lectue 8: Solutos to = ad Nomed Lea Spaces 73 Fdg N To fd N, we eed to chaacteze all solutos to = 0 Recall that ow opeatos peseve N, so that = 0 = 0 We ca solve = 0 ecusvel backwads

More information

Analysis of a Stochastic Lotka-Volterra Competitive System with Distributed Delays

Analysis of a Stochastic Lotka-Volterra Competitive System with Distributed Delays Ieraoal Coferece o Appled Maheac Sulao ad Modellg (AMSM 6) Aaly of a Sochac Loa-Volerra Copeve Sye wh Drbued Delay Xagu Da ad Xaou L School of Maheacal Scece of Togre Uvery Togre 5543 PR Cha Correpodg

More information

Single-Plane Auto-Balancing of Rigid Rotors

Single-Plane Auto-Balancing of Rigid Rotors TECHNISCHE MECHANIK Bad 4 Hef (4) -4 Mauspegag: 4. Novebe 3 Sgle-Plae Auo-Balacg of gd oos L. Spelg B. h H. Ducse Ths pape peses a aalcal sud of sgle-plae auoac balacg of sacall ad dacall ubalaced gd oos

More information

African Journal of Science and Technology (AJST) Science and Engineering Series Vol. 4, No. 2, pp GENERALISED DELETION DESIGNS

African Journal of Science and Technology (AJST) Science and Engineering Series Vol. 4, No. 2, pp GENERALISED DELETION DESIGNS Af Joul of See Tehology (AJST) See Egeeg See Vol. 4, No.,. 7-79 GENERALISED DELETION DESIGNS Mhel Ku Gh Joh Wylff Ohbo Dee of Mhe, Uvey of Nob, P. O. Bo 3097, Nob, Key ABSTRACT:- I h e yel gle ele fol

More information

Outline. Review Homework Problem. Review Homework Problem II. Review Dimensionless Problem. Review Convection Problem

Outline. Review Homework Problem. Review Homework Problem II. Review Dimensionless Problem. Review Convection Problem adial diffsio eqaio Febay 4 9 Diffsio Eqaios i ylidical oodiaes ay aeo Mechaical Egieeig 5B Seia i Egieeig Aalysis Febay 4, 9 Olie eview las class Gadie ad covecio boday codiio Diffsio eqaio i adial coodiaes

More information

Math 7409 Homework 2 Fall from which we can calculate the cycle index of the action of S 5 on pairs of vertices as

Math 7409 Homework 2 Fall from which we can calculate the cycle index of the action of S 5 on pairs of vertices as Math 7409 Hoewok 2 Fall 2010 1. Eueate the equivalece classes of siple gaphs o 5 vetices by usig the patte ivetoy as a guide. The cycle idex of S 5 actig o 5 vetices is 1 x 5 120 1 10 x 3 1 x 2 15 x 1

More information

Maximum Likelihood Estimation

Maximum Likelihood Estimation Mau Lkelhood aon Beln Chen Depaen of Copue Scence & Infoaon ngneeng aonal Tawan oal Unvey Refeence:. he Alpaydn, Inoducon o Machne Leanng, Chape 4, MIT Pe, 4 Saple Sac and Populaon Paaee A Scheac Depcon

More information

Cameras and World Geometry

Cameras and World Geometry Caeas ad Wold Geoe How all is his woa? How high is he caea? Wha is he caea oaio w. wold? Which ball is close? Jaes Has Thigs o eebe Has Pihole caea odel ad caea (pojecio) ai Hoogeeous coodiaes allow pojecio

More information

Solution set Stat 471/Spring 06. Homework 2

Solution 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 information

A GENERAL CLASS OF ESTIMATORS UNDER MULTI PHASE SAMPLING

A GENERAL CLASS OF ESTIMATORS UNDER MULTI PHASE SAMPLING TATITIC IN TRANITION-ew sees Octobe 9 83 TATITIC IN TRANITION-ew sees Octobe 9 Vol. No. pp. 83 9 A GENERAL CLA OF ETIMATOR UNDER MULTI PHAE AMPLING M.. Ahed & Atsu.. Dovlo ABTRACT Ths pape deves the geeal

More information

Increasing the Image Quality of Atomic Force Microscope by Using Improved Double Tapered Micro Cantilever

Increasing the Image Quality of Atomic Force Microscope by Using Improved Double Tapered Micro Cantilever Rece Reseaces Teecocaos foacs Eecocs a Sga Pocessg ceasg e age Qa of oc Foce Mcope Usg pove oe Tapee Mco aeve Saeg epae of Mecaca Egeeg aava Bac sac za Uves aava Tea a a_saeg@aavaa.ac. sac: Te esoa feqec

More information

Op Amp Noise in Dynamic Range Maximization of Integrated Active-RC Filters

Op Amp Noise in Dynamic Range Maximization of Integrated Active-RC Filters Op Amp Nose Dyamc Rage Maxmzao of Iegaed Acve-R Fles N MARAOS* AND M MLADENO** * Naoal echcal Uvesy of Ahes Dep of Eleccal ad ompue Egeeg 9 Ioo Polyechou S ogafou 577 Ahes eece ** Depame of heoecal Elecoechcs

More information

Overview. Solving PDEs. Solving PDEs II. Midterm Exam. Review Spherical Laplace. The wave equation February 23, ME 501B Engineering Analysis 1

Overview. Solving PDEs. Solving PDEs II. Midterm Exam. Review Spherical Laplace. The wave equation February 23, ME 501B Engineering Analysis 1 The wave eqao ebay 3 9 aplae Eqao Colso ad The Wave Eqao ay Caeo Mehaal Egeeg 5 Sea Egeeg alyss ebay 3 9 Ovevew evew aeal o dae eeal appoah fo solvg PDEs Ohe deas abo aplae s Eqao Devao ad physal eag of

More information

Chapter 1 - Free Vibration of Multi-Degree-of-Freedom Systems - I

Chapter 1 - Free Vibration of Multi-Degree-of-Freedom Systems - I CEE49b Chaper - Free Vbrao of M-Degree-of-Freedo Syses - I Free Udaped Vbrao The basc ype of respose of -degree-of-freedo syses s free daped vbrao Aaogos o sge degree of freedo syses he aayss of free vbrao

More information

The Poisson Process Properties of the Poisson Process

The 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 information

Handling Fuzzy Constraints in Flow Shop Problem

Handling Fuzzy Constraints in Flow Shop Problem Handlng Fuzzy Consans n Flow Shop Poblem Xueyan Song and Sanja Peovc School of Compue Scence & IT, Unvesy of Nongham, UK E-mal: {s sp}@cs.no.ac.uk Absac In hs pape, we pesen an appoach o deal wh fuzzy

More information

-HYBRID LAPLACE TRANSFORM AND APPLICATIONS TO MULTIDIMENSIONAL HYBRID SYSTEMS. PART II: DETERMINING THE ORIGINAL

-HYBRID LAPLACE TRANSFORM AND APPLICATIONS TO MULTIDIMENSIONAL HYBRID SYSTEMS. PART II: DETERMINING THE ORIGINAL UPB Sc B See A Vo 72 I 3 2 ISSN 223-727 MUTIPE -HYBRID APACE TRANSORM AND APPICATIONS TO MUTIDIMENSIONA HYBRID SYSTEMS PART II: DETERMININ THE ORIINA Ve PREPEIŢĂ Te VASIACHE 2 Ace co copeeă oă - pce he

More information

Algebraic Properties of Modular Addition Modulo a Power of Two

Algebraic Properties of Modular Addition Modulo a Power of Two Agebac Popees of Moda Addo Modo a Powe of Two S M Dehav Aeza Rahmpo Facy of Mahemaca ad Compe Sceces Khaazm Uvesy Teha Ia sd_dehavsm@hac Facy of Sceces Qom Uvesy Qom Ia aahmpo@sqomac Absac; Moda addo modo

More information

Overview. Review Superposition Solution. Review Superposition. Review x and y Swap. Review General Superposition

Overview. Review Superposition Solution. Review Superposition. Review x and y Swap. Review General Superposition ylcal aplace Soltos ebay 6 9 aplace Eqato Soltos ylcal Geoety ay aetto Mechacal Egeeg 5B Sea Egeeg Aalyss ebay 6 9 Ovevew evew last class Speposto soltos tocto to aal cooates Atoal soltos of aplace s eqato

More information

Modeling Multibody Dynamic Systems With Uncertainties. Part I: Theoretical and Computational Aspects

Modeling Multibody Dynamic Systems With Uncertainties. Part I: Theoretical and Computational Aspects Modelg Mulbod Dac ses Wh Uceaes. Pa I: Theoecal ad Copuaoal Aspecs Ada adu* Coa adu ad Mehd Ahada ga Polechc Isue ad ae Ues *Copue cece Depae sadu@cs..edu Mechacal Egeeg Depae {csadu ahada}@.edu Absac

More information

β A Constant-G m Biasing

β A Constant-G m Biasing p 2002 EE 532 Anal IC Des II Pae 73 Cnsan-G Bas ecall ha us a PTAT cuen efeence (see p f p. 66 he nes) bas a bpla anss pes cnsan anscnucance e epeaue (an als epenen f supply lae an pcess). Hw h we achee

More information

NUMERICAL EVALUATION of DYNAMIC RESPONSE

NUMERICAL EVALUATION of DYNAMIC RESPONSE NUMERICAL EVALUATION of YNAMIC RESPONSE Aalycal solo of he eqao of oo for a sgle degree of freedo syse s sally o ossble f he excao aled force or grod accelerao ü g -vares arbrarly h e or f he syse s olear.

More information

2/20/2013. Topics. Power Flow Part 1 Text: Power Transmission. Power Transmission. Power Transmission. Power Transmission

2/20/2013. Topics. Power Flow Part 1 Text: Power Transmission. Power Transmission. Power Transmission. Power Transmission /0/0 Topcs Power Flow Part Text: 0-0. Power Trassso Revsted Power Flow Equatos Power Flow Proble Stateet ECEGR 45 Power Systes Power Trassso Power Trassso Recall that for a short trassso le, the power

More information

Parameter Estimation and Hypothesis Testing of Two Negative Binomial Distribution Population with Missing Data

Parameter Estimation and Hypothesis Testing of Two Negative Binomial Distribution Population with Missing Data Avlble ole wwwsceceeccom Physcs Poce 0 475 480 0 Ieol Cofeece o Mecl Physcs Bomecl ee Pmee smo Hyohess es of wo Neve Boml Dsbuo Poulo wh Mss D Zhwe Zho Collee of MhemcsJl Noml UvesyS Ch zhozhwe@6com Absc

More information

ESTIMATION OF PARAMETERS AND VERIFICATION OF STATISTICAL HYPOTHESES FOR GAUSSIAN MODELS OF STOCK PRICE

ESTIMATION OF PARAMETERS AND VERIFICATION OF STATISTICAL HYPOTHESES FOR GAUSSIAN MODELS OF STOCK PRICE Lhuaa Joual of Sascs Leuvos sasos daba 06, vol 55, o, pp 9 0 06, 55,, 9 0 p wwwsascsjouall ESTIMATIO OF PARAMETERS AD VERIFICATIO OF STATISTICAL YPOTESES FOR GAUSSIA MODELS OF STOCK PRICE Dmyo Maushevych,

More information

SIMULATIUON OF SEISMIC ACTION FOR TBILISI CITY WITH LOCAL SEISMOLOGICAL PARTICULARITIES AND SITE EFFECTS

SIMULATIUON OF SEISMIC ACTION FOR TBILISI CITY WITH LOCAL SEISMOLOGICAL PARTICULARITIES AND SITE EFFECTS SIMULTIUON OF SEISMIC CTION FOR TBILISI CITY WITH LOCL SEISMOLOGICL PRTICULRITIES ND SITE EFFECTS Paaa REKVV ad Keeva MDIVNI Geoga Naoal ssocao fo Egeeg Sesmology ad Eahquake Egeeg Tbls Geoga ekvavapaaa@yahoo.com

More information

Real-Time Systems. Example: scheduling using EDF. Feasibility analysis for EDF. Example: scheduling using EDF

Real-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 information

Linear Perturbation Bounds of the Continuous-Time LMI-Based H Quadratic Stability Problem for Descriptor Systems

Linear Perturbation Bounds of the Continuous-Time LMI-Based H Quadratic Stability Problem for Descriptor Systems UGRN DE OF ENE ERNE ND NFORON EHNOOGE Volu No 4 ofa a ubao ouds of h ouous- -asd H uadac ably obl fo Dscpo yss dy ochv chcal Uvsy of ofa Faculy of uoacs Dpa of yss ad ool 756 ofa Eal ayochv@u-sofa.bg bsac

More information

Laplace Transform. Definition of Laplace Transform: f(t) that satisfies The Laplace transform of f(t) is defined as.

Laplace 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 information

The Finite Difference Time Domain (FDTD) Method Discretizing Maxwell s equation in space and time

The Finite Difference Time Domain (FDTD) Method Discretizing Maxwell s equation in space and time Chape 3 The Fe Dffeece Te Doa FDTD Mehod I hs chape we povde a evew of heoecal echques of FDTD ehod eploed he cue wo. Ou sulaos ae based o he well-ow fe-dffeece e-doa FDTD[] echque. The FDTD ehod s a goous

More information

X-Ray Notes, Part III

X-Ray Notes, Part III oll 6 X-y oe 3: Pe X-Ry oe, P III oe Deeo Coe oupu o x-y ye h look lke h: We efe ue of que lhly ffee efo h ue y ovk: Co: C ΔS S Sl o oe Ro: SR S Co o oe Ro: CR ΔS C SR Pevouly, we ee he SR fo ye hv pxel

More information

FALL HOMEWORK NO. 6 - SOLUTION Problem 1.: Use the Storage-Indication Method to route the Input hydrograph tabulated below.

FALL 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 information

FORCED VIBRATION of MDOF SYSTEMS

FORCED VIBRATION of MDOF SYSTEMS FORCED VIBRAION of DOF SSES he respose of a N DOF sysem s govered by he marx equao of moo: ] u C] u K] u 1 h al codos u u0 ad u u 0. hs marx equao of moo represes a sysem of N smulaeous equaos u ad s me

More information

Maximum likelihood estimate of phylogeny. BIOL 495S/ CS 490B/ MATH 490B/ STAT 490B Introduction to Bioinformatics April 24, 2002

Maximum 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 information

Applications of force vibration. Rotating unbalance Base excitation Vibration measurement devices

Applications of force vibration. Rotating unbalance Base excitation Vibration measurement devices Applicaios of foce viaio Roaig ualace Base exciaio Viaio easuee devices Roaig ualace 1 Roaig ualace: Viaio caused y iegulaiies i he disiuio of he ass i he oaig copoe. Roaig ualace 0 FBD 1 FBD x x 0 e 0

More information

Sharif University of Technology - CEDRA By: Professor Ali Meghdari

Sharif University of Technology - CEDRA By: Professor Ali Meghdari Shaif Univesiy of echnology - CEDRA By: Pofesso Ali Meghai Pupose: o exen he Enegy appoach in eiving euaions of oion i.e. Lagange s Meho fo Mechanical Syses. opics: Genealize Cooinaes Lagangian Euaion

More information