Quadrature Signals: Complex, But Not Complicated

Save this PDF as:

Size: px
Start display at page:

Transcription

2 Th Dvlopmn and Noaion of Complx Numbrs To sablish our rminology, w dfin a ral numbr o b hos numbrs w us in vry day lif, lik a volag, a mpraur on h Fahrnhi scal, or h balanc of your chcking accoun. Ths on-dimnsional numbrs can b ihr posiiv or ngaiv as dpicd in Figur 1(a). In ha figur w show a on-dimnsional and say ha a singl ral numbr can b rprsnd by a poin on ha. Ou of radiion, l's call his, h. This poin rprsns h ral numbr a = -. (j) This poin rprsns h complx numbr c =.5 + j (a) +j lin (b) lin Figur 1. An graphical inrpraion of a ral numbr and a complx numbr. A complx numbr, c, is shown in Figur 1(b) whr i's also rprsnd as a poin. Howvr, complx numbrs ar no rsricd o li on a on-dimnsional lin, bu can rsid anywhr on a wo-dimnsional plan. Tha plan is calld h complx plan (som mahmaicians lik o call i an Argand diagram), and i nabls us o rprsn complx numbrs having boh ral and imaginary pars. For xampl in Figur 1(b), h complx numbr c =.5 + j is a poin lying on h complx plan on nihr h ral nor h imaginary. W loca poin c by going +.5 unis along h ral and up + unis along h imaginary. Think of hos ral and imaginary axs xacly as you hink of h Eas-Ws and Norh-Souh dircions on a road map. W'll us a gomric viwpoin o hlp us undrsand som of h arihmic of complx numbrs. Taking a look a Figur, w can us h rigonomry of righ riangls o dfin svral diffrn ways of rprsning h complx numbr c. b (j) c = a + jb M φ a Figur Th phasor rprsnaion of complx numbr c = a + jb on h complx plan. Our complx numbr c is rprsnd in a numbr of diffrn ways in h liraur, such as: Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd

3 Noaion Nam: Rcangular form: Mah Exprssion: c = a + jb Rmarks: Usd for xplanaory purposs. Easis o undrsand. [Also calld h Carsian form.] Trigonomric c = M[cos(φ) + jsin(φ)] Commonly usd o dscrib form: quadraur signals in communicaions sysms. Polar form: c = M jφ Mos puzzling, bu h primary form usd in mah quaions. [Also calld h Exponnial form. Somims wrin as Mxp(jφ).] Magniudangl c = M φ Usd for dscripiv purposs, form: bu oo cumbrsom for us in algbraic quaions. [Essnially a shorhand vrsion of Eq. (3).] (1) () (3) (4) Eqs. (3) and (4) rmind us ha c can also b considrd h ip of a phasor on h complx plan, wih magniud M, in h dircion of φ dgrs rlaiv o h posiiv ral as shown in Figur. Kp in mind ha c is a complx numbr and h variabls a, b, M, and φ ar all ral numbrs. Th magniud of c, somims calld h modulus of c, is M = c = a + b (5) [Trivia qusion: In wha 1939 movi, considrd by many o b h gras movi vr mad, did a main characr amp o quo Eq. (5)?] Back o businss. Th phas angl φ, or argumn, is h arcangn of h raio imaginary par ral par, or φ = an -1 b a (6) If w s Eq. (3) qual o Eq. (), M jφ = M[cos(φ) + jsin(φ)], w can sa wha's namd in his honor and now calld on of Eulr's idniis as: jφ = cos(φ) + jsin(φ) (7) Th suspicious radr should now b asking, "Why is i valid o rprsn a complx numbr using Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd

5 cubic quaions in h sixnh cnury. [1], [] Mahmaicians rlucanly bgan o accp h absrac concp of -1, wihou having o visualiz i, bcaus is mahmaical propris wr consisn wih h arihmic of normal ral numbrs. I was Eulr's quaing complx numbrs o ral sins and cosins, and Gauss' brillian inroducion of h complx plan, ha finally lgiimizd h noion of -1 o Europ's mahmaicians in h ighnh cnury. Eulr, going byond h provinc of ral numbrs, showd ha complx numbrs had a clan consisn rlaionship o h wll-known ral rigonomric funcions of sins and cosins. As Einsin showd h quivalnc of mass and nrgy, Eulr showd h quivalnc of ral sins and cosins o complx numbrs. Jus as modrn-day physiciss don know wha an lcron is bu hy undrsand is propris, w ll no worry abou wha 'j' is and b saisfid wih undrsanding is bhavior. For our purposs, h j-opraor mans roa a complx numbr by 9 o counrclockwis. (For you good folk in h UK, counrclockwis mans ani-clockwis.) L's s why. W'll g comforabl wih h complx plan rprsnaion of imaginary numbrs by xamining h mahmaical propris of h j = -1 opraor as shown in Figur 4. j8 = muliply by "j" j8 Figur 4. Wha happns o h ral numbr 8 whn you sar muliplying i by j. Muliplying any numbr on h ral by j rsuls in an imaginary produc ha lis on h imaginary. Th xampl in Figur 4 shows ha if +8 is rprsnd by h do lying on h posiiv ral, muliplying +8 by j rsuls in an imaginary numbr, +j8, whos posiion has bn road 9 o counrclockwis (from +8) puing i on h posiiv imaginary. Similarly, muliplying +j8 by j rsuls in anohr 9 o roaion yilding h -8 lying on h ngaiv ral bcaus j = -1. Muliplying -8 by j rsuls in a furhr 9 o roaion giving h -j8 lying on h ngaiv imaginary. Whnvr any numbr rprsnd by a do is muliplid by j h rsul is a counrclockwis roaion of 9 o. (Convrsly, muliplicaion by -j rsuls in a clockwis roaion of -9 o on h complx plan.) If w l φ = π/ in Eq. 7, w can say ha jπ/ = cos( π/) + jsin(π/) = + j1, or jπ/ = j (9) Hr's h poin o rmmbr. If you hav a singl complx numbr, rprsnd by a poin on h complx plan, muliplying ha numbr by j or by jπ/ will rsul in a nw complx numbr ha's road 9 o counrclockwis (CCW) on h complx plan. Don' forg his, as i will b Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd

6 usful as you bgin rading h liraur of quadraur procssing sysms! L's paus for a momn hr o cach our brah. Don' worry if h idas of imaginary numbrs and h complx plan sm a lil mysrious. I's ha way for vryon a firs you'll g comforabl wih hm h mor you us hm. (Rmmbr, h j-opraor puzzld Europ's havywigh mahmaicians for hundrds of yars.) Grand, no only is h mahmaics of complx numbrs a bi srang a firs, bu h rminology is almos bizarr. Whil h rm imaginary is an unforuna on o us, h rm complx is downrigh wird. Whn firs ncounrd, h phras complx numbrs maks us hink 'complicad numbrs'. This is rgrabl bcaus h concp of complx numbrs is no rally all ha complicad. Jus know ha h purpos of h abov mahmaical rigmarol was o valida Eqs. (), (3), (7), and (8). Now, l's (finally!) alk abou im-domain signals. R prsning Signals Using Complx Phasors OK, w now urn our anion o a complx numbr ha is a funcion im. Considr a numbr whos magniud is on, and whos phas angl incrass wih im. Tha complx numbr is h jπfo poin shown in Figur 5(a). (Hr h π rm is frquncy in radians/scond, and i corrsponds o a frquncy of cycls/scond whr is masurd in Hrz.) As im gs largr, h complx numbr's phas angl incrass and our numbr orbis h origin of h complx plan in a CCW dircion. Figur 5(a) shows h numbr, rprsnd by h black do, frozn a som arbirary insan in im. If, say, h frquncy = Hz, hn h do would roa around h circl wo ims pr scond. W can also hink of anohr complx numbr -jπfo (h whi do) orbiing in a clockwis dircion bcaus is phas angl gs mor ngaiv as im incrass. j = im in sconds, = frquncy in Hrz j jπ jπ -1 φ = π 1 φ = -πf o -1 φ = π 1 φ = -πf o -jπ -jπ -j -j (a) (b) Figur 5. A snapsho, in im, of wo complx numbrs whos xponns chang wih im. L's now call our wo jπfo and -jπfo complx xprssions quadraur signals. Thy ach m. Thos jπfo and -jπ hav boh ral and imaginary pars, and hy ar boh funcions of i xprssions ar ofn calld complx xponnials in h liraur. W can also hink of hos wo quadraur signals, jπfo and -jπfo, as h ips of wo phasors roaing in opposi dircions as shown in Figur 5(b). W'r going o sick wih his phasor noaion for now bcaus i'll allow us o achiv our goal of rprsning ral sinusoids in h conx of h complx plan. Don' ouch ha dial! Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd

7 To nsur ha w undrsand h bhavior of hos phasors, Figur 6(a) shows h hrdimnsional pah of h jπfo phasor as im passs. W'v addd h im, coming ou of h pag, o show h spiral pah of h phasor. Figur 6(b) shows a coninuous vrsion of jus h ip of h jπfo phasor. Tha jπfo complx numbr, or if you wish, h phasor's ip, follows a corkscrw pah spiraling along, and cnrd abou, h im. Th ral and imaginary pars of jπfo ar shown as h sin and cosin projcions in Figur 6(b). ( j ) o o 9 Imag 1-1 jπf o sin(π ) 18 o o 7-36 o -1 Tim 1 Tim cos(π ) (a) (b) Figur 6. Th moion of h jπ phasor (a), and phasor 's ip (b). Rurn o Figur 5(b) and ask yourslf: "Slf, wha's h vcor sum of hos wo phasors as hy roa in opposi dircions?" Think abou his for a momn... Tha's righ, h phasors' ral pars will always add consrucivly, and hir imaginary pars will always cancl. This mans ha h summaion of hs jπfo and -jπfo phasors will always b a purly ral numbr. Implmnaions of modrn-day digial communicaions sysms ar basd on his propry! To mphasiz h imporanc of h ral sum of hs wo complx sinusoids w'll draw y anohr picur. Considr h wavform in h hr-dimnsional Figur 7 gnrad by h sum of wo half-magniud complx phasors, jπfo / and -jπfo /, roaing in opposi dircions abou, and moving down along, h im. ( j ) 1 = cos(π ) jπ Tim Figur 7. A cosin rprsnd by h sum of wo roaing complx phasors. Thinking abou hs phasors, i's clar now why h cosin wav can b quad o h sum of -jπ Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd

9 Ngaiv frquncy j -jπ Dircion along h imaginary jπ - j Posiiv frquncy Magniud is 1/ Figur 8. Inrpraion of complx xponnials. W'll rprsn a singl complx xponnial as a narrowband impuls locad a h frquncy spcifid in h xponn. In addiion, w'll show h phas rlaionships bwn hos complx xponnials along h ral and imaginary axs. To illusra hos phas rlaionships, a complx frquncy domain rprsnaion is ncssary. Wih all ha said, ak a look a Figur cos(π ) Imag Par... Par Tim cos(π ) = jπ + -jπ - Imag Par Par... sin(π ) Imag Par... Par Tim sin(π ) = j -jπ - j jπ - Imag Par Par Figur 9. Complx frquncy domain rprsnaion of a cosin wav and sinwav. S how a ral cosin wav and a ral sinwav ar dpicd in our complx frquncy domain rprsnaion on h righ sid of Figur 9. Thos bold arrows on h righ of Figur 9 ar no roaing phasors, bu insad ar frquncy-domain impuls symbols indicaing a singl spcral lin for singl a complx xponnial jπ. Th dircions in which h spcral impulss ar poining mrly indica h rlaiv phass of h spcral componns. Th ampliud of hos spcral impulss ar 1/. OK... why ar w bohring wih his 3-D frquncy-domain rprsnaion? Bcaus i's h ool w'll us o undrsand h gnraion (modulaion) and dcion (dmodulaion) of quadraur signals in digial (and som analog) communicaions sysms, and hos ar wo of h goals of his uorial. Bfor w considr hos procsss howvr, l's valida his frquncy-domain rprsnaion wih a lil xampl. Figur 1 is a sraighforward xampl of how w us h complx frquncy domain. Thr w bgin wih a ral sinwav, apply h j opraor o i, and hn add h rsul o a ral cosin wav of h sam frquncy. Th final rsul is h singl complx xponnial jπ illusraing graphically Eulr's idniy ha w sad mahmaically in Eq. (7). Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd

12 Original Coninuous Spcrum -f c X(f) B f c Dsird Digiizd "Basband" Spcrum -f s X(m) f s (m) Figur 13. Th 'bfor and afr' spcra of a quadraur-sampld signal. Our goal in quadraur-sampling is o obain a digiizd vrsion of h analog bandpass signal, bu w wan ha digiizd signal's discr spcrum cnrd abou zro Hz, no f c Hz. Tha is, w wan o mix a im signal wih -jπf c o prform complx down-convrsion. Th frquncy f s is h digiizr's sampling ra in sampls/scond. W show rplicad spcra a h boom of Figur 13 jus o rmind ourslvs of his ffc whn A/D convrsion aks plac. OK,... ak a look a h following quadraur-sampling block diagram known as I/Q dmodulaion (or 'Wavr dmodulaion' for hos folk wih xprinc in communicaions hory) shown a h op of Figur 14. Tha arrangmn of wo sinusoidal oscillaors, wih hir rlaiv 9 o phas, is ofn calld a 'quadraur-oscillaor'. Thos jπf c and -jπf c rms in ha busy Figur 14 rmind us ha h consiun complx xponnials comprising a ral cosin duplicas ach par of X bp (f) spcrum o produc h X i (f) spcrum. Th Figur shows how w g h filrd coninuous in-phas porion our dsird complx quadraur signal. By dfiniion, hos X i (f) and I(f) spcra ar rad as 'ral only'. Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd

13 Coninuous Discr x () bp Coninuous spcrum In-phas coninuous spcrum cos(πf ) c sin(πf ) c o 9 x () i x q () -f c LPF LPF X (f) i i() q() X bp (f) f s A/D A/D par i(n) q(n) f c -jπf c jπf c complx squnc is: i(n) - jq(n) -f -f -B/ +B/ c c f c f c B LP filrd in-phas coninuous spcrum I(f) -B/ B/ Filrd ral par Figur 14. Quadraur-sampling block diagram and spcra wihin h in-phas (uppr) signal pah. Likwis, Figur 15 shows how w g h filrd coninuous quadraur phas porion our complx quadraur signal by mixing x bp () wih sin(πf c ). Coninuous spcrum -f c X bp (f) B f c Quadraur coninuous spcrum par X (f) q -f c -f c f c Ngaiv du o h minus sign of h sin's f c -j jπf Q(f) LP filrd quadraur coninuous spcrum -B/ B/ Filrd imaginary par Figur 15. Spcra wihin h quadraur phas (lowr) signal pah of h block diagram. Hr's whr w'r going: I(f) - jq(f) is h spcrum of a complx rplica our original bandpass signal x bp (). W show h addiion of hos wo spcra in Figur 16. Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd

14 I(f) Filrd coninuous in-phas (ral only) -B/ B/ -B/ Q(f) Filrd coninuous quadraur (imaginary only) B/ -B/ I(f) - jq(f) Spcrum of coninuous complx signal: i() - jq() B/ Figur 16. Combining h I(f) and Q(f) spcra o obain h dsird 'I(f) - jq(f)' spcra. This ypical dpicion of quadraur-sampling sms lik mumbo jumbo unil you look a his siuaion from a hr-dimnsional sandpoin, as in Figur 17, whr h -j facor roas h 'imaginary-only' Q(f) by -9 o, making i 'ral-only'. This -jq(f) is hn addd o I(f). ( j ) ( j ) I(f) A hrdimnsional viw ( j ) Q(f) -jq(f) ( j ) I(f) - jq(f) Figur D viw of combining h I(f) and Q(f) spcra o obain h I(f) - jq(f) spcra. Th complx spcrum a h boom Figur 18 shows wha w wand; a digiizd vrsion of h complx bandpass signal cnrd abou zro Hz. Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd

16 gnraors, s o h sam frquncy. (Howvr, somhow w hav o synchroniz hos wo hardwar gnraors so ha hir rlaiv phas shif is fixd a 9 o.) Nx w connc coax cabls o h gnraors' oupu conncors and run hos wo cabls, labld 'i()' for our cosin signal and 'q()' for our sinwav signal, down h hall o hir dsinaion Now for a wo-qusion pop quiz. In h ohr lab, wha would w s on h scrn of an oscilloscop if h coninuous i() and q() signals wr conncd o h horizonal and vrical inpu channls, rspcivly, of h scop? (Rmmbring, of cours, o s h scop's Horizonal Swp conrol o h 'Exrnal' posiion.) q() = sin(π ) i() = cos(π ) O-scop Vr. In Horiz. In Figur. Displaying a quadraur signal using an oscilloscop. Nx, wha would b sn on h scop's display if h cabls wr mislabld and h wo signals wr inadvrnly swappd? Th answr o h firs qusion is ha w d s a brigh 'spo' roaing counrclockwis in a circl on h scop's display. If h cabls wr swappd, w'd s anohr circl, bu his im i would b orbiing in a clockwis dircion. This would b a na lil dmonsraion if w s h signal gnraors' frquncis o, say, 1 Hz. This oscilloscop xampl hlps us answr h imporan qusion, "Whn w work wih quadraur signals, how is h j-opraor implmnd in hardwar?" Th answr is ha h j- opraor is implmnd by how w ra h wo signals rlaiv o ach ohr. W hav o ra hm orhogonally such ha h in-phas i() signal rprsns an Eas-Ws valu, and h quadraur phas q() signal rprsns an orhogonal Norh-Souh valu. (By orhogonal, I man ha h Norh-Souh dircion is orind xacly 9 o rlaiv o h Eas-Ws dircion.) So in our oscilloscop xampl h j-opraor is implmnd mrly by how h conncions ar mad o h scop. Th in-phas i() signal conrols horizonal dflcion and h quadraur phas q() signal conrols vrical dflcion. Th rsul is a wo-dimnsional quadraur signal rprsnd by h insananous posiion of h do on h scop's display. Th prson in h lab down h hall who's rciving, say, h discr squncs i(n) and q(n) has h abiliy o conrol h orinaion of h final complx spcra by adding or subracing h jq(n) squnc as shown in Figur 1. i(n) q(n) -1 i(n) - jq(n) -B/ B/ i(n) + jq(n) -B/ B/ Figur 1. Using h sign of q(n) o conrol spcral orinaion. Th op pah in Figur 1 is quivaln o muliplying h original x bp () by -jπf c, and h boom pah is quivaln o muliplying h x bp () by jπf c. Thrfor, had h quadraur porion our Copyrigh Novmbr 8, Richard Lyons, All Righs Rsrvd

DIFFERENTIAL EQUATIONS with TI-89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations

DIFFERENTIAL EQUATIONS wih TI-89 ABDUL HASSEN and JAY SCHIFFMAN W will assum ha h radr is familiar wih h calculaor s kyboard and h basic opraions. In paricular w hav assumd ha h radr knows h funcions of

Chap 5. Continuous-Time Fourier Transform and Applications

77 Chap 5 Coninuous-im ourir ransform and Applicaions 5 Illusraiv Dfiniion of ourir ransform In his chapr, ill dvlop h basis for ourir analysis of non-priodic signals, hich is h only group of signals maningful

First Order Linear Differential Equations

Firs Ordr Linar Diffrnial Equaions A firs ordr ordinary diffrnial quaion is linar if i can b wrin in h form y p() y g() whr p and g ar arbirary funcions of. This is calld h sandard or canonical form of

Estimating Powers with Base Close to Unity and Large Exponents

Divulgacions Mamáicas Vol. 3 No. 2005), pp. 2 34 Esimaing Powrs wih Bas Clos o Uniy and Larg Exponns Esimacón d Poncias con Bas Crcana a la Unidad y Grands Exponns Vio Lampr Vio.Lampr@fgg.uni-lj.si) FGG,

EECE 301 Signals & Systems Prof. Mark Fowler

EECE 3 Signals & Sysms Prof. Mark Fowlr No S #4 C-T Signals: Fourir Transform (for Non-Priodic Signals) Rading Assignmn: Scion 3.4 & 3.5 of Kamn and Hck /27 Cours Flow Diagram Th arrows hr show concpual

QUALITY OF DYING AND DEATH QUESTIONNAIRE FOR NURSES VERSION 3.2A

UNIVERSITY OF WASHINGTON SCHOOL OF MEDICINE QUALITY OF DYING AND DEATH QUESTIONNAIRE FOR NURSES VERSION 3.2A Plas rurn your compld qusionnair in h nclosd nvlop o: [Rurn Addrss] RNID PID Copyrigh by h Univrsiy

MTBF: Understanding Its Role in Reliability

Modul MTBF: Undrsanding Is Rol in Rliabiliy By David C. Wilson Foundr / CEO March 4, Wilson Consuling Srvics, LLC dav@wilsonconsulingsrvics.n www.wilsonconsulingsrvics.n Wilson Consuling Srvics, LLC Pag

Numerical Algorithm for the Stochastic Present Value of Aggregate Claims in the Renewal Risk Model

Gn. Mah. Nos, Vol. 9, No. 2, Dcmbr, 23, pp. 4- ISSN 229-784; Copyrigh ICSRS Publicaion, 23 www.i-csrs.org Availabl fr onlin a hp://www.gman.in Numrical Algorihm for h Sochasic Prsn Valu of Aggrga Claims

Transfer function, Laplace transform, Low pass filter

Lab No PH-35 Porland Sa Univriy A. La Roa Tranfr funcion, Laplac ranform, Low pa filr. INTRODUTION. THE LAPLAE TRANSFORMATION L 3. TRANSFER FUNTIONS 4. ELETRIAL SYSTEMS Analyi of h hr baic paiv lmn R,

Many quantities are transduced in a displacement and then in an electric signal (pressure, temperature, acceleration). Prof. B.

Displacmn snsors Many quaniis ar ransducd in a displacmn and hn in an lcric signal (prssur, mpraur, acclraion). Poniomrs Poniomrs i p p i o i p A poniomr is basd on a sliding conac moving on a rsisor.

Unit 2. Unit 2: Rhythms in Mexican Music. Find Our Second Neighborhood (5 minutes) Preparation

Uni 2 Prparaion Uni 2: Rhyhms in Mxican Music Find Our Scond Nighborhood (5 minus) Th Conducor now aks us on a journy from Morningsid Highs, Manhaan, o Eas Harlm, Manhaan, o m our nx singr, Clso. Hav sudns

INTEGRATED SKILLS TEACHER S NOTES

TEACHER S NOTES INTEGRATED SKILLS TEACHER S NOTES LEVEL: Inrmdia + AGE: Tnagrs / Aduls TIME NEEDED: 90 minus + projc LANGUAGE FOCUS: Passiv; undrsand vocabulary in conx; opic words vrbs and nouns LEAD-IN

RDIOCTIVITY DOE-HDBK-1019/1-93 omic and Nuclar Physics RDIOCTIVITY Th ra a which a sampl of radioaciv marial dcays is no consan. s individual aoms of h marial dcay, hr ar fwr of hos yps of aoms rmaining.

RC (Resistor-Capacitor) Circuits. AP Physics C

(Rsisr-Capacir Circuis AP Physics C Circui Iniial Cndiins An circui is n whr yu hav a capacir and rsisr in h sam circui. Supps w hav h fllwing circui: Iniially, h capacir is UNCHARGED ( = 0 and h currn

GENETIC ALGORITHMS IN SEASONAL DEMAND FORECASTING

forcasing, dmand, gnic algorihm Grzgorz Chodak*, Wiold Kwaśnicki* GENETIC ALGORITHMS IN SEASONAL DEMAND FORECASTING Th mhod of forcasing sasonal dmand applying gnic algorihm is prsnd. Spcific form of usd

The 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

Solutions 1.4-Page 40

Soluions.4-Pag 40 Problm Find gnral soluions (implici if ncssary, plici if convnin) of h diffrnial quaions. dy = d ( 4y) / Sparaing h variabls yilds: / / / y dy = 4 d y / dy = 4 / d Ingraing boh sids o

Transient Thermoelastic Behavior of Semi-infinite Cylinder by Using Marchi-Zgrablich and Fourier Transform Technique

Inrnaional Journal of Mahmaical Enginring and Scinc ISSN : 77-698 Volum 1 Issu 5 (May 01) hp://www.ijms.com/ hps://sis.googl.com/si/ijmsjournal/ Transin Thrmolasic Bhavior of Smi-infini Cylindr by Using

Krebs (1972). A group of organisms of the same species occupying a particular space at a particular time

FW 662 Lcur 1 - Dnsiy-indpndn populaion modls Tx: Golli, 21, A Primr of Ecology Wha is a populaion? Krbs (1972). A group of organisms of h sam spcis occupying a paricular spac a a paricular im Col (1957).

The 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 wll-suitd as a first introduction to Cnts. Th CN modl is dscribd in grat dtail, xplaining th basic concpts of C-nts. Hnc, it can b rad by popl

Term Structure of Interest Rates: The Theories

Handou 03 Econ 333 Abdul Munasb Trm Srucur of Inrs Ras: Th Thors Trm Srucur Facs Lookng a Fgur, w obsrv wo rm srucur facs Fac : Inrs ras for dffrn maurs nd o mov oghr ovr m Fac : Ylds on shor-rm bond mor

The Sensitivity of Beta to the Time Horizon when Log Prices follow an Ornstein- Uhlenbeck Process

T Snsiiviy of Ba o Tim Horizon wn Log Prics follow an Ornsin- Ulnbck Procss Oc 8, 00) KiHoon Jimmy Hong Dparmn of Economics, Cambridg Univrsiy Ocobr 4, 00 Sv Sacll Triniy Collg, Cambridg Univrsiy Prsnaion

Non-Homogeneous Systems, Euler s Method, and Exponential Matrix

Non-Homognous Systms, Eulr s Mthod, and Exponntial Matrix W carry on nonhomognous first-ordr linar systm of diffrntial quations. W will show how Eulr s mthod gnralizs to systms, giving us a numrical approach

Option Pricing with Constant & Time Varying Volatility

Opion Pricing wih Consan & im arying olailiy Willi mmlr Cnr for Empirical Macroconomics Bilfld Grmany; h Brnard chwarz Cnr for Economic Policy Analysis Nw York NY UA and Economics Dparmn h Nw chool for

SPECIAL VOWEL SOUNDS

SPECIAL VOWEL SOUNDS Plas consult th appropriat supplmnt for th corrsponding computr softwar lsson. Rfr to th 42 Sounds Postr for ach of th Spcial Vowl Sounds. TEACHER INFORMATION: Spcial Vowl Sounds (SVS)

QUANTITATIVE 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.

Step Functions; and Laplace Transforms of Piecewise Continuous Functions

Sp Fnion; and Lapla Tranform of Piwi Conino Fnion Th prn objiv i o h Lapla ranform o olv diffrnial qaion wih piwi onino foring fnion ha i, foring fnion ha onain dioninii Bfor ha old b don, w nd o larn

Interest Parity Conditions. Interest parity conditions are no-arbitrage profit conditions for capital. The easiest way to

World Economy - Inrs Ra Pariy (nry wrin for h Princon Encyclopdia of h World Economy) Inrs Pariy Condiions Th pariy condiions Inrs pariy condiions ar no-arbirag profi condiions for capial. Th asis way

5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:

.4 Eponntial Functions: Diffrntiation an Intgration TOOTLIFTST: Eponntial functions ar of th form f ( ) Ab. W will, in this sction, look at a spcific typ of ponntial function whr th bas, b, is.78.... This

5 2 index. e e. Prime numbers. Prime factors and factor trees. Powers. worked example 10. base. power

Prim numbrs W giv spcial nams to numbrs dpnding on how many factors thy hav. A prim numbr has xactly two factors: itslf and 1. A composit numbr has mor than two factors. 1 is a spcial numbr nithr prim

You can recycle all your cans, plastics, paper, cardboard, garden waste and food waste at home.

Your 4 bin srvic You can rcycl all your cans, plasics, papr, cardboard, gardn was and food was a hom. This guid conains imporan informaion abou wha can b rcycld in your bins. Plas ak a momn o rad i. for

Question 3: How do you find the relative extrema of a function?

ustion 3: How do you find th rlativ trma of a function? Th stratgy for tracking th sign of th drivativ is usful for mor than dtrmining whr a function is incrasing or dcrasing. It is also usful for locating

CPS 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:

Mathematics. 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

New 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 ral-valud Fourir sris is xplaind and formula ar givn for convrting

HANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.

Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can

Free ACA SOLUTION (IRS 1094&1095 Reporting)

Fr ACA SOLUTION (IRS 1094&1095 Rporting) Th Insuranc Exchang (301) 279-1062 ACA Srvics Transmit IRS Form 1094 -C for mployrs Print & mail IRS Form 1095-C to mploys HR Assist 360 will gnrat th 1095 s for

Subject: Quality Management System Requirements SOP 4-1-0-1 -

SOP 4-1-0-1 - 1.) Scop: This SOP covrs h rquirmns for Pharmco s Qualiy Managmn Sysm. 2.) Normaiv Rfrncs: SOP 4-4-2: Qualiy Manual. SOP 4-2-3-A: Masr Lis of Conrolld Documns SOP 5-6-0 Managmn Rsponsibiliy

Fourier Series and Fourier Transform

Fourier Series and Fourier ransform Complex exponenials Complex version of Fourier Series ime Shifing, Magniude, Phase Fourier ransform Copyrigh 2007 by M.H. Perro All righs reserved. 6.082 Spring 2007

Estimating Private Equity Returns from Limited Partner Cash Flows

Esimaing Priva Equiy Rurns from Limid Parnr Cash Flows Andrw Ang, Bingxu Chn, William N. Gozmann, and Ludovic Phalippou* Novmbr 25, 2013 W inroduc a mhodology o sima h hisorical im sris of rurns o invsmn

Damon s Newark is hosting an October Charity Fest to help raise money for The Fallen Heroes and Big Brothers and Big Sisters of Licking County.

Rgarding: Ocobr Chariy Fs on Saurday Ocobr 15, 2011. Damon s Nwark is hosing an Ocobr Chariy Fs o hlp rais mony for Th and Big Brohrs and Big Sisrs of Licking Couny. Th purpos of BBBS is o organiz, undr

14.3 Area Between Curves

14. Ara Btwn Curvs Qustion 1: How is th ara btwn two functions calculatd? Qustion : What ar consumrs and producrs surplus? Earlir in this chaptr, w usd dfinit intgrals to find th ara undr a function and

e = C / electron Q = Ne

Physics 0 Modul 01 Homwork 1. A glass rod that has bn chargd to +15.0 nc touchs a mtal sphr. Aftrword, th rod's charg is +8.00 nc. What kind of chargd particl was transfrrd btwn th rod and th sphr, and

Introduction to Measurement, Error Analysis, Propagation of Error, and Reporting Experimental Results

Inroducion o Masurmn, Error Analysis, Propagaion of Error, and Rporing Exprimnal Rsuls AJ Pinar, TD Drummr, D Caspary Dparmn of Chmical Enginring Michigan Tchnological Univrsiy Houghon, MI 4993 Spmbr,

Rapid Estimation of Water Flooding Performance and Optimization in EOR by Using Capacitance Resistive Model

Iranian Journal of Chmical Enginring Vol. 9, No. 4 (Auumn), 22, IAChE Rapid Esimaion of War Flooding Prformanc and Opimizaion in EOR by Using Capacianc Rsisiv Modl A.R. Basami, M. Dlshad 2, P. Pourafshary

Lecture 13-14: The Trade-off between Inflation and Unemployment

EC201 Inrmdia Macroconomics EC201 Inrmdia Macroconomics Lcr 13-14: Th Trad-off bwn Inflaion and Unmploymn Lcr Olin: - Th hillips Crv: - Th rlaionship bwn inflaion and nmploymn; - Th hillips crv as an aggrga

Uniplan REIT Portfolio Fiduciary Services Uniplan Investment Counsel, Inc.

Uniplan EIT Porfolio Uniplan Invsmn Counsl, Inc. 22939 W Ovrson d Union Grov, Wisconsin 53182 Syl: Sub-Syl: Firm AUM: Firm Sragy AUM: EITs EITs Domsic \$2.2 billion \$1.3 billion Yar Foundd: GIMA Saus: Firm

EXTRACTION OF FINANCIAL MARKET EXPECTATIONS ABOUT INFLATION AND INTEREST RATES FROM A LIQUID MARKET. Documentos de Trabajo N.

ETRCTION OF FINNCIL MRKET EPECTTIONS OUT INFLTION ND INTEREST RTES FROM LIQUID MRKET 2009 Ricardo Gimno and José Manul Marqués Documnos d Trabajo N.º 0906 ETRCTION OF FINNCIL MRKET EPECTTIONS OUT INFLTION

Lecture 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

ISSeG EGEE07 Poster Ideas for Edinburgh Brainstorming

SSG EGEE07 Pos das fo Edinbugh Bainsoming 3xposs, plus hoizonal and vical banns (A0=841mm x 1189mm) Why SSG: anion gabbing: hadlins/shock phoos/damaic ycaching imag Wha is SSG: pojc ovviw: SSG ino, diffnc

Econ 371: Answer Key for Problem Set 1 (Chapter 12-13)

con 37: Answr Ky for Problm St (Chaptr 2-3) 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

Brussels, February 28th, 2013 WHAT IS

Brussls, Fbruary 28h, 2013 WHAT IS 1 OPEN SOURCE 2 CLOUD 3 SERVICES 4 BROKER 5 INTERMEDIATION AGGREGATION ARBITRAGE Cloud Srvics Brokr provids a singl consisn inrfac o mulipl diffring providrs, whhr h

CHARGE AND DISCHARGE OF A CAPACITOR

REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

Lecture 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.

Basis risk. When speaking about forward or futures contracts, basis risk is the market

Basis risk Whn spaking about forward or futurs contracts, basis risk is th markt risk mismatch btwn a position in th spot asst and th corrsponding futurs contract. Mor broadly spaking, basis risk (also

Hopkins Elementary School. Improvement Plan Hopkins Elementary School

Hopkins Elmnary School Improvmn Plan 2015-16 Hopkins Elmnary School Hopkins Public Schools Mr. Sco VanBonn 400 Clark Sr Hopkins, MI 49328-0278 Documn Gnrad On April 19, 2016 TABLE OF CONTENTS Ovrviw 1

CFD-Calculation of Fluid Flow in a Pressurized Water Reactor

Journal of Scincs, Islamic Rpublic of Iran 19(3): 73-81 (008) Univrsiy of Thran, ISSN 1016-1104 hp://jscincs.u.ac.ir CFD-Calculaion of Fluid Flow in a Prssurizd War Racor H. Farajollahi, * A. Ghasmizad,

CONTINUOUS TIME KALMAN FILTER MODELS FOR THE VALUATION OF COMMODITY FUTURES AND OPTIONS

CONTINUOUS TIME KALMAN FILTER MODELS FOR THE VALUATION OF COMMODITY FUTURES AND OPTIONS ANDRÉS GARCÍA MIRANTES DOCTORAL THESIS PhD IN QUANTITATIVE FINANCE AND BANKING UNIVERSIDAD DE CASTILLA-LA MANCHA

Voltage level shifting

rek Applicaion Noe Number 1 r. Maciej A. Noras Absrac A brief descripion of volage shifing circuis. 1 Inroducion In applicaions requiring a unipolar A volage signal, he signal may be delivered from a bi-polar

State Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University

Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween

4# FEAP(First Aid & Emergency Situations)

Projc Na Projc Ovrviw 4# FEAP(Firs Aid & Ergncy Siuaions) Sudns will b dvloping knowldg and skills in rcognising and rsponding o rgncy siuaions. Thy will donsra hs skills wihin workshop basd scnarios.

Genetic Drift and Gene Flow Illustration

Gntic Drift and Gn Flow Illustration This is a mor dtaild dscription of Activity Ida 4, Chaptr 3, If Not Rac, How do W Explain Biological Diffrncs? in: How Ral is Rac? A Sourcbook on Rac, Cultur, and Biology.

Logo Design/Development 1-on-1

Logo Dsign/Dvlopmnt 1-on-1 If your company is looking to mak an imprssion and grow in th marktplac, you ll nd a logo. Fortunatly, a good graphic dsignr can crat on for you. Whil th pric tags for thos famous

intro Imagine that someone asked you to describe church using only the bible. What would you say to them?

intro Imagin that somon askd you to dscrib church using only th bibl. What would you say to thm? So many of th things w'v mad church to b arn't ssntial in scriptur. W'r on a journy of r-imagining what

Dept. of Heating, Ventilation and Air-Conditioning. Zentralschweizerisches Technikum Luzern Ingenieurschule HTL

Znralshwizrishs Thnikum Luzrn Ingniurshul HTL Dp. o Haing, Vnilaion Elkrohnik - Mashinnhnik - Hizungs-, Lüungs-, Klimahnik - Arhikur - Bauingniurwsn Dvlopd in h proj Low Tmpraur Low Cos Ha Pump Haing Sysm

RC, RL and RLC circuits

Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.

by John Donald, Lecturer, School of Accounting, Economics and Finance, Deakin University, Australia

Studnt Nots Cost Volum Profit Analysis by John Donald, Lcturr, School of Accounting, Economics and Financ, Dakin Univrsity, Australia As mntiond in th last st of Studnt Nots, th ability to catgoris costs

Representing Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1

Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

LG has introduced the NeON 2, with newly developed Cello Technology which improves performance and reliability. Up to 320W 300W

Cllo Tchnology LG has introducd th NON 2, with nwly dvlopd Cllo Tchnology which improvs prformanc and rliability. Up to 320W 300W Cllo Tchnology Cll Connction Elctrically Low Loss Low Strss Optical Absorption

17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur

Module 4 Single-phase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,

Ch 2.1: Linear Equations; Method of Integrating Factors

Ch.: Linar Equaions; Mhod of Ingraing Facors A linar firs ordr ODE has h gnral form d d f whr f is linar in. Eampls includ quaions wih consan cofficins such as hos in Chapr a b or quaions wih variabl cofficins:

Factorials! Stirling s formula

Author s not: This articl may us idas you havn t larnd yt, and might sm ovrly complicatd. It is not. Undrstanding Stirling s formula is not for th faint of hart, and rquirs concntrating on a sustaind mathmatical

union scholars program APPLICATION DEADLINE: FEBRUARY 28 YOU CAN CHANGE THE WORLD... AND EARN MONEY FOR COLLEGE AT THE SAME TIME!

union scholars YOU CAN CHANGE THE WORLD... program AND EARN MONEY FOR COLLEGE AT THE SAME TIME! AFSCME Unitd Ngro Collg Fund Harvard Univrsity Labor and Worklif Program APPLICATION DEADLINE: FEBRUARY 28

Uniplan REIT Portfolio Select UMA Uniplan Investment Counsel, Inc.

Uniplan EIT Porfolio Uniplan Invsmn Counsl, Inc. 22939 W Ovrson d Union Grov, Wisconsin 53182 Syl: Sub-Syl: Firm AUM: Firm Sragy AUM: EITs EITs Domsic \$2.2 billion \$1.3 billion Yar Foundd: GIMA Saus: Firm

Use a high-level conceptual data model (ER Model). Identify objects of interest (entities) and relationships between these objects

Chaptr 3: Entity Rlationship Modl Databas Dsign Procss Us a high-lvl concptual data modl (ER Modl). Idntify objcts of intrst (ntitis) and rlationships btwn ths objcts Idntify constraints (conditions) End

Motion Along a Straight Line

Moion Along a Sraigh Line On Sepember 6, 993, Dave Munday, a diesel mechanic by rade, wen over he Canadian edge of Niagara Falls for he second ime, freely falling 48 m o he waer (and rocks) below. On his

IT Update - August 2006

IT Nws Saus: No Aciv Til: Da: 7726 Summay (Opional): Body: Wlcom Back! Offic of Infomaion Tchnology Upda: IT Upda - Augus 26 Rob K. Blchman, Ph.D. Associa Dico, Offic of Infomaion Tchnology Whil You W

Fourier Series and Spectrum

EE54 Signls nd Sysms Fourir Sris nd Spcrum Yo Wng Polychnic Univrsiy Mos of h slids includd r xrcd from lcur prsnions prprd by McCllln nd Schfr Licns Info for SPFirs Slids his wor rlsd undr Criv Commons

The Land Partnerships Handbook. The Land Partnerships Handbook. Using land to unlock business innovation. Second Edition

Th Land Parnrships Handbook Using land o unlock businss innovaion Scond Ediion Using land o unlock businss innovaion Conns 04 Wha is h Land Parnrships approach? 06 Sp 1: Taking sock 08 Sp 2: Finding h

Small Cap Fiduciary Services

Sragy Saus: Closd Sragy closd o nw accouns and opn o addiional asss Th London Company 1800 Baybrry Cour, Sui 301 ichmond, Virginia 23226 Syl: Sub-Syl: Firm AUM: Firm Sragy AUM: US Valu-orind \$10.0 billion

AP Calculus AB 2008 Scoring Guidelines

AP Calculus AB 8 Scoring Guidlins Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos mission is to connct studnts to collg succss and opportunity.

Lateef Investment Management, L.P. 300 Drakes Landing Road, Suite 210 Greenbrae, California 94904

Laf Invsmn Managmn, L.P. 300 Draks Landing oad, Sui 210 Grnbra, California 94904 Sragy Saus: Closd Sragy closd o nw accouns and opn o addiional asss Syl: Sub-Syl: Firm AUM: Firm Sragy AUM: US Larg Cap

Remember you can apply online. It s quick and easy. Go to www.gov.uk/advancedlearningloans. Title. Forename(s) Surname. Sex. Male Date of birth D

24+ Advancd Larning Loan Application form Rmmbr you can apply onlin. It s quick and asy. Go to www.gov.uk/advancdlarningloans About this form Complt this form if: you r studying an ligibl cours at an approvd

Density Forecasting of Intraday Call Center Arrivals. using Models Based on Exponential Smoothing

Dnsiy Forcasing of Inraday Call Cnr Arrivals using Modls Basd on Exponnial Soohing Jas W. Taylor Saïd Businss School Univrsiy of Oxford Managn Scinc, 0, Vol. 58, pp. 534-549. Addrss for Corrspondnc: Jas

Modulation and Filtering

Modulaion and Filering Wireless communicaion applicaion Impulse uncion deiniion and properies Fourier Transorm o Impulse, Sine, Cosine Picure analysis using Fourier Transorms Copyrigh 27 by M.H. Perro

ME 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.

Technological Entrepreneurship : Modeling and Forecasting the Diffusion of Innovation in LCD Monitor Industry

0 Inrnaional Confrnc on Economics and Financ Rsarch IPEDR vol.4 (0 (0 IACSIT Prss, Singaor Tchnological Enrrnurshi : Modling and Forcasing h Diffusion of Innovaion in LCD Monior Indusry Li-Ming Chuang,

4 Convolution. Recommended Problems. x2[n] 1 2[n]

4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.1-1.

EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS

25 Vol. 3 () January-March, pp.37-5/tripathi EFFECT OF GEOMETRICAL PARAMETERS ON HEAT TRANSFER PERFORMACE OF RECTANGULAR CIRCUMFERENTIAL FINS *Shilpa Tripathi Dpartmnt of Chmical Enginring, Indor Institut

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.

Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa

Duration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is \$613.

Graduae School of Business Adminisraion Universiy of Virginia UVA-F-38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised

Enterprise. Welcome. Enterprise. Highlights. Issue 8 - Spring 2014. That s Zen thinking

Enrpris Issu 8 - Spring 2014 Wlcom Wlcom o anohr diion of Zn Inrn s Enrpris Nwslr. I s an xciing im for UK businsss, wih nw opions for connciviy, nw applicaions in h cloud, and mos of all nw opporuniis

4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay

324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find

Investment Grade Fixed Income Select UMA Cincinnati Asset Management

Invsmn Grad Fixd Incom Cincinnai Ass Managmn 8845 Govrnor's Hill Driv Cincinnai, Ohio 45249 Syl: US Taxabl Cor Sub-Syl: Taxabl Corpora Firm AUM: \$2.7 billion Firm Sragy AUM: \$2.0 billion Yar Foundd: GIMA

9. Capacitor and Resistor Circuits

ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren

Chapter 28 Magnetic Induction

Chapr 8 Magnic nducion Forad Concpual Probls (a) Th agnic quaor is a lin on h surfac of Earh on which Earh s agnic fild is horizonal. A h agnic quaor, how would you orin a fla sh of papr so as o cra h

Entity-Relationship Model

Entity-Rlationship Modl Kuang-hua Chn Dpartmnt of Library and Information Scinc National Taiwan Univrsity A Company Databas Kps track of a company s mploys, dpartmnts and projcts Aftr th rquirmnts collction