Design of Infinite Impulse Response (IIR) digital filters

Size: px
Start display at page:

Download "Design of Infinite Impulse Response (IIR) digital filters"

Transcription

1 Dign of Infinit Imul Ron (IIR) digital filtr Outut from a digital filtr i mad u from rviou inut and rviou outut, uing th oration of convolution: wo convolutionar involvd: on with th rviou inut, and on with th rviou outut. In ach ca th convolving function i calld th filtr cofficint. If uch a filtr i ubjctd to an imul (a ignal coniting of on valu followd by ro) thn it outut nd not ncarily bcom ro aftr th imul ha run through th ummation. So th imul ron of uch a filtr can b infinit in duration. Such a filtr i calld an Infinit Imul Ron filtr or IIR filtr. Not that th imul ron nd not ncarily b infinit: if it wr, th filtr would b untabl. In fact for mot ractical filtr, th imul ron will di away to a ngligibly mall lvl. On might argu that mathmatically th ron can go on for vr, gtting mallr and mallr: but in a digital world onc a lvl gt blow on bit it might a wll b ro. h Infinit Imul Ron rfr to th ability of th filtr to hav an infinit imul ron and do not imly that it ncarily will hav on: it rv a a warning that thi ty of filtr i ron to fdbac and intability.

2 h filtr can b drawn a a bloc diagram: officint calculation mthod for IIR filtr Whil trying to comut th filtr cofficint on ha to lct a mthod from a numbr of aroximation mthod in ordr to calculat th valu of d and c. A iml way to obtain th IIR filtr cofficint i to lac ol and ro judiciouly in th -lan uch that th rulting filtr ha th dird frquncy ron. hi aroach i nown a th ol-ro-lacmnt mthod. I only uful for vry iml filtr, i.. notch filtring, whr th filtr aramtr (aband ril) nd not b cifid rcily. A mor fficint aroach i firt to dign an analog filtr atifying th dird cification and thn to convrt it into an quivalnt digital filtr. Mot IIR digital filtr ar dignd thi way. hr ar thr main mthod for convrting an analog filtr into a digital filtr. ) imul invarianc ) th matchd -tranform 3) bi-linar -tranform Baic onct and illutrativ dign xaml: Whn a ro i lacd at a oint on th -lan, th frquncy ron will b ro at th corronding oint. A ol on th othr hand roduc a a at th corronding frquncy oint. Pol that ar clo to th unit circl giv ri to larg a whra ro clo to or on th circl through or minima. nc by tratgically lacing ol and ro on th -lan w can obtain iml lowa or othr frquncy lctiv filtr.

3 An imortant oint to rmmbr i that : In ordr for th filtr cofficint to b ral, th ol and ro mut b ithr ral (i.. li on th oitiv or ngativ ral axi) or occur in comlx conjugat air. F /4 F / 0 3F /4 0 F /4 F / 3F /4 Frquncy (a) (b) (a) Pol ro diagram of a iml filtr (b) A tch of it frquncy ron Examl: Illutrating th iml ol-ro mthod of calculating th filtr cofficint a banda digital filtr i rquird to mt th following cification: (i) (ii) (iii) comlt ignal rjction at D and 50. A narrow aband cntrd at 5 A 3dB bandwidth of 0. Auming a amling frquncy of 500, obtain th tranfr function of th filtr by uitably lacing -lan ol and ro, and it diffrnc quation. Solution: Sinc a comlt rjction i rquird at ro and 50 w nd to lac ro at th corronding oint on th -lan. h ar at angl of 0 and /50080 on th unit circl. o hava th aband cntrd at 5 rquir u to lac ol at ±360 5/500±90. o nur that th cofficint ar ral, it i ncary to hav comlx conjugat ol air.

4 Im R (a) (b) (a) Pol-ro dıagram (b) Bloc diagram rrntation of filtr h radiu of th ol i dtrmind by th dird bandwidth. An aroximat rlationhi btwn r, for r> 0.9, and bandwidth bw i: r ( bw / )π for our cification bw 0 and F500. hi lad to an r valu of r F [] ( )( ) jπ / jπ / ( r )( r ) h diffrnc quation i y [] n y[ n ] x[] n x[ n ] b 0, b 0, b - and a 0, a

5 Examl : (Uing th ol-ro lacmnt mthod to calculat cofficint of a notch filtr) Obtain by th ol-ro lacmnt mthod th tranfr function and th diffrnc quation of a iml digital notch filtr that mt th following cification. Solution: Notch frquncy 50 3dB width of notch ±5 amling frquncy 500 o rjct th comonnt at 50 w lac a air of comlx ro at oint on th unit circl corronding to 50 that i at angl of /500±36. o achiv a har notch filtr and imrovd amlitud ron on ithr id of th notch frquncy, a air of comlx conjugat ol ar lacd at a radiu r <. h width of th notch i dtrmind by th location of th ol. h rlationhi btwn th bandwidth and radiu i am a in th rviou xaml. Im [ f ] R Frquncy h rlationhi btwn th bandwidth and th rdiu i alicabl. hu th radiu of th ol i From th tranfr function of th filtr w hav: j36 j36 [] [ ][ ] [ ][ ]

6 h diffrnc quation will b: y [] n x[] n.680x[ n ] x[ n ].564y[ n ] y[ n ] b 0, b -.680, b and a -.564, a Imul Invariant Mthod of officint alculation In thi mthod, tarting with a uitabl analog tranfr function () th imul ron h(t) i obtaind uing th Lalac tranform. h h(t) o obtaind i uitably amld to roduc h(n), and th dird tranfr function () i thn obtaind by -tranforming h(n) whr i th amling intrval. Examl: (thortic) Digiti an analog filtr uing imul invarianc mthod. h analog filtr ha a tranfr function givn by: () Solution: If w aly th invr Lalac tranform w can gt h(t) a blow: h ( t) L t { ( ) } L According to th imul invariant mthod, th imul ron of th quivalnt digital filtr, h(n), i qual to h(t) at th dicrt tim tn, n 0,,,... n ( n ) h( t) h t n h tranfr function of [] i obtaind by -tranforming h(n) : n n 0 n 0 [] h( n ) n n hu from th rult o far w can writ : o aly th imul invariant mthod to a high ordr IIR filtr with iml ol, th tranfr function () i firt xandd uing artial fraction a th um of ingl-ol filtr:

7 M M M... ) ( whr ar th ol of ().Sinc ach trm ha th am form thn w can ay that: M M igh ordr IIR filtr ar normally ralid a cacad or aralll combination of tandard cond ordr filtr ction. For ca M ( ) ( ) ( ) If th ol and ar comlx conjugat th and will alo b comlx conjugat ( ) ( ) [ ] ( ) * co in co * i i i i r r r r r r r and i ar th ral and imaginary art of and r and i ar th ral and imaginary art of. Examl: Dign a digital filtr to aroximat th following normalid analog tranfr function: ) ( Uing th imul invariant mthod obtain th tranfr function, (), of th digital filtr, auming a 3dB cutoff frquncy of 50 and a amling frquncy of.8. Solution: Firt w nd to frquncy cal th normalid tranfr function. hi i achivd by rlacing by /α whr α π

8 ' () () α / α α α whr; ( j) * α ( j), α j j, Sinc th ol ar comlx conjugat, th tranformation i ud to obtain th dicrt-tim tranfr function, (). * r 0, i , i 0.507, r 0.507, r 0.594, in( ) , i co( ) i and r Uing th valu in th rviouly obtaind xrion w gt () a: [] If w ubtitut jw in th abov quation th valu of [] at w0 i 3, aroximatly qual to th amling frquncy. Such larg gain i charactritic of imul invariant filtr. o th gain down and to avoid ovrflow whn th filtr i imlmntd, it i common ractic to multily [] by (or quivalntly divid it by th amling frquncy) [] hrfor, w hav b 0 0 b a.0308 a

9 h bilinar tranformation Lt u conidr th idntity ( ) whr, and ar comlx variabl. hi i what i nown a th bilinar tranformation or utin tranformation. h invr of thi tranformation i aily dtrmind a jw In th dicrt domain w ar intrtd in valu on th unit circl whra in th Lalac domain valu on th imaginary axi (jω) i of imortanc. h tranformation and it invr ar wll dfind for all on th imaginary axi whr jω. r Ω dnot th analog frqunci in rad/c. h bilinar tranformation ti th two frquncy cal togthr a follow. Starting from ( ) jw and ltting w obtain: jω jw jw / jw / ( ) ( ) in( w/ ) jw jw / jw / i co ( w/ ) i tan( w/ ) and conquntly th rlation btwn th two frquncy cal i givn by Ω tan w ( / ) W not that w0 imli Ω0 and for incraing w th continuou frqunci will alo incra and a w π th Ω. nc th ntir continuou tim frquncy rgion Ω 0, i mad to th ur half of th unit circl. [ )

10 ranformd ranfr Function On imortant alication of th bilinar tranformation i for th convrion of an analog filtr to a digital on. h tranformation i alid a follow. Start from a Lalac domain filtr a () which i a rational function in th fr variabl. A rational tranfr function i th quotint of two olynomial. h bilinarly tranformd dicrt tim (Z-domain) tranfr function d [] i dfind a: d [] () ( ) [] a Not that d [] i alo a rational tranfr function. h MALAB command bilinar rform th bilinar tranformation. h following good rorti can b idntifid: It i ay to vrify that th numbr of ol of d [] and a () coincid. hi man that th ordr of th filtr i unchangd. If a () i a tabl tranfr function o i alo d [] h comlx filtr ron, i.. both magnitud and ha, coincid for frqunci rlatd by: a jw ( jω ) [ ] h lat fatur imli that th bilinar-tranformation rrv th ty of filtr. If a () i a low-a filtr thn d [] will alo b a low-a filtr. d d Analog IIR filtr dign o dign any of th four ty analog IIR filtr it i ufficint to ba th dign on an analog IIR lowa filtr. hi ction brifly ummari th dign of analog LP IIR filtr bad on filtr rototy. An analog rototy LP-filtr i givn by : LP ( ) n n ( )... n whr () i a rototy olynomial. h ty of LP filtr dfin th olynomial cofficint and th variou rototy olynomial can b found in tandard txtboo. h MALAB command butta and chba giv th ro and ol of th rototy filtr. Examl ar givn in th tabl blow:

11 Ordr Prototy olynomial () Gain abl : Buttrworth rototy olynomial Ordr Prototy olynomial () Gain abl : hbyhv ty rototy olynomial wit a-band ril 0.5dB. All rototy olynomial hav th 3dB cutoff frquncy normalid to Ωrad/. For a articular dign with a dird cut-off frquncy of Ω LPc th filtr i modifid by caling th variabl with th invr of th dird cut-off frquncy. h final analog dign i thn; LP ( ) LP ( ) n n Ω LPc... Ω LPc Ω LPc In MALAB ll rform thi dign from a givn rototy. n Ω LPc n P,BP and BS filtr dign hr othr ty of dign, high-a, band-a, and band-to can all b gnratd from a low-a filtr uing cial frquncy tranformation. W aum that LP () i a low-a filtr with -3dB cutoff frquncy Ω LPc and to band frquncy at Ω LP a illutratd in figur. Figur : h four tandard filtr : LP low-a, Phigh-a, BPband-a, BSband-to

12 igh-a dign A high-a dign i achivd from a low-a filtr by th tranformation P ( ) ( ) LP Ω I LP Ω I whr Ω Ω Pc P Ω Ω Ω Ω I LPc I LP r Ω I i a valid caling. In MALAB th function lh rform thi dign from a givn rototy. Analog IIR dign ummary Aum th rulting filtr ordr i givn. hi immdiatly contrain th width of th tranition on and w only hav to dign th oition of th cut-off frqunci. ) Idntify th cut-off frqunci (Ω XXlc and Ω XXhc ) of th dign cification. From thm dtrmin Ω and th cut-off frquncy Ω of th low-a filtr. I ) Dign a low-a filtr with th dird ΩLPc uing a low-a rototy filtr (.g. Buttrworth or hbyhv). If th final dign i a BP or BS filtr, thn thi low-a filtr hould hav half th ordr of th final dird dign. 3) U th corronding frquncy tranformation to finally obtain th dird filtr. LPc

13 Digital IIR filtr dign h dign of an IIR filtr i rathr traightforward whn uing th bilinar tranformation tchniqu. Mot of th wor i don whn digning th analog filtr. Rcall that if an analog filtr i convrtd to a digital on via th bilinar tranformation w obtain th rlation: A th bilinar tranformation rrv th ty of filtr (LP, BP tc.) w only nd to ma ur that th cut-off frqunci gt mad to th corrct lac. h dign tchniqu i ummarid hr:. Start by idntifying th dicrt tim frqunci dcribing th cutoff frqunci. U normalid frqunci in radian whr w π rrnt half th amlingfrquncy ( w πf/f ).. alculat th corronding analog frqunci uing Ω tan( w/ ) 3. Dign an analog filtr a () bad on th cification a dcribd in ction bfor. 4. U th bilinar tranformation to obtain th final digital (Z-domain) filtr.

14

15 Examl

16 Examl:

CHAPTER 4c. ROOTS OF EQUATIONS

CHAPTER 4c. ROOTS OF EQUATIONS CHAPTER c. ROOTS OF EQUATIONS A. J. Clark School o Enginring Dpartmnt o Civil and Environmntal Enginring by Dr. Ibrahim A. Aakka Spring 00 ENCE 03 - Computation Mthod in Civil Enginring II Dpartmnt o Civil

More information

Section 7.4: Exponential Growth and Decay

Section 7.4: Exponential Growth and Decay 1 Sction 7.4: Exponntial Growth and Dcay Practic HW from Stwart Txtbook (not to hand in) p. 532 # 1-17 odd In th nxt two ction, w xamin how population growth can b modld uing diffrntial quation. W tart

More information

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

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

More information

Parallel and Distributed Programming. Performance Metrics

Parallel and Distributed Programming. Performance Metrics Paralll and Distributd Programming Prformanc! wo main goals to b achivd with th dsign of aralll alications ar:! Prformanc: th caacity to rduc th tim to solv th roblm whn th comuting rsourcs incras;! Scalability:

More information

Chapter 10 Function of a Matrix

Chapter 10 Function of a Matrix EE448/58 Vrsion. John Stnsby Chatr Function of a atrix t f(z) b a comlx-valud function of a comlx variabl z. t A b an n n comlxvalud matrix. In this chatr, w giv a dfinition for th n n matrix f(a). Also,

More information

New Basis Functions. Section 8. Complex Fourier Series

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

More information

5.4 Exponential Functions: Differentiation and Integration TOOTLIFTST:

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

More information

A Note on Approximating. the Normal Distribution Function

A Note on Approximating. the Normal Distribution Function Applid Mathmatical Scincs, Vol, 00, no 9, 45-49 A Not on Approimating th Normal Distribution Function K M Aludaat and M T Alodat Dpartmnt of Statistics Yarmouk Univrsity, Jordan Aludaatkm@hotmailcom and

More information

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

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

More information

QUANTITATIVE METHODS CLASSES WEEK SEVEN

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.

More information

ME 612 Metal Forming and Theory of Plasticity. 6. Strain

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.

More information

http://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force

http://www.wwnorton.com/chemistry/tutorials/ch14.htm Repulsive Force ctivation nrgis http://www.wwnorton.com/chmistry/tutorials/ch14.htm (back to collision thory...) Potntial and Kintic nrgy during a collision + + ngativly chargd lctron cloud Rpulsiv Forc ngativly chargd

More information

Foreign Exchange Markets and Exchange Rates

Foreign Exchange Markets and Exchange Rates Microconomics Topic 1: Explain why xchang rats indicat th pric of intrnational currncis and how xchang rats ar dtrmind by supply and dmand for currncis in intrnational markts. Rfrnc: Grgory Mankiw s Principls

More information

Mathematics. Mathematics 3. hsn.uk.net. Higher HSN23000

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

More information

CPS 220 Theory of Computation REGULAR LANGUAGES. Regular expressions

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:

More information

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

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

More information

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

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

More information

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

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

More information

Lecture 3: Diffusion: Fick s first law

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

More information

THE FUNDAMENTALS OF CURRENT SENSE TRANSFORMER DESIGN. Patrick A. Cattermole, Senior Applications Engineer MMG 10 Vansco Road, Toronto Ontario Canada

THE FUNDAMENTALS OF CURRENT SENSE TRANSFORMER DESIGN. Patrick A. Cattermole, Senior Applications Engineer MMG 10 Vansco Road, Toronto Ontario Canada , Snior Alications nginr MMG 10 Vansco Road, Toronto Ontario Canada Abstract Th following ar will first rviw th basic rincils of oration of a Currnt Sns Transformr and thn follow a simlifid dsign rocdur.

More information

ESCI 241 Meteorology Lesson 6 Humidity

ESCI 241 Meteorology Lesson 6 Humidity ESCI 41 Mtorology Lsson 6 Humiity Raing: MT Chatr 5 PARTIAL PRESSURE In a mixtur of gass, ach gas scis contributs to th total rssur. ο Th rssur xrt by a singl gas scis is known as th artial rssur for that

More information

A Novel Approach For Generating Rules For SMS Spam Filtering Using Rough Sets

A Novel Approach For Generating Rules For SMS Spam Filtering Using Rough Sets A Novl Approach For Gnrating Rul For SMS Spam Filtring Uing Rough St Ahima Wadhawan, Nrja Ngi Abtract: Spam i dfind a unwantd commrcial mag to many rcipint. Email Spamming i a univral problm with which

More information

Version 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final.

Version 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC3. (Specification 6360) Pure Core 3. Final. Vrsion.0 Gnral Crtificat of Education (A-lvl) January 0 Mathmatics MPC (Spcification 660) Pur Cor Final Mark Schm Mark schms ar prpard by th Principal Eaminr and considrd, togthr with th rlvant qustions,

More information

Transistor is a semiconductor device with fast respond and accuracy. There are two types

Transistor is a semiconductor device with fast respond and accuracy. There are two types Tranitor Amplifir Prpard y: Poa Xuan Yap Thory: Tranitor i a miondutor dvi with fat rpond and auray. Thr ar two typ of tranitor, a Bipolar Juntion Tranitor and a Fild Efft Tranitor. Hr, w will looking

More information

Lecture 20: Emitter Follower and Differential Amplifiers

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.

More information

The example is taken from Sect. 1.2 of Vol. 1 of the CPN book.

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

More information

Factorials! Stirling s formula

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

More information

Who uses our services? We have a growing customer base. with institutions all around the globe.

Who uses our services? We have a growing customer base. with institutions all around the globe. not taking xpr Srvic Guid 2013 / 2014 NTE i an affordabl option for audio to txt convrion. Our rvic includ not or dirct trancription rvic from prviouly rcordd audio fil. Our rvic appal pcially to tudnt

More information

Traffic Flow Analysis (2)

Traffic Flow Analysis (2) Traffic Flow Analysis () Statistical Proprtis. Flow rat distributions. Hadway distributions. Spd distributions by Dr. Gang-Ln Chang, Profssor Dirctor of Traffic safty and Oprations Lab. Univrsity of Maryland,

More information

[ ] These are the motor parameters that are needed: Motor voltage constant. J total (lb-in-sec^2)

[ ] These are the motor parameters that are needed: Motor voltage constant. J total (lb-in-sec^2) MEASURING MOOR PARAMEERS Fil: Motor paramtrs hs ar th motor paramtrs that ar ndd: Motor voltag constant (volts-sc/rad Motor torqu constant (lb-in/amp Motor rsistanc R a (ohms Motor inductanc L a (Hnris

More information

Using Selective Laser Sintering to produce Scaled Structural Models

Using Selective Laser Sintering to produce Scaled Structural Models Procding of th SEM Annual Cfrnc Jun 1-4, 29 Albuqurqu Nw Mxico USA 29 Socity for Exrintal Mchanic Inc. Uing Slctiv Lar Sintring to roduc Scald Structural Modl Cook, W R* and Tolin, R A; Th Univrity of

More information

CPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List.

CPU. Rasterization. Per Vertex Operations & Primitive Assembly. Polynomial Evaluator. Frame Buffer. Per Fragment. Display List. Elmntary Rndring Elmntary rastr algorithms for fast rndring Gomtric Primitivs Lin procssing Polygon procssing Managing OpnGL Stat OpnGL uffrs OpnGL Gomtric Primitivs ll gomtric primitivs ar spcifid by

More information

Vector Network Analyzer

Vector Network Analyzer Cours on Microwav Masurmnts Vctor Ntwork Analyzr Prof. Luca Prrgrini Dpt. of Elctrical, Computr and Biomdical Enginring Univrsity of Pavia -mail: luca.prrgrini@unipv.it wb: microwav.unipv.it Microwav Masurmnts

More information

Financial Mathematics

Financial Mathematics Financial Mathatics A ractical Guid for Actuaris and othr Businss rofssionals B Chris Ruckan, FSA & Jo Francis, FSA, CFA ublishd b B rofssional Education Solutions to practic qustions Chaptr 7 Solution

More information

(Analytic Formula for the European Normal Black Scholes Formula)

(Analytic Formula for the European Normal Black Scholes Formula) (Analytic Formula for th Europan Normal Black Schols Formula) by Kazuhiro Iwasawa Dcmbr 2, 2001 In this short summary papr, a brif summary of Black Schols typ formula for Normal modl will b givn. Usually

More information

A tutorial for laboratory determination of Planck s constant from the Planck radiation law

A tutorial for laboratory determination of Planck s constant from the Planck radiation law A tutorial for laboratory dtrmination of Planck s constant from th Planck radiation law Adam Usman, John Dogari, M. idwan Enuwa and sa Sambo Dartmnt of Physics, Fdral Univrsity of Tchnology, P. M. B. 076,

More information

HOMEWORK FOR UNIT 5-1: FORCE AND MOTION

HOMEWORK FOR UNIT 5-1: FORCE AND MOTION Nam Dat Partnrs HOMEWORK FOR UNIT 51: FORCE AND MOTION 1. You ar givn tn idntial springs. Dsrib how you would dvlop a sal of for (i., a mans of produing rpatabl fors of a varity of sizs) using ths springs.

More information

81-1-ISD Economic Considerations of Heat Transfer on Sheet Metal Duct

81-1-ISD Economic Considerations of Heat Transfer on Sheet Metal Duct Air Handling Systms Enginring & chnical Bulltin 81-1-ISD Economic Considrations of Hat ransfr on Sht Mtal Duct Othr bulltins hav dmonstratd th nd to add insulation to cooling/hating ducts in ordr to achiv

More information

AP Calculus AB 2008 Scoring Guidelines

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.

More information

Cumulative effects of idalopirdine, a 5-HT 6 antagonist in advanced development for the treatment of mild and moderate Alzheimer s disease

Cumulative effects of idalopirdine, a 5-HT 6 antagonist in advanced development for the treatment of mild and moderate Alzheimer s disease Cumulativ ffct of idalopirdin, a 5-HT 6 antagonit in advancd dvlopmnt for th tratmnt of mild and modrat Alzhimr dia Congrè National d unité d oin, d évaluation t d pri n charg Alzhimr (USPLAZ) productio

More information

A negotiation-based Multi-agent System for Supply Chain Management

A negotiation-based Multi-agent System for Supply Chain Management ngotiation-bad Multi-agnt Sytm for Supply Chain Managmnt Y Chn, Yun Png, Tim Finin, Yanni Labrou, Scott Cot Bill Chu, Jian Yao, Rongming Sun, Bob Wilhlm Computr Scinc and Elctronic Enginring Univrity of

More information

Journal of Engineering and Natural Sciences Mühendislik ve Fen Bilimleri Dergisi

Journal of Engineering and Natural Sciences Mühendislik ve Fen Bilimleri Dergisi Journal of Enginring and Natural Scincs Mühndisli v Fn Bilimlri Drgisi Sigma 4/ Invitd Rviw Par OPTIMAL DESIGN OF NONLINEAR MAGNETIC SYSTEMS USING FINITE ELEMENTS Lvnt OVACIK * Istanbul Tchnical Univrsity,

More information

AP Calculus Multiple-Choice Question Collection 1969 1998. connect to college success www.collegeboard.com

AP Calculus Multiple-Choice Question Collection 1969 1998. connect to college success www.collegeboard.com AP Calculus Multipl-Choic Qustion Collction 969 998 connct to collg succss www.collgboard.com Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos

More information

Why An Event App... Before You Start... Try A Few Apps... Event Management Features... Generate Revenue... Vendors & Questions to Ask...

Why An Event App... Before You Start... Try A Few Apps... Event Management Features... Generate Revenue... Vendors & Questions to Ask... Mo b i l E v ntap pgui d : Ho wt op ur c ha t hb te v ntap p f o ry o ura o c i a t i o n T he nt i a l Gui d t oe v ntap p E v nt nt i a l b y Tabl of Contnt Why An Evnt App......... o Whr to Start With

More information

Adverse Selection and Moral Hazard in a Model With 2 States of the World

Adverse Selection and Moral Hazard in a Model With 2 States of the World Advrs Slction and Moral Hazard in a Modl With 2 Stats of th World A modl of a risky situation with two discrt stats of th world has th advantag that it can b natly rprsntd using indiffrnc curv diagrams,

More information

E X C H A N G E R U L E S A N D C L E A R I N G R U L E S O F N A S D A Q O M X D E R I V A T I V E S M A R K E T S

E X C H A N G E R U L E S A N D C L E A R I N G R U L E S O F N A S D A Q O M X D E R I V A T I V E S M A R K E T S E X C H A N G E R U L E S A N D C L E A R I N G R U L E S O F N A S D A Q O M X D E R I V A T I V E S M A R K E T S Fair Valu 1 Valuation Variabls Tabl 1 blow shows th variabls us in th rspctiv valuation

More information

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

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

More information

Long run: Law of one price Purchasing Power Parity. Short run: Market for foreign exchange Factors affecting the market for foreign exchange

Long run: Law of one price Purchasing Power Parity. Short run: Market for foreign exchange Factors affecting the market for foreign exchange Lctur 6: Th Forign xchang Markt xchang Rats in th long run CON 34 Mony and Banking Profssor Yamin Ahmad xchang Rats in th Short Run Intrst Parity Big Concpts Long run: Law of on pric Purchasing Powr Parity

More information

FACULTY SALARIES FALL 2004. NKU CUPA Data Compared To Published National Data

FACULTY SALARIES FALL 2004. NKU CUPA Data Compared To Published National Data FACULTY SALARIES FALL 2004 NKU CUPA Data Compard To Publishd National Data May 2005 Fall 2004 NKU Faculty Salaris Compard To Fall 2004 Publishd CUPA Data In th fall 2004 Northrn Kntucky Univrsity was among

More information

Free ACA SOLUTION (IRS 1094&1095 Reporting)

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

More information

SPECIAL VOWEL SOUNDS

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)

More information

Effect of Design Parameter on the Performance of Lithium Ion Battery

Effect of Design Parameter on the Performance of Lithium Ion Battery Aadil Ahmad, Mohd. Parvz / Intrnational Journal of Enginring Rarch and Application Vol. 3, Iu 4, Jul-Aug 2013, pp.1196-1201 Effct of Dign Paramtr on th Prformanc of Lithium Ion Battry Aadil Ahmad 1, Mohd.

More information

Current and Resistance

Current and Resistance Chaptr 6 Currnt and Rsistanc 6.1 Elctric Currnt...6-6.1.1 Currnt Dnsity...6-6. Ohm s Law...6-4 6.3 Elctrical Enrgy and Powr...6-7 6.4 Summary...6-8 6.5 Solvd Problms...6-9 6.5.1 Rsistivity of a Cabl...6-9

More information

Category 7: Employee Commuting

Category 7: Employee Commuting 7 Catgory 7: Employ Commuting Catgory dscription This catgory includs missions from th transportation of mploys 4 btwn thir homs and thir worksits. Emissions from mploy commuting may aris from: Automobil

More information

Magic Message Maker Amaze your customers with this Gift of Caring communication piece

Magic Message Maker Amaze your customers with this Gift of Caring communication piece Magic Mssag Makr maz your customrs with this Gift of aring communication pic Girls larn th powr and impact of crativ markting with this attntion grabbing communication pic that will hlp thm o a World of

More information

Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means

Sharp bounds for Sándor mean in terms of arithmetic, geometric and harmonic means Qian t al. Journal of Inqualitis and Applications (015) 015:1 DOI 10.1186/s1660-015-0741-1 R E S E A R C H Opn Accss Sharp bounds for Sándor man in trms of arithmtic, gomtric and harmonic mans Wi-Mao Qian

More information

AC Circuits Three-Phase Circuits

AC Circuits Three-Phase Circuits AC Circuits Thr-Phs Circuits Contnts Wht is Thr-Phs Circuit? Blnc Thr-Phs oltgs Blnc Thr-Phs Connction Powr in Blncd Systm Unblncd Thr-Phs Systms Aliction Rsidntil Wiring Sinusoidl voltg sourcs A siml

More information

Constraint-Based Analysis of Gene Deletion in a Metabolic Network

Constraint-Based Analysis of Gene Deletion in a Metabolic Network Constraint-Basd Analysis of Gn Dltion in a Mtabolic Ntwork Abdlhalim Larhlimi and Alxandr Bockmayr DFG-Rsarch Cntr Mathon, FB Mathmatik und Informatik, Fri Univrsität Brlin, Arnimall, 3, 14195 Brlin, Grmany

More information

Circuits with Transistors

Circuits with Transistors ircuits with Transistors ontnts 1 Transistors 1 2 Amplifirs 2 2.1 h paramtrs.................................... 3 3 Bipolar Junction Transistor (BJT) 3 3.1 BJT as a switch...................................

More information

Projections - 3D Viewing. Overview Lecture 4. Projection - 3D viewing. Projections. Projections Parallel Perspective

Projections - 3D Viewing. Overview Lecture 4. Projection - 3D viewing. Projections. Projections Parallel Perspective Ovrviw Lctur 4 Projctions - 3D Viwing Projctions Paralll Prspctiv 3D Viw Volum 3D Viwing Transformation Camra Modl - Assignmnt 2 OFF fils 3D mor compl than 2D On mor dimnsion Displa dvic still 2D Analog

More information

Keywords Cloud Computing, Service level agreement, cloud provider, business level policies, performance objectives.

Keywords Cloud Computing, Service level agreement, cloud provider, business level policies, performance objectives. Volum 3, Issu 6, Jun 2013 ISSN: 2277 128X Intrnational Journal of Advancd Rsarch in Computr Scinc and Softwar Enginring Rsarch Papr Availabl onlin at: wwwijarcsscom Dynamic Ranking and Slction of Cloud

More information

Introduction to Finite Element Modeling

Introduction to Finite Element Modeling Introduction to Finit Elmnt Modling Enginring analysis of mchanical systms hav bn addrssd by driving diffrntial quations rlating th variabls of through basic physical principls such as quilibrium, consrvation

More information

Consulting. Creating value through HR HR Strategy

Consulting. Creating value through HR HR Strategy onulting rating valu through HR HR Stratgy HR Stratgy On of th ky ucc factor of high prforming organiation i putting thir popl firt. oday HR xcutiv ar challngd to dvlop fficint and ffctiv, impl yt powrful

More information

Mandatory Standards and Organizational Information Security

Mandatory Standards and Organizational Information Security Mandatory Standard and Organizational Information Scurity ull Papr ubmittd to WISE 2012. Word count: 9,846 Chul Ho L, Xianjun Gng, Srinivaan Raghunathan Th Univrity of Txa at Dalla {irontigr, gng, raghu}@utdalla.du

More information

BASIC DEFINITIONS AND TERMINOLOGY OF SOILS

BASIC DEFINITIONS AND TERMINOLOGY OF SOILS 1 BASIC DEFINITIONS AND TERMINOLOGY OF SOILS Soil i a thr pha atrial hich coit of olid particl hich ak up th oil klto ad void hich ay b full of atr if th oil i aturatd, ay b full of air if th oil i dry,

More information

Problem Set 6 Solutions

Problem Set 6 Solutions 6.04/18.06J Mathmatics for Computr Scic March 15, 005 Srii Dvadas ad Eric Lhma Problm St 6 Solutios Du: Moday, March 8 at 9 PM Problm 1. Sammy th Shar is a fiacial srvic providr who offrs loas o th followig

More information

Analog IIR Filter Design

Analog IIR Filter Design Filter Deign - IIR cwliu@twin.ee.nctu.edu.tw Analog IIR Filter Deign Commonly ued analog filter : Lowpa Butterworth filter all-pole filter characterized by magnitude repone. Nfilter order Pole of HH- are

More information

Production Costing (Chapter 8 of W&W)

Production Costing (Chapter 8 of W&W) Production Costing (Chaptr 8 of W&W).0 Introduction Production costs rfr to th oprational costs associatd with producing lctric nrgy. Th most significant componnt of production costs ar th ful costs ncssary

More information

Lecture notes: 160B revised 9/28/06 Lecture 1: Exchange Rates and the Foreign Exchange Market FT chapter 13

Lecture notes: 160B revised 9/28/06 Lecture 1: Exchange Rates and the Foreign Exchange Market FT chapter 13 Lctur nots: 160B rvisd 9/28/06 Lctur 1: xchang Rats and th Forign xchang Markt FT chaptr 13 Topics: xchang Rats Forign xchang markt Asst approach to xchang rats Intrst Rat Parity Conditions 1) Dfinitions

More information

Noise Power Ratio (NPR) A 65-Year Old Telephone System Specification Finds New Life in Modern Wireless Applications.

Noise Power Ratio (NPR) A 65-Year Old Telephone System Specification Finds New Life in Modern Wireless Applications. TUTORIL ois Powr Ratio (PR) 65-Yar Old Tlphon Systm Spcification Finds w Lif in Modrn Wirlss pplications ITRODUTIO by Walt Kstr Th concpt of ois Powr Ratio (PR) has bn around sinc th arly days of frquncy

More information

A Formal Model for Data Flow Diagram Rules

A Formal Model for Data Flow Diagram Rules Volum No. MAY 0 ARPN Journal o Sytm and Sotwar 00- AJSS Journal. All rght rrvd htt://www.cntc-ournal.org A Formal Modl or Data Flow Dagram Rul Rozat Ibrahm Sow Yn Yn Dartmnt o Sotwar Engnrng Unvrty Tun

More information

LAB 3: VELOCITY AND ACCELERATION GRAPHS

LAB 3: VELOCITY AND ACCELERATION GRAPHS Goas: LAB 3: ELOCITY AND ACCELERATION GRAPHS Invstigat accration vs. tim graphs Prdict accration graphs from vocity graphs Invstigat accration as sop of vocity vs. tim graph Part 1 - Making ocity- Graphs

More information

Gold versus stock investment: An econometric analysis

Gold versus stock investment: An econometric analysis Intrnational Journal of Dvlopmnt and Sustainability Onlin ISSN: 268-8662 www.isdsnt.com/ijds Volum Numbr, Jun 202, Pag -7 ISDS Articl ID: IJDS20300 Gold vrsus stock invstmnt: An conomtric analysis Martin

More information

Higher. Exponentials and Logarithms 160

Higher. Exponentials and Logarithms 160 hsn uknt Highr Mthmtics UNIT UTCME Eponntils nd Logrithms Contnts Eponntils nd Logrithms 6 Eponntils 6 Logrithms 6 Lws of Logrithms 6 Eponntils nd Logrithms to th Bs 65 5 Eponntil nd Logrithmic Equtions

More information

Incomplete 2-Port Vector Network Analyzer Calibration Methods

Incomplete 2-Port Vector Network Analyzer Calibration Methods Incomplt -Port Vctor Ntwork nalyzr Calibration Mthods. Hnz, N. Tmpon, G. Monastrios, H. ilva 4 RF Mtrology Laboratory Instituto Nacional d Tcnología Industrial (INTI) Bunos irs, rgntina ahnz@inti.gov.ar

More information

CAFA DIVERSITY JURISDICTION

CAFA DIVERSITY JURISDICTION Cla Action 101: CAFA Divrity Juridiction at a Glanc By Kathryn Honckr Jun 20, 2013 In thi dition of Cla Action 101, w giv a viual guid to th Cla Action Fairn Act (CAFA), 28 U.S.C. 1332(d)(2), to hlp you

More information

stop super ripoffs By using this bargaining kit to protect your members superannuation in their EA up for WitH Your union

stop super ripoffs By using this bargaining kit to protect your members superannuation in their EA up for WitH Your union top upr ripoff By uing thi bargaining kit to protct your mmbr uprannuation in thir EA Union hav a lot to b proud of whn it com to uprannuation. Not only did w invnt it, but jut thi yar w hlpd win a tagd

More information

I. INTRODUCTION. Figure 1, The Input Display II. DESIGN PROCEDURE

I. INTRODUCTION. Figure 1, The Input Display II. DESIGN PROCEDURE Ballast Dsign Softwar Ptr Grn, Snior ighting Systms Enginr, Intrnational Rctifir, ighting Group, 101S Spulvda Boulvard, El Sgundo, CA, 9045-438 as prsntd at PCIM Europ 0 Abstract: W hav dvlopd a Windows

More information

THE STUDY OF BARRIERS TO ENTREPRENEURSHIP IN MEN AND WOMEN

THE STUDY OF BARRIERS TO ENTREPRENEURSHIP IN MEN AND WOMEN THE STUDY OF BARRIERS TO ENTREPRENEURSHIP IN MEN AND WOMEN Mohammad Baghr Gorji: Corrponding Author Aitant Profor, Dpartmnt of Managmnt, Aliabad Katoul Branch, Ilamic Azad Univrity, Aliabad Katoul, Iran

More information

Design of IIR Digital Filters. Ch. 5 - Lecture 18 DSP ECE 623 IIR DIGITAL FILTERS. Andreas Spanias. Chapter 5 IIR Filter Design. spanias@asu.

Design of IIR Digital Filters. Ch. 5 - Lecture 18 DSP ECE 623 IIR DIGITAL FILTERS. Andreas Spanias. Chapter 5 IIR Filter Design. spanias@asu. Title/Number: DSP ECE 623 By Andreas Sanias Chater 5 IIR Filter Design sanias@asu.edu Phone: 48 965 837, Fax: 48 965 8325 htt://www.eas.asu.edu/~sanias Coyright Andreas Sanias Ch. 5 - Design of IIR Digital

More information

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

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

More information

Ohm s Law. Ohmic relationship V=IR. Electric Power. Non Ohmic devises. Schematic representation. Electric Power

Ohm s Law. Ohmic relationship V=IR. Electric Power. Non Ohmic devises. Schematic representation. Electric Power Ohm Law Ohmic relationhip V=IR Ohm law tate that current through the conductor i directly proportional to the voltage acro it if temperature and other phyical condition do not change. In many material,

More information

SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM

SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM RESEARCH PAPERS IN MANAGEMENT STUDIES SPREAD OPTION VALUATION AND THE FAST FOURIER TRANSFORM M.A.H. Dmpstr & S.S.G. Hong WP 26/2000 Th Judg Institut of Managmnt Trumpington Strt Cambridg CB2 1AG Ths paprs

More information

Double Integrals in Polar Coordinates

Double Integrals in Polar Coordinates Double Integrals in Polar Coorinates Part : The Area Di erential in Polar Coorinates We can also aly the change of variable formula to the olar coorinate transformation x = r cos () ; y = r sin () However,

More information

Physics 106 Lecture 12. Oscillations II. Recap: SHM using phasors (uniform circular motion) music structural and mechanical engineering waves

Physics 106 Lecture 12. Oscillations II. Recap: SHM using phasors (uniform circular motion) music structural and mechanical engineering waves Physics 6 Lctur Oscillations II SJ 7 th Ed.: Chap 5.4, Rad only 5.6 & 5.7 Rcap: SHM using phasors (unifor circular otion) Physical pndulu xapl apd haronic oscillations Forcd oscillations and rsonanc. Rsonanc

More information

June 2012. Enprise Rent. Enprise 1.1.6. Author: Document Version: Product: Product Version: SAP Version: 8.81.100 8.8

June 2012. Enprise Rent. Enprise 1.1.6. Author: Document Version: Product: Product Version: SAP Version: 8.81.100 8.8 Jun 22 Enpris Rnt Author: Documnt Vrsion: Product: Product Vrsion: SAP Vrsion: Enpris Enpris Rnt 88 88 Enpris Rnt 22 Enpris Solutions All rights rsrvd No parts of this work may b rproducd in any form or

More information

Van der Waals Forces Between Atoms

Van der Waals Forces Between Atoms Van dr Waals Forcs twn tos Michal Fowlr /8/7 Introduction Th prfct gas quation of stat PV = NkT is anifstly incapabl of dscribing actual gass at low tpraturs, sinc thy undrgo a discontinuous chang of volu

More information

Hardware Modules of the RSA Algorithm

Hardware Modules of the RSA Algorithm SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol. 11, No. 1, Fbruary 2014, 121-131 UDC: 004.3`142:621.394.14 DOI: 10.2298/SJEE140114011S Hardwar Moduls of th RSA Algorithm Vlibor Škobić 1, Branko Dokić 1,

More information

Entity-Relationship Model

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

More information

A Theoretical Model of Public Response to the Homeland Security Advisory System

A Theoretical Model of Public Response to the Homeland Security Advisory System A Thortical Modl of Public Rspons to th Homland Scurity Advisory Systm Amy (Wnxuan) Ding Dpartmnt of Information and Dcision Scincs Univrsity of Illinois Chicago, IL 60607 wxding@uicdu Using a diffrntial

More information

C H A P T E R 1 Writing Reports with SAS

C H A P T E R 1 Writing Reports with SAS C H A P T E R 1 Writing Rports with SAS Prsnting information in a way that s undrstood by th audinc is fundamntally important to anyon s job. Onc you collct your data and undrstand its structur, you nd

More information

Coordinate Transformation

Coordinate Transformation Coordinate Transformation Coordinate Transformations In this chater, we exlore maings where a maing is a function that "mas" one set to another, usually in a way that reserves at least some of the underlyign

More information

The Cubic Formula. The quadratic formula tells us the roots of a quadratic polynomial, a polynomial of the form ax 2 + bx + c. The roots (if b 2 b+

The Cubic Formula. The quadratic formula tells us the roots of a quadratic polynomial, a polynomial of the form ax 2 + bx + c. The roots (if b 2 b+ The Cubic Formula The quadratic formula tells us the roots of a quadratic olynomial, a olynomial of the form ax + bx + c. The roots (if b b+ 4ac 0) are b 4ac a and b b 4ac a. The cubic formula tells us

More information

Analyzing the Economic Efficiency of ebaylike Online Reputation Reporting Mechanisms

Analyzing the Economic Efficiency of ebaylike Online Reputation Reporting Mechanisms A rsarch and ducation initiativ at th MIT Sloan School of Managmnt Analyzing th Economic Efficincy of Baylik Onlin Rputation Rporting Mchanisms Papr Chrysanthos Dllarocas July For mor information, plas

More information

Job shop scheduling with unit processing times

Job shop scheduling with unit processing times Job shop schduling with unit procssing tims Nikhil Bansal Tracy Kimbrl Maxim Sviridnko Abstract W considr randomizd algorithms for th prmptiv job shop problm, or quivalntly, th cas in which all oprations

More information

An Broad outline of Redundant Array of Inexpensive Disks Shaifali Shrivastava 1 Department of Computer Science and Engineering AITR, Indore

An Broad outline of Redundant Array of Inexpensive Disks Shaifali Shrivastava 1 Department of Computer Science and Engineering AITR, Indore Intrnational Journal of mrging Tchnology and dvancd nginring Wbsit: www.ijta.com (ISSN 2250-2459, Volum 2, Issu 4, pril 2012) n road outlin of Rdundant rray of Inxpnsiv isks Shaifali Shrivastava 1 partmnt

More information

Fundamentals: NATURE OF HEAT, TEMPERATURE, AND ENERGY

Fundamentals: NATURE OF HEAT, TEMPERATURE, AND ENERGY Fundamntals: NATURE OF HEAT, TEMPERATURE, AND ENERGY DEFINITIONS: Quantum Mchanics study of individual intractions within atoms and molculs of particl associatd with occupid quantum stat of a singl particl

More information

Outside Cut 1 of fabric Cut 1 of interfacing

Outside Cut 1 of fabric Cut 1 of interfacing a a Outsi Cut o abric Cut o intracing a a b b Outsi Cut o abric Cut o intracing Placmnt lin or Mony Pockts Dix Not: F. Cut Fol b. Pin t /8 in 5. Nx bottom pics sw th 6. For t Prss, 7. Lay togth on th 8.

More information

Vibrational Spectroscopy

Vibrational Spectroscopy Vibrational Spctroscopy armonic scillator Potntial Enrgy Slction Ruls V( ) = k = R R whr R quilibrium bond lngth Th dipol momnt of a molcul can b pandd as a function of = R R. µ ( ) =µ ( ) + + + + 6 3

More information

WORKERS' COMPENSATION ANALYST, 1774 SENIOR WORKERS' COMPENSATION ANALYST, 1769

WORKERS' COMPENSATION ANALYST, 1774 SENIOR WORKERS' COMPENSATION ANALYST, 1769 08-16-85 WORKERS' COMPENSATION ANALYST, 1774 SENIOR WORKERS' COMPENSATION ANALYST, 1769 Summary of Dutis : Dtrmins City accptanc of workrs' compnsation cass for injurd mploys; authorizs appropriat tratmnt

More information