Narrow-band Receiver Radio Architectures

Size: px
Start display at page:

Download "Narrow-band Receiver Radio Architectures"

Transcription

1 Now-bnd Receive Rdio chiecues

2 l l l l l Double-convesion single-qud Supeheodyne Diec-convesion Single-convesion singlequd, hoodyne, Zeo- Double-convesion double-qud Low- Refeences

3 Down-convesion; The ige poble: Desied signl LN ~ Blocke ige 3 Blocke level cn be odes of gniude lge hn desied signl f no blocke pesen, you sill fold he ige noise ~3dB SNR hi Feq Feq cos cos cos cos fe he lowpss file following ixe, he oupu is: oupu cos cos cos cos cos

4 The Supeheodyne chiecue: 4 OFF chip ge file file Pssive BPF LN The choice of he fequency: Relively low equies vey high-q ige file o povide decen ejecion he ige fequency, which nsles ino highe file pssbnd loss nd so highe Rx NF, nd highe cos. Relively high esuls in oe cuen consupion fo he subsequen sge cicuis, especilly he VG. ypicl fequency is 00-00MHz. One of he os obus nd low-cuen eceive chiecues equies sevel bulky off-chip coponens ~ ge file eoves boh blocke nd noise fo ige befoe downconvesion

5 plnning in Supehe chiecue: 5 OFF chip ge file file VG PLL ~ VCO Pssive BPF LN Q PLL ~ VCO Two sepe VCO s nd synhesizes e usully needed. The is usully fixed, while he is vible o down conve he desied signl o fll wihin he SW file bndwidh. The soluion is 3-4 chip soluion wih hee exenl SW files. should neve be de close o be n inege uliple of fo ny chnnel. The N h honic of he fixed could lek ino ixe cusing unwned ineod. blocke f n leks o ixe n

6 The ½ poble in supehe chiecue: blocke Supehe Rx 6 0 ½ Le us ssue hee is n undesied blocke hlf wy beween he desied signl nd he, s shown. Thee e wo echnis his blocke cn fold ono he desied bnd. The fis if he fon-end hs song second ode non-lineiy. ½ cos 0.5 cos blocke blocke blocke f he hs lso lge second ode coponen, he esul would be coponen of he blocke folding ono he bnd, igh on op of he desied bnd cos cos cos... blocke blocke Noe h he blocke folds wih wice is signl bndwidh due o he sque.

7 7 0 0 ½ ½ The second ens by which he blocke cn fold ono he desied bnd if he sge hs song second ode nonlineiy. cos0.5 cos bloc ke blocke blocke Noe gin hee h he blocke folds wih wice is signl bndwidh due o he sque. is heefoe n exeely ipon decision o chose he fequency so h hee is no song hlf blocke nywhee in he syse, o if hee is ny, he desied second ode nonlineiy of he eceive is cefully clculed.

8 Supehe; Dul convesion single-qud: 8 OFF chip ge file file VG ~ Pssive BPF LN Q Disdvnges: ~ equies sevel bulky off-chip SW files. genelly wo diffeen synhesizes e equied hee-chip soluion:, nd synhesize dvnges: obus. high-dynic nge SW files elxes h of he cive cicuis, dsiclly educing hei equied dynic nge.. Esie o design low cuen consupion <0 fo WCDM

9 9

10 0 ge suppession by pope phsing Hilbe chiecue: Pssive BPF LN 90 ~ 90 Σ B C C C C B cos cos cos cos cos cos sin sin sin sin sin cos cos cos cos cos cos cos ' file

11 Hilbe chiecue: Disdvnges: equies exeely good phse nd gin ching. Only ~35dB of ige ejecion is possible wihou iing o uning. This is no enough in soe pplicions whee he ige is 60dB highe hn he desied signl. consues oe cuen.. bi powe hungy dvnges: eliines he exenl ige SW file. offes bee inegion, bu sill second chip is needed fo finl down convesion 45 Pssive BPF LN ~ Σ D file odified bee cking chiecue B 35 C

12 pc of gin nd phse iblnce on ige ejecion Le us ssue h signls nd Q expeience non-idenicl gin. Fuheoe, he phse is no excly qudue nd hs soe phse eo. To find he ipc of boh gin nd phse iblnce on ige ejecion, le us e-wie he equions s follows:

13 ge ejecion equion The ige ejecion in db cn hen be wien s: is in d Fo sll : 3 Fo sll nd, he RR cn be ppoxied o: degees 0log db

14 Double convesion double qud chiecue: C 4 ige Pssive BPF Pssive BPF LN B + ige + signl Fis D F E + Σ - + Σ + Q ~ N second The dul convesion double qud chiecue hs he dvnge of de-sensiizing he eceive o gin nd phse iblnce of nd Q phs. Mke sue nd vlues e chosen so ige fo fis down-convesion flls ou of bnd ejeced by exenl SW pssive BP file.

15 5 E D Q F C B F B E D C B sin cos cos cos sin Lowpss sin sin cos Lowpss sin sin sin Lowpss cos cos cos Lowpss sin sin sin Lowpss cos cos cos Lowpss cos cos The bove nlysis ssues idel qudue geneion. Bu wh if hee e soe phse eos?

16 6 E D Q F C B F B E D C B sin cos cos cos sin Lowpss sin sin cos Lowpss sin sin sin Lowpss cos cos cos Lowpss sin sin sin Lowpss cos cos cos Lowpss cos cos gin eo nlysis: s seen he gin isch eo ges educed due o he poduc of wo sll vlues. ypicl gin eo of such chiecue cn be s good s 0.dB -sig.5ghz.

17 Phse eo nlysis: cos Lowpss cos B Lowpss sin C Lowpsscos D Lowpsssin E LowpssB cos F LowpssB sin cos cos cos sin sin cos cos sin sin sin sin cos cos Cn you deive he finl esul? Hoewok!!! 7

18 The diec convesion Zeo- chiecue: Why no eliine he copleely nd ix he signl wih n he exc se fequency s he desied signl? 8 in Pssive BPF LN 90 ~ Q in LPF fo Q LPF cos cos 0.5 cos in 0 sin 0.5 sin in No ige nyoe wih pefec /Q ching! Wh if hee is /Q isch in diec-convesion? Cn hee sill be n ige?

19 pc of /Q isch in diec-convesion chiecue: in LPF fo cos Q LPF in cos, nd finie /Q gin nd phse isch of nd, especively in sin coscos sinsin sincos cossin /Q coss lk ge fo se desied signl coposed of Q signl leking ino nd signl leking ino Q /Q coss lk. This esuls in degdion of SNR. Since he ige coes fo he signl iself, he ige ejecion equieen is llevied se by ge SNR of he eceive 9 -f f 0 fe ixe feq Negive feq ige lis

20 Pobles wih Zeo- convesion: 0. self ixing: Pssive BPF LN 90 ~ Vies wih nenn VSWR B Q p cos p p cos Dynic DC offse self ixing degdes DC-offse in diec-convesion eceive. Sic one cn be clibed ou bu dynic one ie vying cn be poble

21 . DC offse: 60dB gin Pssive BPF LN 90 ~ Q V DC offse V DC offse Since he signl is conveed down o DC, he signl ph pos ixe is DC coupled. This kes sligh DC offse o ppe s vols nd os likely o sue he subsequen bsebnd cicuiy. C coupling is expensive since lge C coupling cps should be used o chieve low highpss cone <khz. lso ny offse nsiens due o gin swiching will esul in lge DC nsien h will ke long ie few s fo khz cone o sele. Two coon echniques fo DC offse eovl: nlog sevo loops B digil DCoffse clibions

22 3. Sensiiviy o second ode disoion P: Le us ssue wo jes seped by Δf s s s s cos cos cos cos cos cos cos cn be seen hn he second ode disoion in he eceive esuls in wo low fequency disoion h folds ino he desied signl fe being downconveed ound DC:. The bsebnd odulion ound he je cie ges folded ino bsebnd wih wice bndwidh. The wo jes be gins ech ohe nd poduce coponen only Δf wy fo DC. f Δf is close enough, he M disoion will fll ino he desied signl bnd. Even if Δf does no fll wihin he desied signl bnd, i cn be close enough o clip he enie eceive if i does no ge fileed ou popely befoe i eches he VG. Noe h s cn be siple CW je in soe syses

23 4. Sensiiviy o /f flicke noise noise densiy 00ni /f, slope 0dB/dec 3 Since he down conveed signl is ceneed ound DC low fequency, he device /f noise becoes ipon. Fo exple, if he /f noise cone of GSM eceive is hlf he desied signl bndwidh s shown bove, he degdion his could cuse o he effecive noise is ssuing noise inegion sing khz: n ve n ve 00k 00kHz khz df f 00kHz 00kHz 00k ln b 00k k ni bdf ; khz whee k 00n 00k 00k 00k.5 00kb 6.5n 00kHz i ; b n This is vey chllenging sk in CMOS design given he elively lge /f noise cone coped o bipol. Theefoe, device sizing is necessy o bing he device /f noise cone o n ccepble level <0.5dB noise ipc. i i 00kHz feq

24 Diec-convesion eceive efeence design 4 dvnges: high inegion ge plfo fo uli- bnd eceive low powe no s good s supehe! Vey popul chiecue fo oden syses such s WiFi, 3G WCDM/UMTS, 4G LTE nd 5G

25 ge ejec files Low- chiecue: Blocke ige Pssive BPF LN 90 ~ Q 5 Why no void signl ound DC by choosing n fequency h is low enough so files nd cicuiy cn be ineged on chip, ye high enough o void he pobles ound DC. ypicl low fequency is one o wice he signl bndwidh. he single qud chiecue bove sill suffes sensiiviy of ige ejecion o phse nd gin isch. ypicl 35dB ige ejecion is no enough fo os syses o be ble o hndle lge ige signl wih finie dynic nge ineged cicuiy. Need RR clibions he ige ejec files pefo boh chnnel selecion nd ige ejecion he se ie. The design of such files will be discussed in subsequen lecues. Popul chiecue fo syses wih now-bnd signls h hs lge enegy conen close o DC such s Blueooh, GPS, nd GSM

26 The double qud low- chiecue: 6 Pssive BPF LN B D + Σ - + Σ + Q Cn be finie o zeo second C ~ poved ige ejecion due o desensiizion o qudue gin nd phse eo.

27 7 D C B Q q q q sin cos cos cos cos cos sin sin sin sin sin cos sin sin sin sin sin cos cos cos cos cos sin sin cos cos Signl ph nlysis: Se nlysis s befoe!

28 Refeences: [].. bidi, Rdio Fequency neged Cicuis fo Poble Counicions, Poc. CCC, Sn Diego, pp 5-58, My 994. [].. bidi, Diec Convesion Rdio Tnsceives fo Digil Counicions, Poc. SSCC, pp 86-87, Feb 995. [3] J. Clos, M. Seye Single-Chip 900MHz CMOS Receive Fon-end wih High Pefonce Low- Topology, EEE JSSC, Vol. 30, no., pp , Dec [4] B. Rzvi, Micoeleconics, Penice Hll P [5] J. Clos, M. Seye, CMOS Wieless Tnsceive Design, Kluwe cdeic Publishes [6] Pcicl Syse Design, Willi F. Egn, SBN: , Hdcove 46 pges, pil 003, Wiley-EEE Pess. 8

() t. () t r () t or v. ( t) () () ( ) = ( ) or ( ) () () () t or dv () () Section 10.4 Motion in Space: Velocity and Acceleration

() t. () t r () t or v. ( t) () () ( ) = ( ) or ( ) () () () t or dv () () Section 10.4 Motion in Space: Velocity and Acceleration Secion 1.4 Moion in Spce: Velociy nd Acceleion We e going o dive lile deepe ino somehing we ve ledy inoduced, nmely () nd (). Discuss wih you neighbo he elionships beween posiion, velociy nd cceleion you

More information

Circuits 24/08/2010. Question. Question. Practice Questions QV CV. Review Formula s RC R R R V IR ... Charging P IV I R ... E Pt.

Circuits 24/08/2010. Question. Question. Practice Questions QV CV. Review Formula s RC R R R V IR ... Charging P IV I R ... E Pt. 4/08/00 eview Fomul s icuis cice s BL B A B I I I I E...... s n n hging Q Q 0 e... n... Q Q n 0 e Q I I0e Dischging Q U Q A wie mde of bss nd nohe wie mde of silve hve he sme lengh, bu he dimee of he bss

More information

USING LOWER CLASS WEIGHTS TO CORRECT AND CHECK THE NONLINEARITY OF BALANCES

USING LOWER CLASS WEIGHTS TO CORRECT AND CHECK THE NONLINEARITY OF BALANCES USING OWER CSS WEIGHTS TO CORRECT ND CHECK THE NONINERITY OF BNCES Tiohy Chnglin Wng, Qiho Yun, hu Reichuh Mele-Toledo Insuens Shnghi Co. d, Shnghi, P R Chin Mele-Toledo GH, Geifensee, Swizelnd BSTRCT

More information

f(x) dx with An integral having either an infinite limit of integration or an unbounded integrand is called improper. Here are two examples dx x x 2

f(x) dx with An integral having either an infinite limit of integration or an unbounded integrand is called improper. Here are two examples dx x x 2 Impope Inegls To his poin we hve only consideed inegls f() wih he is of inegion nd b finie nd he inegnd f() bounded (nd in fc coninuous ecep possibly fo finiely mny jump disconinuiies) An inegl hving eihe

More information

PHYSICS 102. Intro PHYSICS-ELECTROMAGNETISM

PHYSICS 102. Intro PHYSICS-ELECTROMAGNETISM PHYS 0 Suen Nme: Suen Numbe: FAUTY OF SIENE Viul Miem EXAMINATION PHYSIS 0 Ino PHYSIS-EETROMAGNETISM Emines: D. Yoichi Miyh INSTRUTIONS: Aemp ll 4 quesions. All quesions hve equl weighs 0 poins ech. Answes

More information

Reinforcement learning

Reinforcement learning CS 75 Mchine Lening Lecue b einfocemen lening Milos Huskech milos@cs.pi.edu 539 Senno Sque einfocemen lening We wn o len conol policy: : X A We see emples of bu oupus e no given Insed of we ge feedbck

More information

Technical Vibration - text 2 - forced vibration, rotational vibration

Technical Vibration - text 2 - forced vibration, rotational vibration Technicl Viion - e - foced viion, oionl viion 4. oced viion, viion unde he consn eenl foce The viion unde he eenl foce. eenl The quesion is if he eenl foce e is consn o vying. If vying, wh is he foce funcion.

More information

S Radio transmission and network access Exercise 1-2

S Radio transmission and network access Exercise 1-2 S-7.330 Rdio rnsmission nd nework ccess Exercise 1 - P1 In four-symbol digil sysem wih eqully probble symbols he pulses in he figure re used in rnsmission over AWGN-chnnel. s () s () s () s () 1 3 4 )

More information

P441 Analytical Mechanics - I. Coupled Oscillators. c Alex R. Dzierba

P441 Analytical Mechanics - I. Coupled Oscillators. c Alex R. Dzierba Lecure 3 Mondy - Deceber 5, 005 Wrien or ls upded: Deceber 3, 005 P44 Anlyicl Mechnics - I oupled Oscillors c Alex R. Dzierb oupled oscillors - rix echnique In Figure we show n exple of wo coupled oscillors,

More information

Ans: In the rectangular loop with the assigned direction for i2: di L dt , (1) where (2) a) At t = 0, i1(t) = I1U(t) is applied and (1) becomes

Ans: In the rectangular loop with the assigned direction for i2: di L dt , (1) where (2) a) At t = 0, i1(t) = I1U(t) is applied and (1) becomes omewok # P7-3 ecngul loop of widh w nd heigh h is siued ne ve long wie cing cuen i s in Fig 7- ssume i o e ecngul pulse s shown in Fig 7- Find he induced cuen i in he ecngul loop whose self-inducnce is

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

Circular Motion. Radians. One revolution is equivalent to which is also equivalent to 2π radians. Therefore we can.

Circular Motion. Radians. One revolution is equivalent to which is also equivalent to 2π radians. Therefore we can. 1 Cicula Moion Radians One evoluion is equivalen o 360 0 which is also equivalen o 2π adians. Theefoe we can say ha 360 = 2π adians, 180 = π adians, 90 = π 2 adians. Hence 1 adian = 360 2π Convesions Rule

More information

Physics 201, Lecture 5

Physics 201, Lecture 5 Phsics 1 Lecue 5 Tod s Topics n Moion in D (Chp 4.1-4.3): n D Kinemicl Quniies (sec. 4.1) n D Kinemics wih Consn Acceleion (sec. 4.) n D Pojecile (Sec 4.3) n Epeced fom Peiew: n Displcemen eloci cceleion

More information

4.8 Improper Integrals

4.8 Improper Integrals 4.8 Improper Inegrls Well you ve mde i hrough ll he inegrion echniques. Congrs! Unforunely for us, we sill need o cover one more inegrl. They re clled Improper Inegrls. A his poin, we ve only del wih inegrls

More information

COMP 465: Data Mining More on PageRank

COMP 465: Data Mining More on PageRank COMP 465: Dt Mnng Moe on PgeRnk Sldes Adpted Fo: www.ds.og (Mnng Mssve Dtsets) Powe Iteton: Set = 1/ 1: = 2: = Goto 1 Exple: d 1/3 1/3 5/12 9/24 6/15 = 1/3 3/6 1/3 11/24 6/15 1/3 1/6 3/12 1/6 3/15 Iteton

More information

e t dt e t dt = lim e t dt T (1 e T ) = 1

e t dt e t dt = lim e t dt T (1 e T ) = 1 Improper Inegrls There re wo ypes of improper inegrls - hose wih infinie limis of inegrion, nd hose wih inegrnds h pproch some poin wihin he limis of inegrion. Firs we will consider inegrls wih infinie

More information

, on the power of the transmitter P t fed to it, and on the distance R between the antenna and the observation point as. r r t

, on the power of the transmitter P t fed to it, and on the distance R between the antenna and the observation point as. r r t Lecue 6: Fiis Tansmission Equaion and Rada Range Equaion (Fiis equaion. Maximum ange of a wieless link. Rada coss secion. Rada equaion. Maximum ange of a ada. 1. Fiis ansmission equaion Fiis ansmission

More information

Some algorthim for solving system of linear volterra integral equation of second kind by using MATLAB 7 ALAN JALAL ABD ALKADER

Some algorthim for solving system of linear volterra integral equation of second kind by using MATLAB 7 ALAN JALAL ABD ALKADER . Soe lgoi o solving syse o line vole inegl eqion o second ind by sing MATLAB 7 ALAN JALAL ABD ALKADER College o Edcion / Al- Msnsiiy Univesiy Depen o Meics تقديم البحث :-//7 قبول النشر:- //. Absc ( /

More information

Week 8. Topic 2 Properties of Logarithms

Week 8. Topic 2 Properties of Logarithms Week 8 Topic 2 Popeties of Logithms 1 Week 8 Topic 2 Popeties of Logithms Intoduction Since the esult of ithm is n eponent, we hve mny popeties of ithms tht e elted to the popeties of eponents. They e

More information

r r r r r EE334 Electromagnetic Theory I Todd Kaiser

r r r r r EE334 Electromagnetic Theory I Todd Kaiser 334 lecoagneic Theoy I Todd Kaise Maxwell s quaions: Maxwell s equaions wee developed on expeienal evidence and have been found o goven all classical elecoagneic phenoena. They can be wien in diffeenial

More information

Forms of Energy. Mass = Energy. Page 1. SPH4U: Introduction to Work. Work & Energy. Particle Physics:

Forms of Energy. Mass = Energy. Page 1. SPH4U: Introduction to Work. Work & Energy. Particle Physics: SPH4U: Inroducion o ork ork & Energy ork & Energy Discussion Definiion Do Produc ork of consn force ork/kineic energy heore ork of uliple consn forces Coens One of he os iporn conceps in physics Alernive

More information

Faraday s Law. To be able to find. motional emf transformer and motional emf. Motional emf

Faraday s Law. To be able to find. motional emf transformer and motional emf. Motional emf Objecie F s w Tnsfome Moionl To be ble o fin nsfome. moionl nsfome n moionl. 331 1 331 Mwell s quion: ic Fiel D: Guss lw :KV : Guss lw H: Ampee s w Poin Fom Inegl Fom D D Q sufce loop H sufce H I enclose

More information

Homework 5 for BST 631: Statistical Theory I Solutions, 09/21/2006

Homework 5 for BST 631: Statistical Theory I Solutions, 09/21/2006 Homewok 5 fo BST 63: Sisicl Theoy I Soluions, 9//6 Due Time: 5:PM Thusy, on 9/8/6. Polem ( oins). Book olem.8. Soluion: E = x f ( x) = ( x) f ( x) + ( x ) f ( x) = xf ( x) + xf ( x) + f ( x) f ( x) Accoing

More information

Science Advertisement Intergovernmental Panel on Climate Change: The Physical Science Basis 2/3/2007 Physics 253

Science Advertisement Intergovernmental Panel on Climate Change: The Physical Science Basis   2/3/2007 Physics 253 Science Adeisemen Inegoenmenl Pnel on Clime Chnge: The Phsicl Science Bsis hp://www.ipcc.ch/spmfeb7.pdf /3/7 Phsics 53 hp://www.fonews.com/pojecs/pdf/spmfeb7.pdf /3/7 Phsics 53 3 Sus: Uni, Chpe 3 Vecos

More information

Lecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation

Lecture 18: Kinetics of Phase Growth in a Two-component System: general kinetics analysis based on the dilute-solution approximation Lecue 8: Kineics of Phase Gowh in a Two-componen Sysem: geneal kineics analysis based on he dilue-soluion appoximaion Today s opics: In he las Lecues, we leaned hee diffeen ways o descibe he diffusion

More information

Fig. 1S. The antenna construction: (a) main geometrical parameters, (b) the wire support pillar and (c) the console link between wire and coaxial

Fig. 1S. The antenna construction: (a) main geometrical parameters, (b) the wire support pillar and (c) the console link between wire and coaxial a b c Fig. S. The anenna consucion: (a) ain geoeical paaees, (b) he wie suppo pilla and (c) he console link beween wie and coaial pobe. Fig. S. The anenna coss-secion in he y-z plane. Accoding o [], he

More information

Low-complexity Algorithms for MIMO Multiplexing Systems

Low-complexity Algorithms for MIMO Multiplexing Systems Low-complexiy Algoihms fo MIMO Muliplexing Sysems Ouline Inoducion QRD-M M algoihm Algoihm I: : o educe he numbe of suviving pahs. Algoihm II: : o educe he numbe of candidaes fo each ansmied signal. :

More information

Computer Propagation Analysis Tools

Computer Propagation Analysis Tools Compue Popagaion Analysis Tools. Compue Popagaion Analysis Tools Inoducion By now you ae pobably geing he idea ha pedicing eceived signal sengh is a eally impoan as in he design of a wieless communicaion

More information

D zone schemes

D zone schemes Ch. 5. Enegy Bnds in Cysls 5.. -D zone schemes Fee elecons E k m h Fee elecons in cysl sinα P + cosα cosk α cos α cos k cos( k + π n α k + πn mv ob P 0 h cos α cos k n α k + π m h k E Enegy is peiodic

More information

ENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 9 Solutions [Theorems of Gauss and Stokes]

ENGI 4430 Advanced Calculus for Engineering Faculty of Engineering and Applied Science Problem Set 9 Solutions [Theorems of Gauss and Stokes] ENGI 44 Avance alculus fo Engineeing Faculy of Engineeing an Applie cience Poblem e 9 oluions [Theoems of Gauss an okes]. A fla aea A is boune by he iangle whose veices ae he poins P(,, ), Q(,, ) an R(,,

More information

ME 141. Engineering Mechanics

ME 141. Engineering Mechanics ME 141 Engineeing Mechnics Lecue 13: Kinemics of igid bodies hmd Shhedi Shkil Lecue, ep. of Mechnicl Engg, UET E-mil: sshkil@me.bue.c.bd, shkil6791@gmil.com Websie: eche.bue.c.bd/sshkil Couesy: Veco Mechnics

More information

Lecture 22 Electromagnetic Waves

Lecture 22 Electromagnetic Waves Lecue Elecomagneic Waves Pogam: 1. Enegy caied by he wave (Poyning veco).. Maxwell s equaions and Bounday condiions a inefaces. 3. Maeials boundaies: eflecion and efacion. Snell s Law. Quesions you should

More information

Contraction Mapping Principle Approach to Differential Equations

Contraction Mapping Principle Approach to Differential Equations epl Journl of Science echnology 0 (009) 49-53 Conrcion pping Principle pproch o Differenil Equions Bishnu P. Dhungn Deprmen of hemics, hendr Rn Cmpus ribhuvn Universiy, Khmu epl bsrc Using n eension of

More information

An Open cycle and Closed cycle Gas Turbine Engines. Methods to improve the performance of simple gas turbine plants

An Open cycle and Closed cycle Gas Turbine Engines. Methods to improve the performance of simple gas turbine plants An Open cycle and losed cycle Gas ubine Engines Mehods o impove he pefomance of simple gas ubine plans I egeneaive Gas ubine ycle: he empeaue of he exhaus gases in a simple gas ubine is highe han he empeaue

More information

Solvability of nonlinear Klein-Gordon equation by Laplace Decomposition Method

Solvability of nonlinear Klein-Gordon equation by Laplace Decomposition Method Vol. 84 pp. 37-4 Jly 5 DOI:.5897/JMCSR4.57 icle Nbe: 63F95459 ISSN 6-973 Copyigh 5 hos ein he copyigh of his icle hp://www.cdeicjonls.og/jmcsr ficn Jonl of Mheics nd Cope Science Resech Fll Lengh Resech

More information

Relative and Circular Motion

Relative and Circular Motion Relaie and Cicula Moion a) Relaie moion b) Cenipeal acceleaion Mechanics Lecue 3 Slide 1 Mechanics Lecue 3 Slide 2 Time on Video Pelecue Looks like mosly eeyone hee has iewed enie pelecue GOOD! Thank you

More information

Physics 2A HW #3 Solutions

Physics 2A HW #3 Solutions Chper 3 Focus on Conceps: 3, 4, 6, 9 Problems: 9, 9, 3, 41, 66, 7, 75, 77 Phsics A HW #3 Soluions Focus On Conceps 3-3 (c) The ccelerion due o grvi is he sme for boh blls, despie he fc h he hve differen

More information

ANALYSIS OF KINEMATICS AND KINETOSTATICS OF FOUR-BAR LINKAGE MECHANISM BASED ON GIVEN PROGRAM

ANALYSIS OF KINEMATICS AND KINETOSTATICS OF FOUR-BAR LINKAGE MECHANISM BASED ON GIVEN PROGRAM SINTII POINGS IV INTNTIONL ONGSS "MINS. TNOLОGIS. MTILS." 7 - SUMM SSSION NLSIS O KINMTIS N KINTOSTTIS O OU- LINKG MNISM S ON GIVN POGM W ISSN - PINT ISSN - Pof.soc.. Shl Pof.soc.. Likj. * Pof.ss.. ji.

More information

September 20 Homework Solutions

September 20 Homework Solutions College of Engineering nd Compuer Science Mechnicl Engineering Deprmen Mechnicl Engineering A Seminr in Engineering Anlysis Fll 7 Number 66 Insrucor: Lrry Creo Sepember Homework Soluions Find he specrum

More information

T-Match: Matching Techniques For Driving Yagi-Uda Antennas: T-Match. 2a s. Z in. (Sections 9.5 & 9.7 of Balanis)

T-Match: Matching Techniques For Driving Yagi-Uda Antennas: T-Match. 2a s. Z in. (Sections 9.5 & 9.7 of Balanis) 3/0/018 _mch.doc Pge 1 of 6 T-Mch: Mching Techniques For Driving Ygi-Ud Anenns: T-Mch (Secions 9.5 & 9.7 of Blnis) l s l / l / in The T-Mch is shun-mching echnique h cn be used o feed he driven elemen

More information

Previously. Extensions to backstepping controller designs. Tracking using backstepping Suppose we consider the general system

Previously. Extensions to backstepping controller designs. Tracking using backstepping Suppose we consider the general system 436-459 Advnced contol nd utomtion Extensions to bckstepping contolle designs Tcking Obseves (nonline dmping) Peviously Lst lectue we looked t designing nonline contolles using the bckstepping technique

More information

Ch.4 Motion in 2D. Ch.4 Motion in 2D

Ch.4 Motion in 2D. Ch.4 Motion in 2D Moion in plne, such s in he sceen, is clled 2-dimensionl (2D) moion. 1. Posiion, displcemen nd eloci ecos If he picle s posiion is ( 1, 1 ) 1, nd ( 2, 2 ) 2, he posiions ecos e 1 = 1 1 2 = 2 2 Aege eloci

More information

General Non-Arbitrage Model. I. Partial Differential Equation for Pricing A. Traded Underlying Security

General Non-Arbitrage Model. I. Partial Differential Equation for Pricing A. Traded Underlying Security 1 Geneal Non-Abiage Model I. Paial Diffeenial Equaion fo Picing A. aded Undelying Secuiy 1. Dynamics of he Asse Given by: a. ds = µ (S, )d + σ (S, )dz b. he asse can be eihe a sock, o a cuency, an index,

More information

SOME USEFUL MATHEMATICS

SOME USEFUL MATHEMATICS SOME USEFU MAHEMAICS SOME USEFU MAHEMAICS I is esy o mesure n preic he behvior of n elecricl circui h conins only c volges n currens. However, mos useful elecricl signls h crry informion vry wih ime. Since

More information

Answers to test yourself questions

Answers to test yourself questions Answes to test youself questions opic Descibing fields Gm Gm Gm Gm he net field t is: g ( d / ) ( 4d / ) d d Gm Gm Gm Gm Gm Gm b he net potentil t is: V d / 4d / d 4d d d V e 4 7 9 49 J kg 7 7 Gm d b E

More information

4/3/2017. PHY 712 Electrodynamics 9-9:50 AM MWF Olin 103

4/3/2017. PHY 712 Electrodynamics 9-9:50 AM MWF Olin 103 PHY 7 Eleodnais 9-9:50 AM MWF Olin 0 Plan fo Leue 0: Coninue eading Chap Snhoon adiaion adiaion fo eleon snhoon deies adiaion fo asonoial objes in iula obis 0/05/07 PHY 7 Sping 07 -- Leue 0 0/05/07 PHY

More information

f t f a f x dx By Lin McMullin f x dx= f b f a. 2

f t f a f x dx By Lin McMullin f x dx= f b f a. 2 Accumulion: Thoughs On () By Lin McMullin f f f d = + The gols of he AP* Clculus progrm include he semen, Sudens should undersnd he definie inegrl s he ne ccumulion of chnge. 1 The Topicl Ouline includes

More information

Average & instantaneous velocity and acceleration Motion with constant acceleration

Average & instantaneous velocity and acceleration Motion with constant acceleration Physics 7: Lecure Reminders Discussion nd Lb secions sr meeing ne week Fill ou Pink dd/drop form if you need o swich o differen secion h is FULL. Do i TODAY. Homework Ch. : 5, 7,, 3,, nd 6 Ch.: 6,, 3 Submission

More information

WORK POWER AND ENERGY Consevaive foce a) A foce is said o be consevaive if he wok done by i is independen of pah followed by he body b) Wok done by a consevaive foce fo a closed pah is zeo c) Wok done

More information

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( )

5-1. We apply Newton s second law (specifically, Eq. 5-2). F = ma = ma sin 20.0 = 1.0 kg 2.00 m/s sin 20.0 = 0.684N. ( ) ( ) 5-1. We apply Newon s second law (specfcally, Eq. 5-). (a) We fnd he componen of he foce s ( ) ( ) F = ma = ma cos 0.0 = 1.00kg.00m/s cos 0.0 = 1.88N. (b) The y componen of he foce s ( ) ( ) F = ma = ma

More information

Derivation of the differential equation of motion

Derivation of the differential equation of motion Divion of h iffnil quion of oion Fis h noions fin h will us fo h ivion of h iffnil quion of oion. Rollo is hough o -insionl isk. xnl ius of h ll isnc cn of ll (O) - IDU s cn of gviy (M) θ ngl of inclinion

More information

Addition & Subtraction of Polynomials

Addition & Subtraction of Polynomials Addiion & Sucion of Polynomil Addiion of Polynomil: Adding wo o moe olynomil i imly me of dding like em. The following ocedue hould e ued o dd olynomil 1. Remove enhee if hee e enhee. Add imil em. Wie

More information

Radial geodesics in Schwarzschild spacetime

Radial geodesics in Schwarzschild spacetime Rdil geodesics in Schwzschild spcetime Spheiclly symmetic solutions to the Einstein eqution tke the fom ds dt d dθ sin θdϕ whee is constnt. We lso hve the connection components, which now tke the fom using

More information

e 2t u(t) e 2t u(t) =?

e 2t u(t) e 2t u(t) =? EE : Signals, Sysems, and Transforms Fall 7. Skech he convoluion of he following wo signals. Tes No noes, closed book. f() Show your work. Simplify your answers. g(). Using he convoluion inegral, find

More information

The Production of Polarization

The Production of Polarization Physics 36: Waves Lecue 13 3/31/211 The Poducion of Polaizaion Today we will alk abou he poducion of polaized ligh. We aleady inoduced he concep of he polaizaion of ligh, a ansvese EM wave. To biefly eview

More information

22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 9: The High Beta Tokamak

22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 9: The High Beta Tokamak .65, MHD Theory of Fusion Sysems Prof. Freidberg Lecure 9: The High e Tokmk Summry of he Properies of n Ohmic Tokmk. Advnges:. good euilibrium (smll shif) b. good sbiliy ( ) c. good confinemen ( τ nr )

More information

One of the common descriptions of curvilinear motion uses path variables, which are measurements made along the tangent t and normal n to the path of

One of the common descriptions of curvilinear motion uses path variables, which are measurements made along the tangent t and normal n to the path of Oe of he commo descipios of cuilie moio uses ph ibles, which e mesuemes mde log he ge d oml o he ph of he picles. d e wo ohogol xes cosideed sepely fo eey is of moio. These coodies poide ul descipio fo

More information

BINOMIAL THEOREM OBJECTIVE PROBLEMS in the expansion of ( 3 +kx ) are equal. Then k =

BINOMIAL THEOREM OBJECTIVE PROBLEMS in the expansion of ( 3 +kx ) are equal. Then k = wwwskshieduciocom BINOMIAL HEOREM OBJEIVE PROBLEMS he coefficies of, i e esio of k e equl he k /7 If e coefficie of, d ems i e i AP, e e vlue of is he coefficies i e,, 7 ems i e esio of e i AP he 7 7 em

More information

(a) Counter-Clockwise (b) Clockwise ()N (c) No rotation (d) Not enough information

(a) Counter-Clockwise (b) Clockwise ()N (c) No rotation (d) Not enough information m m m00 kg dult, m0 kg bby. he seesw stts fom est. Which diection will it ottes? ( Counte-Clockwise (b Clockwise ( (c o ottion ti (d ot enough infomtion Effect of Constnt et oque.3 A constnt non-zeo toque

More information

The shortest path between two truths in the real domain passes through the complex domain. J. Hadamard

The shortest path between two truths in the real domain passes through the complex domain. J. Hadamard Complex Analysis R.G. Halbud R.Halbud@ucl.ac.uk Depamen of Mahemaics Univesiy College London 202 The shoes pah beween wo uhs in he eal domain passes hough he complex domain. J. Hadamad Chape The fis fundamenal

More information

Chapter 10. Simple Harmonic Motion and Elasticity. Goals for Chapter 10

Chapter 10. Simple Harmonic Motion and Elasticity. Goals for Chapter 10 Chper 0 Siple Hronic Moion nd Elsiciy Gols or Chper 0 o ollow periodic oion o sudy o siple hronic oion. o sole equions o siple hronic oion. o use he pendulu s prooypicl syse undergoing siple hronic oion.

More information

Today - Lecture 13. Today s lecture continue with rotations, torque, Note that chapters 11, 12, 13 all involve rotations

Today - Lecture 13. Today s lecture continue with rotations, torque, Note that chapters 11, 12, 13 all involve rotations Today - Lecue 13 Today s lecue coninue wih oaions, oque, Noe ha chapes 11, 1, 13 all inole oaions slide 1 eiew Roaions Chapes 11 & 1 Viewed fom aboe (+z) Roaional, o angula elociy, gies angenial elociy

More information

the king's singers And So It Goes the colour of song Words and Vusic by By Joel LEONARD Arranged by Bob Chilcott

the king's singers And So It Goes the colour of song Words and Vusic by By Joel LEONARD Arranged by Bob Chilcott 085850 SATB div cppell US $25 So Goes Wods nd Vusic by By Joel Anged by Bob Chilco he king's singes L he colou of song A H EXCLUSVELY DSTRBUTED BY LEONARD (Fom The King's Singes 25h Annivesy Jubilee) So

More information

Lecture 10. Solution of Nonlinear Equations - II

Lecture 10. Solution of Nonlinear Equations - II Fied point Poblems Lectue Solution o Nonline Equtions - II Given unction g : R R, vlue such tht gis clled ied point o the unction g, since is unchnged when g is pplied to it. Whees with nonline eqution

More information

Lecture 17: Kinetics of Phase Growth in a Two-component System:

Lecture 17: Kinetics of Phase Growth in a Two-component System: Lecue 17: Kineics of Phase Gowh in a Two-componen Sysem: descipion of diffusion flux acoss he α/ ineface Today s opics Majo asks of oday s Lecue: how o deive he diffusion flux of aoms. Once an incipien

More information

MTH 146 Class 11 Notes

MTH 146 Class 11 Notes 8.- Are of Surfce of Revoluion MTH 6 Clss Noes Suppose we wish o revolve curve C round n is nd find he surfce re of he resuling solid. Suppose f( ) is nonnegive funcion wih coninuous firs derivive on he

More information

Read Range Analysis of Passive RFID Systems for Manufacturing Control Systems

Read Range Analysis of Passive RFID Systems for Manufacturing Control Systems Red Rnge Anlysis of Pssive RFID Sysems fo Mnufcuing Conol Sysems M. Keskilmmi, L. Sydänheimo, P. Slonen, M. Kivikoski leconics Tmpee Univesiy of Technology P.O. Box 69, FIN-3311 Tmpee FINLAND Absc: - An

More information

10 m, so the distance from the Sun to the Moon during a solar eclipse is. The mass of the Sun, Earth, and Moon are = =

10 m, so the distance from the Sun to the Moon during a solar eclipse is. The mass of the Sun, Earth, and Moon are = = Chpte 1 nivesl Gvittion 11 *P1. () The un-th distnce is 1.4 nd the th-moon 8 distnce is.84, so the distnce fom the un to the Moon duing sol eclipse is 11 8 11 1.4.84 = 1.4 The mss of the un, th, nd Moon

More information

PHYSICS 1210 Exam 1 University of Wyoming 14 February points

PHYSICS 1210 Exam 1 University of Wyoming 14 February points PHYSICS 1210 Em 1 Uniersiy of Wyoming 14 Februry 2013 150 poins This es is open-noe nd closed-book. Clculors re permied bu compuers re no. No collborion, consulion, or communicion wih oher people (oher

More information

Fundamental Vehicle Loads & Their Estimation

Fundamental Vehicle Loads & Their Estimation Fundaenal Vehicle Loads & Thei Esiaion The silified loads can only be alied in he eliinay design sage when he absence of es o siulaion daa They should always be qualified and udaed as oe infoaion becoes

More information

Mark Scheme (Results) January 2008

Mark Scheme (Results) January 2008 Mk Scheme (Results) Jnuy 00 GCE GCE Mthemtics (6679/0) Edecel Limited. Registeed in Englnd nd Wles No. 4496750 Registeed Office: One90 High Holbon, London WCV 7BH Jnuy 00 6679 Mechnics M Mk Scheme Question

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

How to Prove the Riemann Hypothesis Author: Fayez Fok Al Adeh.

How to Prove the Riemann Hypothesis Author: Fayez Fok Al Adeh. How o Prove he Riemnn Hohesis Auhor: Fez Fok Al Adeh. Presiden of he Srin Cosmologicl Socie P.O.Bo,387,Dmscus,Sri Tels:963--77679,735 Emil:hf@scs-ne.org Commens: 3 ges Subj-Clss: Funcionl nlsis, comle

More information

Motion. Part 2: Constant Acceleration. Acceleration. October Lab Physics. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.

Motion. Part 2: Constant Acceleration. Acceleration. October Lab Physics. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration. Moion Accelerion Pr : Consn Accelerion Accelerion Accelerion Accelerion is he re of chnge of velociy. = v - vo = Δv Δ ccelerion = = v - vo chnge of velociy elpsed ime Accelerion is vecor, lhough in one-dimensionl

More information

0 for t < 0 1 for t > 0

0 for t < 0 1 for t > 0 8.0 Sep nd del funcions Auhor: Jeremy Orloff The uni Sep Funcion We define he uni sep funcion by u() = 0 for < 0 for > 0 I is clled he uni sep funcion becuse i kes uni sep = 0. I is someimes clled he Heviside

More information

Reinforcement learning

Reinforcement learning Lecue 3 Reinfocemen leaning Milos Hauskech milos@cs.pi.edu 539 Senno Squae Reinfocemen leaning We wan o lean he conol policy: : X A We see examples of x (bu oupus a ae no given) Insead of a we ge a feedback

More information

6. Gas dynamics. Ideal gases Speed of infinitesimal disturbances in still gas

6. Gas dynamics. Ideal gases Speed of infinitesimal disturbances in still gas 6. Gs dynmics Dr. Gergely Krisóf De. of Fluid echnics, BE Februry, 009. Seed of infiniesiml disurbnces in sill gs dv d, dv d, Coninuiy: ( dv)( ) dv omenum r r heorem: ( ( dv) ) d 3443 4 q m dv d dv llievi

More information

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 2

ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER 2 ANSWERS TO EVEN NUMBERED EXERCISES IN CHAPTER Seion Eerise -: Coninuiy of he uiliy funion Le λ ( ) be he monooni uiliy funion defined in he proof of eisene of uiliy funion If his funion is oninuous y hen

More information

Lecture 5. Chapter 3. Electromagnetic Theory, Photons, and Light

Lecture 5. Chapter 3. Electromagnetic Theory, Photons, and Light Lecue 5 Chape 3 lecomagneic Theo, Phoons, and Ligh Gauss s Gauss s Faada s Ampèe- Mawell s + Loen foce: S C ds ds S C F dl dl q Mawell equaions d d qv A q A J ds ds In mae fields ae defined hough ineacion

More information

Week 10: DTMC Applications Ranking Web Pages & Slotted ALOHA. Network Performance 10-1

Week 10: DTMC Applications Ranking Web Pages & Slotted ALOHA. Network Performance 10-1 Week : DTMC Alictions Rnking Web ges & Slotted ALOHA etwok efonce - Outline Aly the theoy of discete tie Mkov chins: Google s nking of web-ges Wht ge is the use ost likely seching fo? Foulte web-gh s Mkov

More information

Motion on a Curve and Curvature

Motion on a Curve and Curvature Moion on Cue nd Cuue his uni is bsed on Secions 9. & 9.3, Chpe 9. All ssigned edings nd execises e fom he exbook Objecies: Mke cein h you cn define, nd use in conex, he ems, conceps nd fomuls lised below:

More information

Non-sinusoidal Signal Generators

Non-sinusoidal Signal Generators Non-sinusoidal Signal Geneaos ecangle, iangle, saw ooh, pulse, ec. Muliibao cicuis: asable no sable saes (wo quasi-sable saes; i emains in each sae fo pedeemined imes) monosable one sable sae, one unsable

More information

Sections 3.1 and 3.4 Exponential Functions (Growth and Decay)

Sections 3.1 and 3.4 Exponential Functions (Growth and Decay) Secions 3.1 and 3.4 Eponenial Funcions (Gowh and Decay) Chape 3. Secions 1 and 4 Page 1 of 5 Wha Would You Rahe Have... $1million, o double you money evey day fo 31 days saing wih 1cen? Day Cens Day Cens

More information

Optimization. x = 22 corresponds to local maximum by second derivative test

Optimization. x = 22 corresponds to local maximum by second derivative test Optimiztion Lectue 17 discussed the exteme vlues of functions. This lectue will pply the lesson fom Lectue 17 to wod poblems. In this section, it is impotnt to emembe we e in Clculus I nd e deling one-vible

More information

Generalized Fibonacci-Type Sequence and its Properties

Generalized Fibonacci-Type Sequence and its Properties Geelized Fibocci-Type Sequece d is Popeies Ompsh Sihwl shw Vys Devshi Tuoil Keshv Kuj Mdsu (MP Idi Resech Schol Fculy of Sciece Pcific Acdemy of Highe Educio d Resech Uivesiy Udipu (Rj Absc: The Fibocci

More information

LECTURE 5. is defined by the position vectors r, 1. and. The displacement vector (from P 1 to P 2 ) is defined through r and 1.

LECTURE 5. is defined by the position vectors r, 1. and. The displacement vector (from P 1 to P 2 ) is defined through r and 1. LECTURE 5 ] DESCRIPTION OF PARTICLE MOTION IN SPACE -The displcemen, veloci nd cceleion in -D moion evel hei veco nue (diecion) houh he cuion h one mus p o hei sin. Thei full veco menin ppes when he picle

More information

Motion. ( (3 dim) ( (1 dim) dt. Equations of Motion (Constant Acceleration) Newton s Law and Weight. Magnitude of the Frictional Force

Motion. ( (3 dim) ( (1 dim) dt. Equations of Motion (Constant Acceleration) Newton s Law and Weight. Magnitude of the Frictional Force Insucos: ield/mche PHYSICS DEPARTMENT PHY 48 Em Sepeme 6, 4 Nme pin, ls fis: Signue: On m hono, I he neihe gien no eceied unuhoied id on his eminion. YOUR TEST NUMBER IS THE 5-DIGIT NUMBER AT THE TOP O

More information

LAPLACE TRANSFORMS. 1. Basic transforms

LAPLACE TRANSFORMS. 1. Basic transforms LAPLACE TRANSFORMS. Bic rnform In hi coure, Lplce Trnform will be inroduced nd heir properie exmined; ble of common rnform will be buil up; nd rnform will be ued o olve ome dierenil equion by rnforming

More information

Two-Pion Exchange Currents in Photodisintegration of the Deuteron

Two-Pion Exchange Currents in Photodisintegration of the Deuteron Two-Pion Exchange Cuens in Phoodisinegaion of he Deueon Dagaa Rozędzik and Jacek Goak Jagieonian Univesiy Kaków MENU00 3 May 00 Wiiasbug Conen Chia Effecive Fied Theoy ChEFT Eecoagneic cuen oeaos wihin

More information

ENGR 1990 Engineering Mathematics The Integral of a Function as a Function

ENGR 1990 Engineering Mathematics The Integral of a Function as a Function ENGR 1990 Engineering Mhemics The Inegrl of Funcion s Funcion Previously, we lerned how o esime he inegrl of funcion f( ) over some inervl y dding he res of finie se of rpezoids h represen he re under

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

INTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q).

INTEGRALS. Exercise 1. Let f : [a, b] R be bounded, and let P and Q be partitions of [a, b]. Prove that if P Q then U(P ) U(Q) and L(P ) L(Q). INTEGRALS JOHN QUIGG Eercise. Le f : [, b] R be bounded, nd le P nd Q be priions of [, b]. Prove h if P Q hen U(P ) U(Q) nd L(P ) L(Q). Soluion: Le P = {,..., n }. Since Q is obined from P by dding finiely

More information

Electronic Supplementary Material

Electronic Supplementary Material Electonic Supplementy Mteil On the coevolution of socil esponsiveness nd behvioul consistency Mx Wolf, G Snde vn Doon & Fnz J Weissing Poc R Soc B 78, 440-448; 0 Bsic set-up of the model Conside the model

More information

Primal and Weakly Primal Sub Semi Modules

Primal and Weakly Primal Sub Semi Modules Aein Inenionl Jounl of Conepoy eeh Vol 4 No ; Jnuy 204 Pil nd Wekly Pil ub ei odule lik Bineh ub l hei Depen Jodn Univeiy of iene nd Tehnology Ibid 220 Jodn Ab Le be ouive eiing wih ideniy nd n -ei odule

More information

7 Wave Equation in Higher Dimensions

7 Wave Equation in Higher Dimensions 7 Wave Equaion in Highe Dimensions We now conside he iniial-value poblem fo he wave equaion in n dimensions, u c u x R n u(x, φ(x u (x, ψ(x whee u n i u x i x i. (7. 7. Mehod of Spheical Means Ref: Evans,

More information

Classification of Equations Characteristics

Classification of Equations Characteristics Clssiiion o Eqions Cheisis Consie n elemen o li moing in wo imensionl spe enoe s poin P elow. The ph o P is inie he line. The posiion ile is s so h n inemenl isne long is s. Le he goening eqions e epesene

More information

On The Estimation of Two Missing Values in Randomized Complete Block Designs

On The Estimation of Two Missing Values in Randomized Complete Block Designs Mahemaical Theoy and Modeling ISSN 45804 (Pape ISSN 505 (Online Vol.6, No.7, 06 www.iise.og On The Esimaion of Two Missing Values in Randomized Complee Bloc Designs EFFANGA, EFFANGA OKON AND BASSE, E.

More information

Summary: Binomial Expansion...! r. where

Summary: Binomial Expansion...! r. where Summy: Biomil Epsio 009 M Teo www.techmejcmth-sg.wes.com ) Re-cp of Additiol Mthemtics Biomil Theoem... whee )!!(! () The fomul is ville i MF so studets do ot eed to memoise it. () The fomul pplies oly

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

MATHEMATICAL FOUNDATIONS FOR APPROXIMATING PARTICLE BEHAVIOUR AT RADIUS OF THE PLANCK LENGTH

MATHEMATICAL FOUNDATIONS FOR APPROXIMATING PARTICLE BEHAVIOUR AT RADIUS OF THE PLANCK LENGTH Fundamenal Jounal of Mahemaical Phsics Vol 3 Issue 013 Pages 55-6 Published online a hp://wwwfdincom/ MATHEMATICAL FOUNDATIONS FOR APPROXIMATING PARTICLE BEHAVIOUR AT RADIUS OF THE PLANCK LENGTH Univesias

More information