Problem 1. Solution: = show that for a constant number of particles: c and V. a) Using the definitions of P

Size: px
Start display at page:

Download "Problem 1. Solution: = show that for a constant number of particles: c and V. a) Using the definitions of P"

Transcription

1 rol. Using t dfinitions of nd nd t first lw of trodynis nd t driv t gnrl rltion: wr nd r t sifi t itis t onstnt rssur nd volu rstivly nd nd r t intrnl nrgy nd volu of ol. first lw rlts d dq d t onstnt volu: Q t onstnt rssur: d d d d d d d dq Q or d d d dividing trougout y d t onstnt rssur: tus ro t fundntl diffrntil rltion for d nd t diffrntil rltion for loltz fr nrgy d d d sow tt for onstnt nur of rtils:

2 fundntl diffrntil rltion d d d king rtil drivtiv wit rst to volu wit onst: rrnging ro d d d v nd Diffrntiting wit rst to t otr vrils: nd On rrivs t Mxwll rltion wit wi Us t ov rsults to find for Idl Gs oining rsults fro nd on gts v rssur qution of stt for on ol of idl gs is tus

3 d Us t ov rsults to find n dr Wls gs n / / ow tt t onstnt rssur in t liit t rsult for of n dr Wls gs is t s s for of n idl gs. O /

4 rol. initil stt of onotoi idl gs is dsrid y nd t trtur rssur nd volu rstivly. gs is tkn ovr t t qusisttilly s sown in t skt. volus r rltd s.5. ind: ow u work dos t gs do on t t nd wt is t ng in its intrnl nrgy? xtrnl work: W ng in xtrnl nrgy: n n n n ow u t is sord in going fro? ro t first lw: 5 d dq d Q W Driv t xrssion for t ntroy ng for n ritrry ross. d d d wit n v nd n d n d n d n d n d n v ln n ln n ln n [ ] ln n ln sin onst for diti rvrsil ross. d If is n diti ross find t finl gs rssur nd t ntroy ng fro t gnrl xrssion otind in rt. or diti xnsion of idl gs onst wit 5 /

5 5 n If t ross is rvrsil on xts. [ ] ln ln ln n n n

6 rol. onsidr on-dinsionl in onsisting on >> sgnts s illustrtd in t skt. Lt t lngt of sgnt wn t long dinsion is rlll to t in nd zro wn t sgnt is vrtil i.. long dinsion is rndiulr to t in dirtion. sgnt s just two stts orizontl nd vrtil nd of ts stts is not dgnrt. distn twn t in nds is fixd. or givn lngt Ll <l< of in wt is t totl nur of irostts ssil y t syst nd wt is t ntroy of t syst s funtion of l? or t in lngt to l tr will l/ orizontls sgnt so tt totl nur of ossil irostts is: Ω! tn!! k ln Ω k ln l! l!! Writ down t rorit trodyni idntity for t syst quivlnt to t first lw nd dsri qulittivly ow on ould otin n xrssion for tnsion for nssry to intin t lngt l ssuing t joints turn frly fro t rsult otind in rt rorit trodyni idntity is d d dl It follows tt nd l l l Using t riroity rltion on gts l l lultd fro t rsult for in rt. wi ould l Otin t rltionsi twn tt tnsion for intining t distn l nd t trtur using t nonil nsl dsrition. nrgy ontriution for orizontl link is tus vrg link lngt is 6

7 l k k k for links: k l k k k k l l ln k l l d Undr wi onditions dos your nswr ld to ook s lw nd wt is orrsonding xrssion for t sring onstnt? t ig or low k k k k k k k l k k k k k k l wit sring onstnt rol. syst of two nrgy lvls nd is oultd y rtils t trtur. ssu tt > so tt - is ositiv. Driv n xrssion for t vrg nrgy r rtil s funtion of trtur. Dtrin t liiting vior nd vlu for vrg nrgy r rtil in t liits of nd. or so Wit liiting vlu li 7

8 8 or so Wit liiting vlu li Driv n xrssion for sifi t of t syst. or ol: k d out sifi t in t liits of nd. or : k or : k rol 5. onsidr n idl gs of sin-½ frions is onfind to n r in dinsions. onsidr t ground stt for su syst. Driv t xrssion for t nur of singl rtil irostts in t ontu intrvl d for t givn syst. In D n n Γ L n n k L k k Γ kdk kdk L dk dk k d dk k g Γ k dk d d d g or rions tr r two ossil sin stts for givn nrgy tus t dnsity of stts douls: d d g

9 9 ind t ri ontu nd il otntil of t syst. il otntil t is lld ri nrgy nd it is rltd to ri ontu s follows ε µ. totl nur of ltrons is d g n In t ground stt µ d d g µ µ ind t vrg nrgy r rtil for t syst. µ ε d d d g d g d ow ssu t gs syst is ld in t unifor gnti fild wi ks n dditionl singl rtil nrgy ontriution ±µ dnding on t sin orinttion. ind t ri ont for sin-u nd sin-down frions. singl rtil nrgy in t gnti fild would ± µ ri ontu is xiu ssil ontu so for t sin-u frions: µ tus µ in-down:

10 µ tus µ lult t vrg gntiztion r r for t syst. vrg nur of sin-down rtils: µ µ µ d µ lugin µ µ iilrly d µ µ vrg gntiztion r r µ µ µ

Functions and Graphs 1. (a) (b) (c) (f) (e) (d) 2. (a) (b) (c) (d)

Functions and Graphs 1. (a) (b) (c) (f) (e) (d) 2. (a) (b) (c) (d) Functions nd Grps. () () (c) - - - O - - - O - - - O - - - - (d) () (f) - - O - 7 6 - - O - -7-6 - - - - - O. () () (c) (d) - - - O - O - O - - O - -. () G() f() + f( ), G(-) f( ) + f(), G() G( ) nd G()

More information

HIGHER ORDER DIFFERENTIAL EQUATIONS

HIGHER ORDER DIFFERENTIAL EQUATIONS Prof Enriqu Mtus Nivs PhD in Mthmtis Edution IGER ORDER DIFFERENTIAL EQUATIONS omognous linr qutions with onstnt offiints of ordr two highr Appl rdution mthod to dtrmin solution of th nonhomognous qution

More information

PH427/PH527: Periodic systems Spring Overview of the PH427 website (syllabus, assignments etc.) 2. Coupled oscillations.

PH427/PH527: Periodic systems Spring Overview of the PH427 website (syllabus, assignments etc.) 2. Coupled oscillations. Dy : Mondy 5 inuts. Ovrviw of th PH47 wsit (syllus, ssignnts tc.). Coupld oscilltions W gin with sss coupld y Hook's Lw springs nd find th possil longitudinl) otion of such syst. W ll xtnd this to finit

More information

SAMPLE CSc 340 EXAM QUESTIONS WITH SOLUTIONS: part 2

SAMPLE CSc 340 EXAM QUESTIONS WITH SOLUTIONS: part 2 AMPLE C EXAM UETION WITH OLUTION: prt. It n sown tt l / wr.7888l. I Φ nots orul or pprotng t vlu o tn t n sown tt t trunton rror o ts pproton s o t or or so onstnts ; tt s Not tt / L Φ L.. Φ.. /. /.. Φ..787.

More information

22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 8: Effect of a Vertical Field on Tokamak Equilibrium

22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 8: Effect of a Vertical Field on Tokamak Equilibrium .65, MHD Thory of usion Systms Prof. ridrg Lctur 8: Effct of Vrticl ild on Tokmk Equilirium Toroidl orc lnc y Mns of Vrticl ild. Lt us riw why th rticl fild is imortnt. 3. or ry short tims, th cuum chmr

More information

Rectangular Waveguides

Rectangular Waveguides Rtgulr Wvguids Wvguids tt://www.tllguid.o/wvguidlirit.tl Uss To rdu ttutio loss ig rquis ig owr C ort ol ov rti rquis Ats s ig-ss iltr Norll irulr or rtgulr W will ssu losslss rtgulr tt://www..surr..u/prsol/d.jris/wguid.tl

More information

16.unified Introduction to Computers and Programming. SOLUTIONS to Examination 4/30/04 9:05am - 10:00am

16.unified Introduction to Computers and Programming. SOLUTIONS to Examination 4/30/04 9:05am - 10:00am 16.unii Introution to Computrs n Prormmin SOLUTIONS to Exmintion /30/0 9:05m - 10:00m Pro. I. Kristin Lunqvist Sprin 00 Grin Stion: Qustion 1 (5) Qustion (15) Qustion 3 (10) Qustion (35) Qustion 5 (10)

More information

INTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x)

INTEGRALS. Chapter 7. d dx. 7.1 Overview Let d dx F (x) = f (x). Then, we write f ( x) Chptr 7 INTEGRALS 7. Ovrviw 7.. Lt d d F () f (). Thn, w writ f ( ) d F () + C. Ths intgrls r clld indfinit intgrls or gnrl intgrls, C is clld constnt of intgrtion. All ths intgrls diffr y constnt. 7..

More information

Having a glimpse of some of the possibilities for solutions of linear systems, we move to methods of finding these solutions. The basic idea we shall

Having a glimpse of some of the possibilities for solutions of linear systems, we move to methods of finding these solutions. The basic idea we shall Hvn lps o so o t posslts or solutons o lnr systs, w ov to tos o nn ts solutons. T s w sll us s to try to sply t syst y lntn so o t vrls n so ts qutons. Tus, w rr to t to s lnton. T prry oprton nvolv s

More information

The Z transform techniques

The Z transform techniques h Z trnfor tchniqu h Z trnfor h th rol in dicrt yt tht th Lplc trnfor h in nlyi of continuou yt. h Z trnfor i th principl nlyticl tool for ingl-loop dicrt-ti yt. h Z trnfor h Z trnfor i to dicrt-ti yt

More information

Chapter 16. 1) is a particular point on the graph of the function. 1. y, where x y 1

Chapter 16. 1) is a particular point on the graph of the function. 1. y, where x y 1 Prctic qustions W now tht th prmtr p is dirctl rltd to th mplitud; thrfor, w cn find tht p. cos d [ sin ] sin sin Not: Evn though ou might not now how to find th prmtr in prt, it is lws dvisl to procd

More information

Integration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals

Integration Continued. Integration by Parts Solving Definite Integrals: Area Under a Curve Improper Integrals Intgrtion Continud Intgrtion y Prts Solving Dinit Intgrls: Ar Undr Curv Impropr Intgrls Intgrtion y Prts Prticulrly usul whn you r trying to tk th intgrl o som unction tht is th product o n lgric prssion

More information

Math 61 : Discrete Structures Final Exam Instructor: Ciprian Manolescu. You have 180 minutes.

Math 61 : Discrete Structures Final Exam Instructor: Ciprian Manolescu. You have 180 minutes. Nm: UCA ID Numr: Stion lttr: th 61 : Disrt Struturs Finl Exm Instrutor: Ciprin nolsu You hv 180 minuts. No ooks, nots or lultors r llow. Do not us your own srth ppr. 1. (2 points h) Tru/Fls: Cirl th right

More information

Tangram Fractions Overview: Students will analyze standard and nonstandard

Tangram Fractions Overview: Students will analyze standard and nonstandard ACTIVITY 1 Mtrils: Stunt opis o tnrm mstrs trnsprnis o tnrm mstrs sissors PROCEDURE Skills: Dsriin n nmin polyons Stuyin onrun Comprin rtions Tnrm Frtions Ovrviw: Stunts will nlyz stnr n nonstnr tnrms

More information

a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points.

a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points. Prole 3: Crnot Cyle of n Idel Gs In this prole, the strting pressure P nd volue of n idel gs in stte, re given he rtio R = / > of the volues of the sttes nd is given Finlly onstnt γ = 5/3 is given You

More information

, between the vertical lines x a and x b. Given a demand curve, having price as a function of quantity, p f (x) at height k is the curve f ( x,

, between the vertical lines x a and x b. Given a demand curve, having price as a function of quantity, p f (x) at height k is the curve f ( x, Clculus for Businss nd Socil Scincs - Prof D Yun Finl Em Rviw vrsion 5/9/7 Chck wbsit for ny postd typos nd updts Pls rport ny typos This rviw sht contins summris of nw topics only (This rviw sht dos hv

More information

Divided. diamonds. Mimic the look of facets in a bracelet that s deceptively deep RIGHT-ANGLE WEAVE. designed by Peggy Brinkman Matteliano

Divided. diamonds. Mimic the look of facets in a bracelet that s deceptively deep RIGHT-ANGLE WEAVE. designed by Peggy Brinkman Matteliano RIGHT-ANGLE WEAVE Dv mons Mm t look o ts n rlt tt s ptvly p sn y Py Brnkmn Mttlno Dv your mons nto trnls o two or our olors. FCT-SCON0216_BNB66 2012 Klm Pulsn Co. Ts mtrl my not rprou n ny orm wtout prmsson

More information

Andre Schneider P621

Andre Schneider P621 ndr Schnidr P61 Probl St #03 Novbr 6, 009 1 Srdnicki 10.3 Vrtx for L 1 = gχϕ ϕ. Th vrtx factor is ig. ϕ ig χ ϕ igur 1: ynan diagra for L 1 = gχϕ ϕ. Srdnicki 11.1 a) Dcay rat for th raction ig igur : ynan

More information

Majorana Neutrino Oscillations in Vacuum

Majorana Neutrino Oscillations in Vacuum Journl of odrn Pysis 0 80-84 tt://dx.doi.org/0.46/jm.0.805 Publisd Onlin August 0 (tt://www.sip.org/journl/jm) jorn Nutrino Osilltions in Vuum Yubr Frny Prz Crlos Jos Quimby Esul d Físi Univrsidd Pdgógi

More information

7 ACM FOR FRAME 2SET 6 FRAME 2SET 5 ACM FOR MAIN FRAME 2SET 4 MAIN FRAME 2SET 3 POLE ASSLY 1 2 CROWN STRUCTURE ASSLY 1 1 CROWN ASSLY 1

7 ACM FOR FRAME 2SET 6 FRAME 2SET 5 ACM FOR MAIN FRAME 2SET 4 MAIN FRAME 2SET 3 POLE ASSLY 1 2 CROWN STRUCTURE ASSLY 1 1 CROWN ASSLY 1 7 M OR RM 2ST 6 RM 2ST 5 M OR MIN RM 2ST 4 MIN RM 2ST 3 POL SSLY 1 2 ROWN STRUTUR SSLY 1 1 ROWN SSLY 1 SR.NO. SRIPTION QTY. a LL IMNSIONS R IN mm I N MT Pi IOLMI 1'NTION LT. Tm: XPLO VIW OR POL MOUNT MLM

More information

Equilibrium Composition and Thermodynamic Properties of Hydrogen Plasma

Equilibrium Composition and Thermodynamic Properties of Hydrogen Plasma Chatr- Equilibrium Comosition and Thrmodynami Prortis of ydrogn Plasma It is wll known that th thrmodynami and transort rortis dnd dirtly on th lasma omosition, whih furthr dnds uon th inlusion of ltronially

More information

Study Of Superconductivity And Antiferromagnetism In Rare Earth Nickel Borocarbides (RNi 2 B 2 C)

Study Of Superconductivity And Antiferromagnetism In Rare Earth Nickel Borocarbides (RNi 2 B 2 C) IOSR Journl o Applid Pysis IOSR-JAP -ISS: 78-86.olum 9 Issu r. II y - Jun 7 PP 7-8 www.iosrjournls.org Study O Suprondutivity And Antirromgntism In Rr Ert il ororids Ri C r. Slil s nd Prti Sumn s prtmnt

More information

Elliptical motion, gravity, etc

Elliptical motion, gravity, etc FW Physics 130 G:\130 lctur\ch 13 Elliticl motion.docx g 1 of 7 11/3/010; 6:40 PM; Lst rintd 11/3/010 6:40:00 PM Fig. 1 Elliticl motion, grvity, tc minor xis mjor xis F 1 =A F =B C - D, mjor nd minor xs

More information

1.60± ± ±0.07 S 1.60± ± ±0.30 X

1.60± ± ±0.07 S 1.60± ± ±0.30 X 02M557-01B - Pg 1 of 7 Produt Fmily: Prt Numbr Sris: Multilyr rmi itors Flxibl Trmintion ST Sris onstrution: Flxibl Trmintions NPO, X7R, X5R nd Y5V diltri mtrils Wr round ltrods 100% mtt tin ovr Ni trmintions

More information

VECTOR ANALYSIS APPLICATION IN ROTATING MAGNETIC FIELDS

VECTOR ANALYSIS APPLICATION IN ROTATING MAGNETIC FIELDS 22-578 VECTOR ANALYSIS APPLICATION IN ROTATING MAGNETIC FIELDS runo Osorno Dprtnt of Eltril And Coputr Enginring Cliforni Stt Univrsity Northridg 18111 Nordhoff St Northridg CA 9133-8436 Eil:runo@s.sun.du

More information

OpenMx Matrices and Operators

OpenMx Matrices and Operators OpnMx Mtris n Oprtors Sr Mln Mtris: t uilin loks Mny typs? Dnots r lmnt mxmtrix( typ= Zro", nrow=, nol=, nm="" ) mxmtrix( typ= Unit", nrow=, nol=, nm="" ) mxmtrix( typ= Int", nrow=, nol=, nm="" ) mxmtrix(

More information

Weighted Graphs. Weighted graphs may be either directed or undirected.

Weighted Graphs. Weighted graphs may be either directed or undirected. 1 In mny ppltons, o rp s n ssot numrl vlu, ll wt. Usully, t wts r nonntv ntrs. Wt rps my tr rt or unrt. T wt o n s otn rrr to s t "ost" o t. In ppltons, t wt my msur o t lnt o rout, t pty o ln, t nry rqur

More information

GUC (Dr. Hany Hammad) 9/28/2016

GUC (Dr. Hany Hammad) 9/28/2016 U (r. Hny Hd) 9/8/06 ctur # 3 ignl flow grphs (cont.): ignl-flow grph rprsnttion of : ssiv sgl-port dvic. owr g qutions rnsducr powr g. Oprtg powr g. vill powr g. ppliction to Ntwork nlyzr lirtion. Nois

More information

TOPIC 5: INTEGRATION

TOPIC 5: INTEGRATION TOPIC 5: INTEGRATION. Th indfinit intgrl In mny rspcts, th oprtion of intgrtion tht w r studying hr is th invrs oprtion of drivtion. Dfinition.. Th function F is n ntidrivtiv (or primitiv) of th function

More information

Physics 222 Midterm, Form: A

Physics 222 Midterm, Form: A Pysis 222 Mitrm, Form: A Nm: Dt: Hr r som usul onstnts. 1 4πɛ 0 = 9 10 9 Nm 2 /C 2 µ0 4π = 1 10 7 tsl s/c = 1.6 10 19 C Qustions 1 5: A ipol onsistin o two r point-lik prtils wit q = 1 µc, sprt y istn

More information

Math 166 Week in Review 2 Sections 1.1b, 1.2, 1.3, & 1.4

Math 166 Week in Review 2 Sections 1.1b, 1.2, 1.3, & 1.4 Mt 166 WIR, Sprin 2012, Bnjmin urisp Mt 166 Wk in Rviw 2 Stions 1.1, 1.2, 1.3, & 1.4 1. S t pproprit rions in Vnn irm tt orrspon to o t ollowin sts. () (B ) B () ( ) B B () (B ) B 1 Mt 166 WIR, Sprin 2012,

More information

XV Quantum Electrodynamics

XV Quantum Electrodynamics XV Qnt lctrdynics Fynn Rls fr QD An xl: Sry: iht Sts f Fynn Tchnis Fr rfrnc s: Hlzn&Mrtin s 86,8,9 Intrdctin t Prticl Physics ctr XV Cntnts R. Or Srin 005 Fynn rls sin 0 ty dl sin sin htn xtrnl lin in

More information

Lecture 6 Thermionic Engines

Lecture 6 Thermionic Engines Ltur 6 hrmioni ngins Rviw Rihrdson formul hrmioni ngins Shotty brrir nd diod pn juntion nd diod disussion.997 Copyright Gng Chn, MI For.997 Dirt Solr/hrml to ltril nrgy Convrsion WARR M. ROHSOW HA AD MASS

More information

Ch 1.2: Solutions of Some Differential Equations

Ch 1.2: Solutions of Some Differential Equations Ch 1.2: Solutions of Som Diffrntil Equtions Rcll th fr fll nd owl/mic diffrntil qutions: v 9.8.2v, p.5 p 45 Ths qutions hv th gnrl form y' = y - b W cn us mthods of clculus to solv diffrntil qutions of

More information

Calculating Tank Wetted Area Saving time, increasing accuracy

Calculating Tank Wetted Area Saving time, increasing accuracy Clulting Tnk Wetted Are ving time, inresing ur B n Jones, P.., P.E. C lulting wetted re in rtillfilled orizontl or vertil lindril or ellitil tnk n e omlited, deending on fluid eigt nd te se of te eds (ends)

More information

Complete Solutions for MATH 3012 Quiz 2, October 25, 2011, WTT

Complete Solutions for MATH 3012 Quiz 2, October 25, 2011, WTT Complt Solutions or MATH 012 Quiz 2, Otor 25, 2011, WTT Not. T nswrs ivn r r mor omplt tn is xpt on n tul xm. It is intn tt t mor omprnsiv solutions prsnt r will vlul to stunts in stuyin or t inl xm. In

More information

Solutions for HW11. Exercise 34. (a) Use the recurrence relation t(g) = t(g e) + t(g/e) to count the number of spanning trees of v 1

Solutions for HW11. Exercise 34. (a) Use the recurrence relation t(g) = t(g e) + t(g/e) to count the number of spanning trees of v 1 Solutions for HW Exris. () Us th rurrn rltion t(g) = t(g ) + t(g/) to ount th numr of spnning trs of v v v u u u Rmmr to kp multipl gs!! First rrw G so tht non of th gs ross: v u v Rursing on = (v, u ):

More information

MAT3707. Tutorial letter 201/1/2017 DISCRETE MATHEMATICS: COMBINATORICS. Semester 1. Department of Mathematical Sciences MAT3707/201/1/2017

MAT3707. Tutorial letter 201/1/2017 DISCRETE MATHEMATICS: COMBINATORICS. Semester 1. Department of Mathematical Sciences MAT3707/201/1/2017 MAT3707/201/1/2017 Tutoril lttr 201/1/2017 DISCRETE MATHEMATICS: COMBINATORICS MAT3707 Smstr 1 Dprtmnt o Mtmtil Sins SOLUTIONS TO ASSIGNMENT 01 BARCODE Din tomorrow. univrsity o sout ri SOLUTIONS TO ASSIGNMENT

More information

Chapter 40 Introduction to Quantum

Chapter 40 Introduction to Quantum Catr 0 Introdution to Quantu Pysis 900-90 A nw tory alld quantu anis was igly sussful in xlaining t bavior of artils of irosoi siz. Baus sintists larn t wav and artil naturs of ligt in 9 t ntury, ty roos

More information

Madad Khan, Saima Anis and Faisal Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan

Madad Khan, Saima Anis and Faisal Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad, Pakistan Rsr Jorl o Ali is Eiri Tolo 68: 326-334 203 IN: 2040-7459; -IN: 2040-7467 Mwll itii Oritio 203 itt: M 04 202 At: Frr 0 203 Plis: Jl 0 203 O F- -ils o -Al-Grss's Groois M K i Ais Fisl Drtt o Mttis COMAT

More information

In which direction do compass needles always align? Why?

In which direction do compass needles always align? Why? AQA Trloy Unt 6.7 Mntsm n Eltromntsm - Hr 1 Complt t p ll: Mnt or s typ o or n t s stronst t t o t mnt. Tr r two typs o mnt pol: n. Wrt wt woul ppn twn t pols n o t mnt ntrtons low: Drw t mnt l lns on

More information

Module 2 Motion Instructions

Module 2 Motion Instructions Moul 2 Motion Instrutions CAUTION: Bor you strt this xprimnt, unrstn tht you r xpt to ollow irtions EXPLICITLY! Tk your tim n r th irtions or h stp n or h prt o th xprimnt. You will rquir to ntr t in prtiulr

More information

C-Curves. An alternative to the use of hyperbolic decline curves S E R A F I M. Prepared by: Serafim Ltd. P. +44 (0)

C-Curves. An alternative to the use of hyperbolic decline curves S E R A F I M. Prepared by: Serafim Ltd. P. +44 (0) An ltntiv to th us of hypolic dclin cuvs Ppd y: Sfim Ltd S E R A F I M info@sfimltd.com P. +44 (02890 4206 www.sfimltd.com Contnts Contnts... i Intoduction... Initil ssumptions... Solving fo cumultiv...

More information

Internet Algorithms. (Oblivious) Routing. Lecture 10 06/24/11. Wereferto. demands(requirements), forall vertex pairs,,

Internet Algorithms. (Oblivious) Routing. Lecture 10 06/24/11. Wereferto. demands(requirements), forall vertex pairs,, Intrnt Algoritms Ltur 10 06/24/11 (Olivious) Routing Givn a ntwork, witdglngtsl and dmands(rquirmnts), forall vrtx pairs,, afasilroutingisa multiommodityflow, satisfying t rquirmnts, i..,,,,, Wrfrto,,,,,

More information

Planar convex hulls (I)

Planar convex hulls (I) Covx Hu Covxty Gv st P o ots 2D, tr ovx u s t sst ovx oyo tt ots ots o P A oyo P s ovx or y, P, t st s try P. Pr ovx us (I) Coutto Gotry [s 3250] Lur To Bowo Co ovx o-ovx 1 2 3 Covx Hu Covx Hu Covx Hu

More information

this is called an indeterninateformof-oior.fi?afleleitns derivatives can now differentiable and give 0 on on open interval containing I agree to.

this is called an indeterninateformof-oior.fi?afleleitns derivatives can now differentiable and give 0 on on open interval containing I agree to. hl sidd r L Hospitl s Rul 11/7/18 Pronouncd Loh mtims splld Non p t mtims w wnt vlut limit ii m itn ) but irst indtrnintmori?lltns indtrmint t inn gl in which cs th clld n i 9kt ti not ncssrily snsign

More information

k m The reason that his is very useful can be seen by examining the Taylor series expansion of some potential V(x) about a minimum point:

k m The reason that his is very useful can be seen by examining the Taylor series expansion of some potential V(x) about a minimum point: roic Oscilltor Pottil W r ow goig to stuy solutios to t TIS for vry usful ottil tt of t roic oscilltor. I clssicl cics tis is quivlt to t block srig robl or tt of t ulu (for sll oscilltios bot of wic r

More information

Chapter 6 Perturbation theory

Chapter 6 Perturbation theory Ct 6 Ptutio to 6. Ti-iddt odgt tutio to i o tutio sst is giv to fid solutios of λ ' ; : iltoi of si stt : igvlus of : otool igfutios of ; δ ii Rlig-Södig tutio to ' λ..6. ; : gl iltoi ': tutio λ : sll

More information

HUGO ROSMAN * Gheorghe Asachi Technical University of Iaşi, Faculty of Electrical Engineering, Energetics and Applied Informatics

HUGO ROSMAN * Gheorghe Asachi Technical University of Iaşi, Faculty of Electrical Engineering, Energetics and Applied Informatics BULETNUL NSTTUTULU POLTEHNC DN Ş Pulit d Univrsitt Thniă Ghorgh shi din şi Toul LV L Fs. 0 SŃi ELECTOTEHNCĂ. ENEGETCĂ. ELECTONCĂ THE CTVE ENEGY TNSMSSON EFFCENCY THOUGH LNE NON-UTONOMOUS ND PSSVE TWO-POTS

More information

Lecture 11 Waves in Periodic Potentials Today: Questions you should be able to address after today s lecture:

Lecture 11 Waves in Periodic Potentials Today: Questions you should be able to address after today s lecture: Lctur 11 Wvs in Priodic Potntils Tody: 1. Invrs lttic dfinition in 1D.. rphicl rprsnttion of priodic nd -priodic functions using th -xis nd invrs lttic vctors. 3. Sris solutions to th priodic potntil Hmiltonin

More information

PHY 410. Final Examination, Spring May 4, 2009 (5:45-7:45 p.m.)

PHY 410. Final Examination, Spring May 4, 2009 (5:45-7:45 p.m.) PHY ina amination, Spring 9 May, 9 5:5-7:5 p.m. PLAS WAIT UTIL YOU AR TOLD TO BGI TH XAM. Wi waiting, carfuy fi in t information rqustd bow Your am: Your Studnt umbr: DO OT TUR THIS PAG UTIL TH XAM STARTS

More information

4.1 Interval Scheduling. Chapter 4. Greedy Algorithms. Interval Scheduling: Greedy Algorithms. Interval Scheduling. Interval scheduling.

4.1 Interval Scheduling. Chapter 4. Greedy Algorithms. Interval Scheduling: Greedy Algorithms. Interval Scheduling. Interval scheduling. Cptr 4 4 Intrvl Suln Gry Alortms Sls y Kvn Wyn Copyrt 005 Prson-Ason Wsly All rts rsrv Intrvl Suln Intrvl Suln: Gry Alortms Intrvl suln! Jo strts t s n nss t! Two os omptl ty on't ovrlp! Gol: n mxmum sust

More information

The University of Sydney MATH 2009

The University of Sydney MATH 2009 T Unvrsty o Syny MATH 2009 APH THEOY Tutorl 7 Solutons 2004 1. Lt t sonnt plnr rp sown. Drw ts ul, n t ul o t ul ( ). Sow tt s sonnt plnr rp, tn s onnt. Du tt ( ) s not somorp to. ( ) A onnt rp s on n

More information

Improving Union. Implementation. Union-by-size Code. Union-by-Size Find Analysis. Path Compression! Improving Find find(e)

Improving Union. Implementation. Union-by-size Code. Union-by-Size Find Analysis. Path Compression! Improving Find find(e) POW CSE 36: Dt Struturs Top #10 T Dynm (Equvln) Duo: Unon-y-Sz & Pt Comprsson Wk!! Luk MDowll Summr Qurtr 003 M! ZING Wt s Goo Mz? Mz Construton lortm Gvn: ollton o rooms V Conntons twn t rooms (ntlly

More information

A Study of the Solutions of the Lotka Volterra. Prey Predator System Using Perturbation. Technique

A Study of the Solutions of the Lotka Volterra. Prey Predator System Using Perturbation. Technique Inrnionl hmil orum no. 667-67 Sud of h Soluions of h o Volrr r rdor Ssm Using rurion Thniqu D.Vnu ol Ro * D. of lid hmis IT Collg of Sin IT Univrsi Vishnm.. Indi Y... Thorni D. of lid hmis IT Collg of

More information

b.) v d =? Example 2 l = 50 m, D = 1.0 mm, E = 6 V, " = 1.72 #10 $8 % & m, and r = 0.5 % a.) R =? c.) V ab =? a.) R eq =?

b.) v d =? Example 2 l = 50 m, D = 1.0 mm, E = 6 V,  = 1.72 #10 $8 % & m, and r = 0.5 % a.) R =? c.) V ab =? a.) R eq =? xmpl : An 8-gug oppr wr hs nomnl mtr o. mm. Ths wr rrs onstnt urrnt o.67 A to W lmp. Th nsty o r ltrons s 8.5 x 8 ltrons pr u mtr. Fn th mgntu o. th urrnt nsty. th rt vloty xmpl D. mm,.67 A, n N 8.5" 8

More information

CMPS 2200 Fall Graphs. Carola Wenk. Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk

CMPS 2200 Fall Graphs. Carola Wenk. Slides courtesy of Charles Leiserson with changes and additions by Carola Wenk CMPS 2200 Fll 2017 Grps Crol Wnk Sls ourtsy o Crls Lsrson wt ns n tons y Crol Wnk 10/23/17 CMPS 2200 Intro. to Alortms 1 Grps Dnton. A rt rp (rp) G = (V, E) s n orr pr onsstn o st V o vrts (snulr: vrtx),

More information

The Angular Momenta Dipole Moments and Gyromagnetic Ratios of the Electron and the Proton

The Angular Momenta Dipole Moments and Gyromagnetic Ratios of the Electron and the Proton Journl of Modrn hysics, 014, 5, 154-157 ublishd Onlin August 014 in SciRs. htt://www.scir.org/journl/jm htt://dx.doi.org/.436/jm.014.51415 Th Angulr Momnt Diol Momnts nd Gyromgntic Rtios of th Elctron

More information

Fundamental Algorithms for System Modeling, Analysis, and Optimization

Fundamental Algorithms for System Modeling, Analysis, and Optimization Fundmntl Algorithms for Sstm Modling, Anlsis, nd Optimiztion Edwrd A. L, Jijt Rohowdhur, Snjit A. Sshi UC Brkl EECS 144/244 Fll 2011 Copright 2010-11, E. A. L, J. Rohowdhur, S. A. Sshi, All rights rsrvd

More information

Grade 7/8 Math Circles March 4/5, Graph Theory I- Solutions

Grade 7/8 Math Circles March 4/5, Graph Theory I- Solutions ulty o Mtmtis Wtrloo, Ontrio N ntr or ution in Mtmtis n omputin r / Mt irls Mr /, 0 rp Tory - Solutions * inits lln qustion. Tr t ollowin wlks on t rp low. or on, stt wtr it is pt? ow o you know? () n

More information

5/1/2018. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees

5/1/2018. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees. Huffman Coding Trees /1/018 W usully no strns y ssnn -lnt os to ll rtrs n t lpt (or mpl, 8-t on n ASCII). Howvr, rnt rtrs our wt rnt rquns, w n sv mmory n ru trnsmttl tm y usn vrl-lnt non. T s to ssn sortr os to rtrs tt our

More information

National Parks and Wildlife Service

National Parks and Wildlife Service I - til rk Willif i ti jti ri llitrly rl J Vi f til rk Willif i, rtt f ltr, Hrit t Gltt, ly l, li, Irl. W www.w.i -il tr.ti@..i itti W () ti jti llitrly rl. Vi. til rk Willif i, rtt f ltr, Hrit t Gltt.

More information

d e c b a d c b a d e c b a a c a d c c e b

d e c b a d c b a d e c b a a c a d c c e b FLAT PEYOTE STITCH Bin y mkin stoppr -- sw trou n pull it lon t tr until it is out 6 rom t n. Sw trou t in witout splittin t tr. You soul l to sli it up n own t tr ut it will sty in pl wn lt lon. Evn-Count

More information

Exam 2 Solutions. Jonathan Turner 4/2/2012. CS 542 Advanced Data Structures and Algorithms

Exam 2 Solutions. Jonathan Turner 4/2/2012. CS 542 Advanced Data Structures and Algorithms CS 542 Avn Dt Stutu n Alotm Exm 2 Soluton Jontn Tun 4/2/202. (5 ont) Con n oton on t tton t tutu n w t n t 2 no. Wt t mllt num o no tt t tton t tutu oul ontn. Exln you nw. Sn n mut n you o u t n t, t n

More information

P a g e 5 1 of R e p o r t P B 4 / 0 9

P a g e 5 1 of R e p o r t P B 4 / 0 9 P a g e 5 1 of R e p o r t P B 4 / 0 9 J A R T a l s o c o n c l u d e d t h a t a l t h o u g h t h e i n t e n t o f N e l s o n s r e h a b i l i t a t i o n p l a n i s t o e n h a n c e c o n n e

More information

I. The Connection between Spectroscopy and Quantum Mechanics

I. The Connection between Spectroscopy and Quantum Mechanics I. Th Connction twn Spctroscopy nd Quntum Mchnics On of th postults of quntum mchnics: Th stt of systm is fully dscrid y its wvfunction, Ψ( r1, r,..., t) whr r 1, r, tc. r th coordints of th constitunt

More information

Face Detection and Recognition. Linear Algebra and Face Recognition. Face Recognition. Face Recognition. Dimension reduction

Face Detection and Recognition. Linear Algebra and Face Recognition. Face Recognition. Face Recognition. Dimension reduction F Dtto Roto Lr Alr F Roto C Y I Ursty O solto: tto o l trs s s ys os ot. Dlt to t to ltpl ws. F Roto Aotr ppro: ort y rry s tor o so E.. 56 56 > pot 6556- stol sp A st o s t ps to ollto o pots ts sp. F

More information

- ASSEMBLY AND INSTALLATION -

- ASSEMBLY AND INSTALLATION - - SSEMLY ND INSTLLTION - Sliin Door Stm Mot Importnt! Ti rmwork n ml to uit 100 mm ini wll tikn (75 mm tuwork) or 125 mm ini wll tikn (100 mm tuwork) HOWEVER t uppli jm kit i pii to itr 100 mm or 125 mm

More information

Self-Adjusting Top Trees

Self-Adjusting Top Trees Th Polm Sl-jsting Top Ts ynmi ts: ol: mintin n n-tx ost tht hngs o tim. link(,w): ts n g twn tis n w. t(,w): lts g (,w). pplition-spii t ssoit with gs n/o tis. ont xmpls: in minimm-wight g in th pth twn

More information

Pythagorean Theorem and Trigonometry

Pythagorean Theorem and Trigonometry Ptgoren Teorem nd Trigonometr Te Ptgoren Teorem is nient, well-known, nd importnt. It s lrge numer of different proofs, inluding one disovered merin President Jmes. Grfield. Te we site ttp://www.ut-te-knot.org/ptgors/inde.stml

More information

AP Calculus BC AP Exam Problems Chapters 1 3

AP Calculus BC AP Exam Problems Chapters 1 3 AP Eam Problms Captrs Prcalculus Rviw. If f is a continuous function dfind for all ral numbrs and if t maimum valu of f() is 5 and t minimum valu of f() is 7, tn wic of t following must b tru? I. T maimum

More information

Bayesian belief networks

Bayesian belief networks CS 2750 oundtions of I Lctur 9 ysin lif ntworks ilos Huskrcht ilos@cs.pitt.du 5329 Snnott Squr. Huskrcht odling uncrtinty with proilitis Dfining th full joint distriution ks it possil to rprsnt nd rson

More information

176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s

176 5 t h Fl oo r. 337 P o ly me r Ma te ri al s A g la di ou s F. L. 462 E l ec tr on ic D ev el op me nt A i ng er A.W.S. 371 C. A. M. A l ex an de r 236 A d mi ni st ra ti on R. H. (M rs ) A n dr ew s P. V. 326 O p ti ca l Tr an sm is si on A p ps

More information

I N A C O M P L E X W O R L D

I N A C O M P L E X W O R L D IS L A M I C E C O N O M I C S I N A C O M P L E X W O R L D E x p l o r a t i o n s i n A g-b eanste d S i m u l a t i o n S a m i A l-s u w a i l e m 1 4 2 9 H 2 0 0 8 I s l a m i c D e v e l o p m e

More information

UNIT # 08 (PART - I)

UNIT # 08 (PART - I) . r. d[h d[h.5 7.5 mol L S d[o d[so UNIT # 8 (PRT - I CHEMICL INETICS EXERCISE # 6. d[ x [ x [ x. r [X[C ' [X [[B r '[ [B [C. r [NO [Cl. d[so d[h.5 5 mol L S d[nh d[nh. 5. 6. r [ [B r [x [y r' [x [y r'

More information

Applications of trees

Applications of trees Trs Apptons o trs Orgnzton rts Attk trs to syst Anyss o tr ntworks Prsng xprssons Trs (rtrv o norton) Don-n strutur Mutstng Dstnton-s orwrng Trnsprnt swts Forwrng ts o prxs t routrs Struturs or nt pntton

More information

Intro to QM due: February 8, 2019 Problem Set 12

Intro to QM due: February 8, 2019 Problem Set 12 Intro to QM du: Fbruary 8, 9 Prob St Prob : Us [ x i, p j ] i δ ij to vrify that th anguar ontu oprators L i jk ɛ ijk x j p k satisfy th coutation rations [ L i, L j ] i k ɛ ijk Lk, [ L i, x j ] i k ɛ

More information

Exhibit 2-9/30/15 Invoice Filing Page 1841 of Page 3660 Docket No

Exhibit 2-9/30/15 Invoice Filing Page 1841 of Page 3660 Docket No xhibit 2-9/3/15 Invie Filing Pge 1841 f Pge 366 Dket. 44498 F u v 7? u ' 1 L ffi s xs L. s 91 S'.e q ; t w W yn S. s t = p '1 F? 5! 4 ` p V -', {} f6 3 j v > ; gl. li -. " F LL tfi = g us J 3 y 4 @" V)

More information

G-001 SACO SACO BAY BIDDEFORD INDEX OF NAVIGATION AIDS GENERAL NOTES: GENERAL PLAN A6 SCALE: 1" = 1000' CANADA MAINE STATE PLANE GEOGRAPHIC NO.

G-001 SACO SACO BAY BIDDEFORD INDEX OF NAVIGATION AIDS GENERAL NOTES: GENERAL PLAN A6 SCALE: 1 = 1000' CANADA MAINE STATE PLANE GEOGRAPHIC NO. 2 3 6 7 8 9 0 2 3 20000 230000 220000 ST TORY M 8-OOT W ST 2880000 2880000 L ROOK RL OTS: UKI OR TUR RKWTR (TYP) U O ROOK. SOUIS R I T TTS. T RR PL IS M LOWR LOW WTR (MLLW) IS S O T 983-200 TIL PO. SOUIS

More information

National 5 Mathematics

National 5 Mathematics St Anrw s A Mttis Drtnt Ntionl Mttis PRACTICE EXAM REVISION 0- Ntionl Mttis Sintii Nottion Stnr For Rvision P. Writ o t ollowin nurs in sintii nottion. 00 000 000 000 000 0 000 000 000. For o t ollowin

More information

However, many atoms can combine to form particular molecules, e.g. Chlorine (Cl) and Sodium (Na) atoms form NaCl molecules.

However, many atoms can combine to form particular molecules, e.g. Chlorine (Cl) and Sodium (Na) atoms form NaCl molecules. Lctur 6 Titl: Fundmntls of th Quntum Thory of molcul formtion Pg- In th lst modul, w hv discussd out th tomic structur nd tomic physics to undrstnd th spctrum of toms. Howvr, mny toms cn comin to form

More information

Allowable bearing capacity and settlement Vertical stress increase in soil

Allowable bearing capacity and settlement Vertical stress increase in soil 5 Allwabl barg aaity and ttlmnt Vrtial tr ra il - du t nntratd lad: 3 5 r r x y - du t irularly ladd ara lad:. G t tabl 5..6 Fd / by dtrmg th trm: r/(/) /(/) 3- blw rtangular ladd ara: th t i at th rnr

More information

1 Input-Output Stability

1 Input-Output Stability Inut-Outut Stability Inut-outut stability analysis allows us to analyz th stability of a givn syst without knowing th intrnal stat x of th syst. Bfor going forward, w hav to introduc so inut-outut athatical

More information

CHEM 333 QUANTUM THEORY AND SPECTROSCOPY PROBLEM SET I SOLUTION KEY

CHEM 333 QUANTUM THEORY AND SPECTROSCOPY PROBLEM SET I SOLUTION KEY CHEM QUANTUM THEORY AND SPECTROSCOPY PROLEM SET I SOLUTION KEY. T for onstant of O is N. -. Assuing a aroni osillator odl for t vibrational otion of O, wat is t diatoi vibrational frquny in - units? v

More information

Physics 43 HW #9 Chapter 40 Key

Physics 43 HW #9 Chapter 40 Key Pysics 43 HW #9 Captr 4 Ky Captr 4 1 Aftr many ours of dilignt rsarc, you obtain t following data on t potolctric ffct for a crtain matrial: Wavlngt of Ligt (nm) Stopping Potntial (V) 36 3 4 14 31 a) Plot

More information

# 1 ' 10 ' 100. Decimal point = 4 hundred. = 6 tens (or sixty) = 5 ones (or five) = 2 tenths. = 7 hundredths.

# 1 ' 10 ' 100. Decimal point = 4 hundred. = 6 tens (or sixty) = 5 ones (or five) = 2 tenths. = 7 hundredths. How os it work? Pl vlu o imls rprsnt prts o whol numr or ojt # 0 000 Tns o thousns # 000 # 00 Thousns Hunrs Tns Ons # 0 Diml point st iml pl: ' 0 # 0 on tnth n iml pl: ' 0 # 00 on hunrth r iml pl: ' 0

More information

Lecture contents. Bloch theorem k-vector Brillouin zone Almost free-electron model Bands Effective mass Holes. NNSE 508 EM Lecture #9

Lecture contents. Bloch theorem k-vector Brillouin zone Almost free-electron model Bands Effective mass Holes. NNSE 508 EM Lecture #9 Lctur contnts Bloch thorm -vctor Brillouin zon Almost fr-lctron modl Bnds ffctiv mss Hols Trnsltionl symmtry: Bloch thorm On-lctron Schrödingr qution ch stt cn ccommo up to lctrons: If Vr is priodic function:

More information

Case Study 1 PHA 5127 Fall 2006 Revised 9/19/06

Case Study 1 PHA 5127 Fall 2006 Revised 9/19/06 Cas Study Qustion. A 3 yar old, 5 kg patint was brougt in for surgry and was givn a /kg iv bolus injction of a muscl rlaxant. T plasma concntrations wr masurd post injction and notd in t tabl blow: Tim

More information

c. What is the average rate of change of f on the interval [, ]? Answer: d. What is a local minimum value of f? Answer: 5 e. On what interval(s) is f

c. What is the average rate of change of f on the interval [, ]? Answer: d. What is a local minimum value of f? Answer: 5 e. On what interval(s) is f Essential Skills Chapter f ( x + h) f ( x ). Simplifying the difference quotient Section. h f ( x + h) f ( x ) Example: For f ( x) = 4x 4 x, find and simplify completely. h Answer: 4 8x 4 h. Finding the

More information

Theorem 1. An undirected graph is a tree if and only if there is a unique simple path between any two of its vertices.

Theorem 1. An undirected graph is a tree if and only if there is a unique simple path between any two of its vertices. Cptr 11: Trs 11.1 - Introuton to Trs Dnton 1 (Tr). A tr s onnt unrt rp wt no sp ruts. Tor 1. An unrt rp s tr n ony tr s unqu sp pt twn ny two o ts vrts. Dnton 2. A root tr s tr n w on vrtx s n snt s t

More information

FSA. CmSc 365 Theory of Computation. Finite State Automata and Regular Expressions (Chapter 2, Section 2.3) ALPHABET operations: U, concatenation, *

FSA. CmSc 365 Theory of Computation. Finite State Automata and Regular Expressions (Chapter 2, Section 2.3) ALPHABET operations: U, concatenation, * CmSc 365 Thory of Computtion Finit Stt Automt nd Rgulr Exprssions (Chptr 2, Sction 2.3) ALPHABET oprtions: U, conctntion, * otin otin Strings Form Rgulr xprssions dscri Closd undr U, conctntion nd * (if

More information

Maxwellian Collisions

Maxwellian Collisions Maxwllian Collisions Maxwll ralizd arly on that th particular typ of collision in which th cross-sction varis at Q rs 1/g offrs drastic siplifications. Intrstingly, this bhavior is physically corrct for

More information

Last time: introduced our first computational model the DFA.

Last time: introduced our first computational model the DFA. Lctur 7 Homwork #7: 2.2.1, 2.2.2, 2.2.3 (hnd in c nd d), Misc: Givn: M, NFA Prov: (q,xy) * (p,y) iff (q,x) * (p,) (follow proof don in clss tody) Lst tim: introducd our first computtionl modl th DFA. Tody

More information

Humanistic, and Particularly Classical, Studies as a Preparation for the Law

Humanistic, and Particularly Classical, Studies as a Preparation for the Law University of Michigan Law School University of Michigan Law School Scholarship Repository Articles Faculty Scholarship 1907 Humanistic, and Particularly Classical, Studies as a Preparation for the Law

More information

A physical solution for solving the zero-flow singularity in static thermal-hydraulics

A physical solution for solving the zero-flow singularity in static thermal-hydraulics A ysicl solution for solving t zro-flow singulrity in sttic trml-ydrulics miing modls Dnil Bouskl Blig El Hfni EDF R&D 6, qui Wtir F-784 Ctou Cd, Frnc dnil.ouskl@df.fr lig.l-fni@df.fr Astrct For t D-D

More information

NOTE: ONLY RIGHT IDLER (CONFIGURATION A) ARM SHOWN IN VIEWS ON THIS PAGE

NOTE: ONLY RIGHT IDLER (CONFIGURATION A) ARM SHOWN IN VIEWS ON THIS PAGE 0 PP PP PP PP PP PP NOT: ONLY RIT ILR (ONIURTION ) RM SOWN IN VIWS ON TIS P ITM SRIPTION MTRIL QTY. IL RM - RIT / RIL LIN TUIN / O (0.0 WLL) ISI RT 0 ROMOLY LINR INS IL RM - LT / RIL LIN TUIN / O (0.0

More information

Decimals DECIMALS.

Decimals DECIMALS. Dimls DECIMALS www.mthltis.o.uk ow os it work? Solutions Dimls P qustions Pl vlu o imls 0 000 00 000 0 000 00 0 000 00 0 000 00 0 000 tnths or 0 thousnths or 000 hunrths or 00 hunrths or 00 0 tn thousnths

More information

learning objectives learn what graphs are in mathematical terms learn how to represent graphs in computers learn about typical graph algorithms

learning objectives learn what graphs are in mathematical terms learn how to represent graphs in computers learn about typical graph algorithms rp loritms lrnin ojtivs loritms your sotwr systm sotwr rwr lrn wt rps r in mtmtil trms lrn ow to rprsnt rps in omputrs lrn out typil rp loritms wy rps? intuitivly, rp is orm y vrtis n s twn vrtis rps r

More information

:9 :9. Public Water Crossings - DE NORTHERN PASS PROJECT. Ashland. Bridgewater

:9 :9. Public Water Crossings - DE NORTHERN PASS PROJECT. Ashland. Bridgewater Lgnd ol/ Loction Nothn ss Nothn ss nsission Lins Nub - - - kv Lin Evsouc Lins 5 kv Hight (in ft oss Sction 75-4 -4 5-4 Evsouc 5 kv Lin E5-8 E5- E5- E5-. Rf to Nothn ss nsission LL ublic Wt ossings SE ockt

More information