If I have not seen as far as others, it is because giants were standing on my shoulders. - H. Abelson

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

,. *â â > V>V. â ND * 828.

n

Executive Committee and Officers ( )

~,. :'lr. H ~ j. l' ", ...,~l. 0 '" ~ bl '!; 1'1. :<! f'~.., I,," r: t,... r':l G. t r,. 1'1 [<, ."" f'" 1n. t.1 ~- n I'>' 1:1 , I. <1 ~'..

Colby College Catalogue

4 8 N v btr 20, 20 th r l f ff nt f l t. r t pl n f r th n tr t n f h h v lr d b n r d t, rd n t h h th t b t f l rd n t f th rld ll b n tr t d n R th

z E z *" I»! HI UJ LU Q t i G < Q UJ > UJ >- C/J o> o C/) X X UJ 5 UJ 0) te : < C/) < 2 H CD O O) </> UJ Ü QC < 4* P? K ll I I <% "fei 'Q f

Pg #11-13, 15, 17, 18, AND Pg # 3-5, 12-15, 19, 20, 25, 27

Chapter DEs with Discontinuous Force Functions

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

MPM 2D Final Exam Prep 2, June b) Y = 2(x + 1)2-18. ~..: 2. (xl- 1:'}")( t J') -' ( B. vi::: 2 ~ 1-'+ 4 1<. -t-:2 -( 6! '.

PH.D. PRELIMINARY EXAMINATION MATHEMATICS

Th n nt T p n n th V ll f x Th r h l l r r h nd xpl r t n rr d nt ff t b Pr f r ll N v n d r n th r 8 l t p t, n z n l n n th n rth t rn p rt n f th v

THE USE OF A CALCULATOR, CELL PHONE, OR ANY OTHER ELEC- TRONIC DEVICE IS NOT PERMITTED DURING THIS EXAMINATION.

Speed of light c = m/s. x n e a x d x = 1. 2 n+1 a n π a. He Li Ne Na Ar K Ni 58.

n r t d n :4 T P bl D n, l d t z d th tr t. r pd l

::::l<r/ L- 1-1>(=-ft\ii--r(~1J~:::: Fo. l. AG -=(0,.2,L}> M - &-c ==- < ) I) ~..-.::.1 ( \ I 0. /:rf!:,-t- f1c =- <I _,, -2...

rhtre PAID U.S. POSTAGE Can't attend? Pass this on to a friend. Cleveland, Ohio Permit No. 799 First Class

46 D b r 4, 20 : p t n f r n b P l h tr p, pl t z r f r n. nd n th t n t d f t n th tr ht r t b f l n t, nd th ff r n b ttl t th r p rf l pp n nt n th

PR D NT N n TR T F R 6 pr l 8 Th Pr d nt Th h t H h n t n, D D r r. Pr d nt: n J n r f th r d t r v th tr t d rn z t n pr r f th n t d t t. n


< < or a. * or c w u. "* \, w * r? ««m * * Z * < -4 * if # * « * W * <r? # *» */>* - 2r 2 * j j. # w O <» x <» V X * M <2 * * * *

,.*Hffi;;* SONAI, IUERCANTII,N I,IMITDII REGD- 0FFICE: 105/33, VARDHMAN GotD[N PLNLA,R0AD No.44, pitampura, DELHI *ffigfk"

r 3 > o m > o > z m Z -< Z il r H O O H H i-» 00 a o x3 X M > I- > 1 n 0) l' 1

1(!l. 6(t - c) e~ F(s) 1. sin at. cosat. sinh at. cosh at. (s=~'2+b'2 8-a. uc(t) uc(t)./~_~)

Math 341 Fall 2008 Friday December 12

fur \ \,,^N/ D7,,)d.s) 7. The champion and Runner up of the previous year shall be allowed to play directly in final Zone.

FINAL EXAM, MATH 353 SUMMER I 2015

PHYS 232 QUIZ minutes.

is: 3o>« P 6 Jsgg E 2 si O j= o ocq O o & s r 3 O O K O r " CD r cd as c 3 ^th ^ "OU a 5 " c ?.53 drag S.S pqc O r OT3 4J o >> o.- X h 5 c o o C C :_

Name: Solutions Final Exam

Future Self-Guides. E,.?, :0-..-.,0 Q., 5...q ',D5', 4,] 1-}., d-'.4.., _. ZoltAn Dbrnyei Introduction. u u rt 5,4) ,-,4, a. a aci,, u 4.


A L A BA M A L A W R E V IE W

EXAM 2, MATH 132 WEDNESDAY, OCTOBER 23, 2002

,y. ~ (Lo )-Y2 ') '---~ F( '...J ( '1, 4. \fer-\{:k. ('X -5)'1.-+ :tl\ ~\:,) ~::; fi(~ S:;')'"'--t L. X-lOX t ~5 = IJ~-~+~5.

o C *$ go ! b», S AT? g (i * ^ fc fa fa U - S 8 += C fl o.2h 2 fl 'fl O ' 0> fl l-h cvo *, &! 5 a o3 a; O g 02 QJ 01 fls g! r«'-fl O fl s- ccco

APPH 4200 Physics of Fluids

4 4 N v b r t, 20 xpr n f th ll f th p p l t n p pr d. H ndr d nd th nd f t v L th n n f th pr v n f V ln, r dn nd l r thr n nt pr n, h r th ff r d nd

,., [~== -I ] ~y_/5 =- 21 Y -/ Y. t. \,X ::: 3J ~ - 3. Test: Linear equations and Linear inequalities. At!$fJJ' ~ dt~ - 5 = -7C +4 + re -t~ -+>< 1- )_

Vr Vr

i.ea IE !e e sv?f 'il i+x3p \r= v * 5,?: S i- co i, ==:= SOrq) Xgs'iY # oo .9 9 PE * v E=S s->'d =ar4lq 5,n =.9 '{nl a':1 t F #l *r C\ t-e

Math 308 Exam I Practice Problems


I n t e r n a t i o n a l E l e c t r o n i c J o u r n a l o f E l e m e n t a r y E.7 d u, c ai ts is ou n e, 1 V3 1o-2 l6, I n t h i s a r t

ECE430 Name 5 () ( '-'1-+/~ Or"- f w.s. Section: (Circle One) 10 MWF 12:30 TuTh (Sauer) (Liu) TOTAL: USEFUL INFORMATION

Nrer/ \f l xeaoe Rx RxyrZH IABXAP.qAATTAJI xvbbqaat KOMnAHT1. rvfiqgrrox 3Axl4 Pn br H esep fiyraap: qa/oq YnaaH6aarap xor

Colby College Catalogue

SOLUTION SET. Chapter 8 LASER OSCILLATION ABOVE THRESHOLD "LASER FUNDAMENTALS" Second Edition

P a g e 3 6 of R e p o r t P B 4 / 0 9


0 t b r 6, 20 t l nf r nt f th l t th t v t f th th lv, ntr t n t th l l l nd d p rt nt th t f ttr t n th p nt t th r f l nd d tr b t n. R v n n th r

Chapter PENTAGONAL NUMBERS

Chain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics

Jsee x dx = In Isec x + tanxl + C Jcsc x dx = - In I cscx + cotxl + C

u4 ~ ~~-Jk~ ~~.~h4t -Jk. ~ ~ 1:;tK.t ~..~~ ~.~~ ~~!t-o 0j~~ ~ wd/l ~ C0n~ 1~J4.(kvl ~. NAME:~ ~.. PERIOD: ~~~.&m~.. OVER CH 13 HOMEWORK.

5 questions, 3 points each, 15 points total possible. 26 Fe Cu Ni Co Pd Ag Ru 101.

Statistics 512 Exam! September 25,2007

to", \~~!' LSI'r-=- 5 b H 2-l

T h e C S E T I P r o j e c t

Ordinary Differential Equations II

Ed H. H w H Ed. en 2: Ed. o o o z. o o. Q Ed. Ed Q to. PQ in o c3 o o. Ed P5 H Z. < u z. Ed H H Z O H U Z. > to. <! Ed Q. < Ed > Es.


Electrodynamics I Midterm - Part A - Closed Book KSU 2005/10/17 Electro Dynamic

MANY ELECTRON ATOMS Chapter 15

Math 4263 Homework Set 1

R e p u b lic o f th e P h ilip p in e s. R e g io n V II, C e n tra l V isa y a s. C ity o f T a g b ila ran

Solutions to the Sample Problems for the Final Exam UCLA Math 135, Winter 2015, Professor David Levermore

2. T H E , ( 7 ) 2 2 ij ij. p i s

Mv3" L7-- Art L 31. am rt. ao - M rr. a cn. art O' N. t00. to o( C7 c O. Ort. n ' C ( a ( W 0. z D0) Ln rni 90 O H N rt 0. to0) O mx rt N. W n.

f. D that is, F dr = c c = [2"' (-a sin t)( -a sin t) + (a cos t)(a cost) dt = f2"' dt = 2

On the convergence of solutions of the non-linear differential equation

Math Assignment. Dylan Zwick. Spring Section 2.4-1, 5, 9, 26, 30 Section 3.1-1, 16, 18, 24, 39 Section 3.2-1, 10, 16, 24, 31

o ri fr \ jr~ ^: *^ vlj o^ f?: ** s;: 2 * i i H-: 2 ~ " ^ o ^ * n 9 C"r * ^, ' 1 ^5' , > ^ t g- S T ^ r. L o n * * S* * w 3 ** ~ 4O O.

Drury&Wliitson. Economical. Planning of Buildings. .(Chilecture B. S. DNJVERSITT' OF. 11,1. 1 ibkahy

N V R T F L F RN P BL T N B ll t n f th D p rt nt f l V l., N., pp NDR. L N, d t r T N P F F L T RTL FR R N. B. P. H. Th t t d n t r n h r d r

Waves on 2 and 3 dimensional domains

Physics 342 Lecture 26. Angular Momentum. Lecture 26. Physics 342 Quantum Mechanics I

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

SOLUTION SET. Chapter 9 REQUIREMENTS FOR OBTAINING POPULATION INVERSIONS "LASER FUNDAMENTALS" Second Edition. By William T.

QUALIFYING EXAMINATION, Part 1. 1:00 pm 5:00 pm, Thursday August 31, 2017

1.1 Notes - Reaction Kinetics

Final Exam - Practice

1 A complete Fourier series solution

&GGL S. Vol.» I>l OCESE: S t. H e l e n a PARISH: St. Paul REGI STER : feorla L. D A T E D :

828.^ 2 F r, Br n, nd t h. n, v n lth h th n l nd h d n r d t n v l l n th f v r x t p th l ft. n ll n n n f lt ll th t p n nt r f d pp nt nt nd, th t

I M P O R T A N T S A F E T Y I N S T R U C T I O N S W h e n u s i n g t h i s e l e c t r o n i c d e v i c e, b a s i c p r e c a u t i o n s s h o

Affidavit and Revenue Certification Chimp Haven, Inc. Caddo Parish Keithville, LA

Midterm Solution

-r IdfJ (r-lc) / f\&() + Q sz r I~ (Y\(,J /I\q) +- fy\17. ~~/VL ~ V\ ("I) -:: f\(") e-rn~>,/'(-

OH BOY! Story. N a r r a t iv e a n d o bj e c t s th ea t e r Fo r a l l a g e s, fr o m th e a ge of 9

surface area per unit time is w(x, t). Derive the partial differential

See, Sketch, Shoot. A Study Abroad program offered by Coastline Community College and ACCENT International Consortium for Academic Programs Abroad

Math 251 December 14, 2005 Final Exam. 1 18pt 2 16pt 3 12pt 4 14pt 5 12pt 6 14pt 7 14pt 8 16pt 9 20pt 10 14pt Total 150pt

Applied Differential Equation. November 30, 2012

Časopis pro pěstování matematiky

7. pl. '2. Peraturan Pemerintah Nomor 67 Tahun 2OL3 tentang Statuta Universitas Gadjah Mada (Lembaran Negara Republik Indonesia

r Parametric, Vector, and Polar Functions (BC Only) J

Colby College Catalogue

Transcription:

ChE 230A Final Exam December 10, 2009 f have not seen as far as others, it is because giants were standing on my shoulders. - H. Abelson 1) Show that eigenvalues (λ) in the Sturm-Liouville problem below are positive if q(x) > 0. Hint: consider the inner product of the equation with y(x). y'' + q( x) y = λ y a< x< b ya ( ) = yb ( ) = 0 2) a) For functionals F[y(x)], G[y(x)], and Λ[y(x)] = F[y(x)]/G[y(x)], show that δλ= [δf - ΛδG]/G How did we use this result in class? Hint: F[y(x)]-λG[y(x)]. δ f( x0 ) b) Show that = δ[ x x0 ] δ f( x) c) Maragliano et al. [J. Chem. Phys. 2006] showed that the optimal reaction coordinate in a space of coarse grained variables minimizes the functional F [ q( z)] = dze β zq M( z ) zq (M(z) is symmetric). Express δ/δq(z) as a divergence of a field in z-space. 3) Short answer: a) For a Hermitian operator L, what must be true of the spectrum for a Green's function to exist? b) To reformulate a Sturm-Liouville problem as a matrix problem you choose basis functions {u k (x)} that satisfy the homogeneous boundary conditions. The matrix A is defined by A ij = <u i (x) L u j (x)>. Will A be Hermitian? Explain. 4) n the local density approximation (Thomas and Fermi), the potential φ(r)

around an atom satisfies 2 = φ() r 4 πρ() r 4 π Zδ[] r where ρ(r) is a charge cloud from the electrons and the delta function is from the charged nucleus at the origin. Both ρ(r) and φ(r) approach zero as r. n the homework you showed 2 (1/ r) = 4 πδ[ r ]. Use this result from your homework to give an expression for φ. Do not try to simplify your result. f you don't see how to use the result above, you might be able to give an answer by another approach. 5) Linear Response Theory: Suppose that at time t=0 a unit electric field is applied to a dielectric body. An electric dipole moment of size μ(t) in the direction of the field would be induced in the body. nstead, an oscillating field is applied, E(t) = Ecos(ωt). Give an expression for μ(t) assuming the unit step response is increases continuously from zero. For three bonus points, give the derivation of the principle you are using. 6) Use the relation from Laplace Transforms: to show that 0 st y'( t) e dt = syˆ ( s) y(0) y( ) = lim s yˆ ( s) s 0

(j) f " - Y + toy ~ /t Y <ylly,>::: A <y{y,> \. Sb { ( \/}'r~ j b /) - <Y Y ~ - '" y Y d, =-.~ ~ - ~ Y Y J< ~ <y'ly/> - }{y'(x)((2 '> 0 <Y H> '" t ~ y2~ J t > 0 <Y( y'> -= )f y(><) ''[ > 6 <yl y'/:: A ~ y ( + <yjh'> <YY'. yt >0 A [ Y( ~ }l ~.t [yc(lj GfYllt'lj '/\ - M ~c ~ 1('/, () dr () /- db 0 \?... {... SF r ~- ~ SG G G'l 'vt/(' l{ (l d 'HA ) r(15 u t JD S() /{ V(}., '~ ft'&"1 0 ( Pt) h(t,'n1 shit()j,.(\/c~ [V.f d a Jun c h"t- t1, d (<rrt ~ fro. 'f/ t-. f ()-r ex (yj/y7 plot) /n -lu. rn1h 1 hqrd r:fh r..v!- L,fJ-u.vilfe (l,cjl/tv-d Cw,'fh rt(}r vl1o!t'6(d f t' 1~/p1 {tl/11 ( (,'t'n \ ) nto v (\ (\ (J. h)}-yt 0 ( (J--fl h-m ' t (j h ~ Vi 171 0 b(vyv/ ) /\ p/~yej 111.. (o!e. ~j (1 Aoyr(l,'1'1qc YVU( lrij!;(y'.

@ tt e~ k e 1 10 lit S /Jf J h/~j/yi 12; fo vv r ' kf{k 0 ) d ( G (VA1 ( f,n 0 / '. {{Xi»:: Jdx: ~[)(-x()l /(x) -:) SiC x o)~ Jdx 6[x -x01(r(x) 4- Sf (J() ), o 0 - SdY~[)(-)(olFll<) " == Sdt6[)(-;rolSF[~) s~ ~ ~[x-xol Sf(~) COY{ S!al ~ h/'k [to f S~l - r~-l -d C"( ' (J( ) U t-o C c u ef. A (17 t'l il 1J'- : (') E ~ ;; t/~ ~ M(l) ~ )' ym'11el",(...'1 ~ f,-,,( Alok [~1:: )di f(l, ~ Z 1. :dl r h)~ /-e '/~Pl(1dul(t S~(l) /Y t:. d~k d'6l.k Le k(ll h.'f{1) rt'h 'tl-t) - - V (e- (.>F [M r;~ + (V~tMt q - - v. (2 e -flf M V~ ) (ll, t M\+ _1l V M i ~ n H ( r;:,.(cr",,< (l qd ''1 (. ~ (;( ah~lt :;:: 'V~ r(.. r (t{t C'j' f ; i'( ~ { ~ ( l) f"$: ::: J1 V' ~ ( V (e-fl~~~~)~~;-r ~k~)

@ r! (/ e o~ fi.t!. e"ju VO lu,( 'J ((1 V be 2(.-ro yes.' ~//{ if Wi (~c~t'h~. an ir1 ((';/l1llt k. brr> i) ot ~'(O(J'n7 J1C)f( - e/9u1 ~t ~r S " Ait =: <LA.,~ ~ tj.i,>"" ::: S{i ~ ~ t U; 6l< - Sut i Ur dx ::: <. u; ~ U,- '> ~ A, J l' '1 HLV y<1l{ <:4 y(,rlt d f&ra f --. l(71 E[ ; 1 'feca(( d-eh'vz/fl\~y. of ";1,,-~o\t - 1(1/ 1'0) fj re 641' $ 1"t-1/1-'7 ( h'rh () S r <; 0 - cftlfj'y, fj '\ 1'1 Q,e V 'J ha/v (h i :/.1 /(51', f '2- f rylc1'j {~t' \l. [itt's tjfqq1/( '5 t, H( ~~Yl o(so 50(-,SC,'CS flte r>. C. $. ~. ;(hliefl"i/te~ ~r d(j'rz 'f ~V1 d ~ V tea () V1 ClJJ ih~~( fjl'r((~ let 't ~ [ t)::: Jd;:o ~ ( ;- J ;0 ) ~ l( rr fj u~ tj) - if n l b[r0 1~ ) \

. 9,('f [, f,vrytttf U (f ) -: t;l( ( 1_. 1/C)' ;--5(t-p e l'!at/ 6-,/Jtt,hrvme (/5 f r "vl r ip(e ---_.. _- ----_. - -_.~-- see YLoltr. ~. d(yr1 vah (,1-'1-'1 tv ~YV1 '(rlf/(j( e 7 rfj,-l1) CD-rrY/ S ~j{'~-i! ~ e t/t dt T Sote os! Cult -to")1e- tjrdf 0 4 L " use. r'~ Ll 1-<.9 fa ( :::: 1~'2. 2. S{iJ<, 01.- +tv? S t'n tv f +ta.j '[ L _e- t/ ~ l1vy( " "- ( )7 $-)0 LSyCs) ~ yo j.p. (i M sen Vr ( f) d f - - S Y(sl - y( 0) 0 S 4 0 ~i ~ -;: Q'rv1. 1\ YCcYJ) - ){foj S Y[s) S""'JO ~~ Y ( rv,\~. e('>-n A ( ) vvj- sy.~ S' -;:> C>

~~t{1 - A C)) fad: + ~ 1 el res(j8-vtfit. 1'1-1 c...d fl. : +'-- e [,,)1:' ( i UJ( -,'w r ) --el-,.. ra'vr d ph (J ) e s?t r/t A ei1j~. = 0 2 j ~ e -i~ 3 := 0 prt141vwt (r ), J ~ ~ (c!~ + e-i<p) Lv r ::: i ~ A frr, 0/

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback