LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS DONE RIGHT KYLE ORMSBY
|
|
- Debra James
- 5 years ago
- Views:
Transcription
1 LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS DONE RIGHT KYLE ORMSBY INTRODUCTION TO THE PROBLEM Considr a continous function F : R n R n W will think of F as a vctor fild, and can think of F x as a vlocity vctor positiond at x R n Our goal is to find a path γ : R R n which starts at a point p R n so γ0 p, our initial condition and, for ach tim t R, satisfis th diffrntial quation γ t F γt If γ x, x 2,, x n, thn w may rwrit as th systm of scalar quations x F x,, x n x 2 F 2 x,, x n x n F n x,, x n W may think of this systm as th tim-drivativs of n dpndnt variabls simultanously satisfying n position-dpndnt quations A solution γ is thn a flow lin, th path of a particl pushd around by th vctor fild F so that its vlocity matchs F at any givn point Exampl As an xampl, considr th vctor fild F : R 2 R 2 taking x, y to F x, y x, y If our initial position is p, p 2, thn w ar sking γ : R R 2 taking t to γt xt, yt satisfying x0 p, y0 p 2, x t xt, and y t yt Figur dpicts th vctor fild F by drawing scald down vrsions of th vctors F x, y at a sampling of points in th squar [0, 2] [0, 2] Th curv drawn within th figur is a solution satisfying th initial condition γ0, This particular systm of diffrntial quations may b solvd on variabl at a tim sinc x only dpnds on x and y only on y Indd, sparation of variabls quickly givs th solution x p t, y p 2 t W may thn not that xy p p 2 is a constant, so th curv tracd by γ is a hyprbola Th abov xampl fits into a class of xampls of th form x a x + a 2 y + b y a 2 x + a 22 y + b in which th a ij and b k ar all scalar constants This may thn b rphrasd in matrix form as x a a y 2 x b + a 22 y b 2 a 2 This admits th furthr gnralization to n variabls x Ax + b
2 FIGURE Th vctor fild F x, y x, y and a flow lin with initial position, whr x a a 2 a n b x 2 x, A a 2 a 22 a 2n, b b 2, x n a n a n2 a nn b n and x is th column vctor consisting of th tim drivativs of x,, x n This is a vry spcial sort of diffrntial quation in which th vctor fild F is an affin transformation If w spcializ yt furthr and tak b 0 so that F is linar, thn w arriv at our currnt objct of study Dfinition 2 A homognous linar systm of diffrntial quations is on of th form whr x x,, x n and A is an n n matrix x Ax Rmark 3 Whn b 0, th litratur calls th systm x Ax + b an inhomognous linar systm of diffrntial quations From our point of viw, it would b nicr to call this an affin systm of diffrntial quations, but w shall quail in th fac of cnturis worth of tradition In ths nots but prhaps not lswhr! w shall assum that all linar systms ar homognous unlss xplicitly notd othrwis Exampl 4 Hr is on of thos patntly ridiculous practical xampls that should nonthlss illustrat th typs of problms which fit into th framwork of linar systms of diffrntial quations: A picklmakr has two brin tanks, on containing 00 gallons of brin, th othr containing 200 gallons of brin A systm of pips and pumps conncts th tanks so that th following procsss happn: a frsh watr ntrs th first tank at a rat of 20 gallons pr minut, b solution movs from th first tank to th scond at a rat of 30 gallons pr minut, on who maks pickls 2
3 c solution movs via a sparat pip from th scond tank to th first at a rat of 0 gallons pr minut, and d solution is dumpd from th scond tank at a rat of 20 gallons pr minut Not that total solution volum is consrvd in this systm If th first tank contains x pounds of salt and th scond tank contains y pounds of salt, thn th dvlopmnt of th systm may b dscribd by th following systm of quations: x 30 x y 200 y 30 x y 200 Simplifying fractions and translating into matrics, w hav x 3/0 /20 x y 3/0 3/20 y Th picklmakr rsts this systm at midnight on January, 206 so that th first tank contains 0 pounds of salt and th scond contains 5 pounds of salt If h lts th systm run for a yar, how much salt will b in th tanks? 2 THE OPERATOR NORM AND MATRIX EXPONENTIATION In ordr to hlp th picklmakr, w ar going to dvlop a thory of matrix xponntiation In ordr to nsur that ths mthods ar lgitimat th picklmakr apprciats our hlp but dmands authnticity, w will tak a dtour through th world of oprator norms Rcall that th complx xponntial function taks any complx numbr t to t k! tk If L : R n R n is a linar transformation, dfin L 0 id, th idntity linar transformation, and inductivly dfin L k L L k for k > 0 Thus L L, L 2 L L, L 3 L L L, tc Dfinition 5 Th xponntial of a linar transformation L : R n R n is L : k! Lk Som commnts on intrprtation ar ncssary, som trivial, som dp First, not that ach factor /k! is a scalar, and if M is a linar transformation and λ is a scalar, thn λm taks v to λmv; this givs maning to th trms k! Lk Also rcall that w ar prfctly comfortabl adding togthr finitly many linar transformations: L + M : v Lv + Mv Thus for any intgr N, th sum N k! Lk maks sns as a linar transformation What, though, is mant whn w lt N? In ordr to mak sns of such a limit, w will nd a notion of distanc btwn linar transformations This notion is providd by th oprator norm In ordr to motivat it, rcall th following lmma which w provd arlir in th trm 3
4 Lmma 6 For any linar transformation L : R n R m, thr xists a constant c R such that for all v R n, Lv c v By taking th smallst possibl such c, w masur th maximal factor by which L dilats vctors; this is th oprator norm, dfind formally blow Dfinition 7 Th oprator norm of a linar transformation L : R n R m is It follows immdiatly that for any v R n, L : inf{c 0 Lv c v for all v R n } Lv L v In your homwork, you will prov that L may b computd in th following ways as wll Proposition 8 For any linar transformation L : R n R m, L sup{ Lv v } sup{ Lv v } sup{ Lv / v v R n {0}} Th oprator norm is, in fact, a norm, maning that it satisfis th following proprtis as wll Proposition 9 For any linar transformations L, M : R n R m and any scalar λ R, i L 0 with quality if and only if L 0, ii λl λ L, iii L + M L + M Proof Th first proprty is obvious, and you will prov th scond on in your homwork For part iii, obsrv that for any v R n, L + Mv Lv + Mv Lv + Mv L v + M v L + M v Sinc L+M is th gratst lowr bound on all c 0 such that L+Mv c v and L + M is such a c, w s that w must hav L + M L + M Whnvr w hav a norm, w gt a notion of distanc This is xactly how w dvlopd th notion of distanc in R n Indd, with th following dfinition and th proprtis listd blow in Proposition, w hav all th ncssary notions ndd to talk intlligntly about convrgnc of th sris dfining L Dfinition 0 Th oprator distanc btwn two linar transformations L, M : R n R m is dl, M : L M Proposition For any linar transformations L, M, N : R n R m w hav i dl, M 0 with quality if and only if L M, ii dl, M dm, L, and iii dl, N dl, M + dm, N Proof Ths proprtis follow immdiatly from thir analogus in Proposition 9 4
5 Additionally, th oprator norm has an xtra fatur that will prov spcially usful in our analysis of th xponntial on linar oprators; namly, th oprator norm is submultiplicativ Thorm 2 For any linar transformations L : R n R m and M : R k R n, L M L M Thinking in trms of dilation factors, this thorm should sm fairly obvious Th proof is a on-linr using Proposition 8 Proof W hav L M sup LMv sup L Mv L sup Mv L M v v v Hr th first inquality follows from our old obsrvation that Lw L w with w Mv Th othr manipulations ar standard W now com upon a surprising proprty of convrgnc with rspct to th oprator distanc W will only sktch th proof sinc a full tratmnt dpnds on Cauchy compltnss of th oprator mtric spac, and w can only do so much in th short tim w v bn givn Proposition 3 Lt L k : R n R n k 0 b a squnc of linar transformations If k 0 L k convrgs as a squnc of ral numbrs, thn k 0 L k convrgs with rspct to th oprator distanc Sktch of proof W would lik to show that thr is a linar transformation L : R n R n such that for all ε > 0 thr xists N N such that n L k L < ε whnvr n > N Lacking a candidat for L, w will instad show that n n L k L k < ε whnvr n, n ar sufficintly larg In othr words, th squnc of partial sums gts arbitrarily clos to itslf for larg indics This is in fact an quivalnt condition by Cauchy compltnss of th oprator mtric spac, but w will not prov it Tak ε > 0 By hypothsis, k 0 L k convrgs, so by Cauchy compltnss of th ral numbrs, w may choos N N such that whn n n > N, For th sam n, n, n L k ε > L k n n L k n L k n kn+ L k so L k convrgs with rspct to th oprator distanc n kn+ n kn+ L k L k < ε, W ar now wll-poisd to prov convrgnc of th oprator xponntial 5
6 Thorm 4 For any linar transformation L : R n R n, L k! Lk convrgs to a linar transformation Proof By Proposition 3, it suffics to prov that k 0 k! Lk convrgs By submultiplicativity Thorm 2 and Proposition 9ii, k! L k But k! Lk k! L k L convrgs so th comparison tst for sris implis that k 0 k! Lk convrgs and w ar don Implicit in our sktch proof of Proposition 3 is th fact that linar transformation R n R m form a complt mtric spac undr th distanc function inducd by th oprator norm In othr words, Cauchy squncs rlativ to th oprator norm of linar transformations convrg to linar transformations Givn Thorm 4, this mans that L is a linar transformation R n R n for any linar transformation L : R n R n As such, L has a matrix, which w provid notation for blow Dfinition 5 If L : R n R n is a linar transformation with matrix A, thn w lt A dnot th matrix associatd with L and call it th xponntial matrix of A Of cours, A has a sris dscription as A k! Ak whr A k dnots itratd matrix multiplication But w must b carful in how w intrprt this sris as it stands, it only maks sns whn w translat back to linar transformations and considr convrgnc rlativ to th oprator norm It is natural to ask for mor Qustion 6 Suppos L k is a squnc of linar transformations R n R m with associatd squnc of m n matrics A k Furthr suppos that L k L as k rlativ to th oprator norm Lt a ij k dnot th ij-ntry of A k, and lt a ij dnot th ij-ntry of th matrix A associatd with L Undr ths circumstancs, is lim k aij k aij? Answr 7 Ys It turns out that all norms on finit dimnsional vctor spacs induc th sam topology, and thus hav th sam convrgnt squncs W will rturn to ths idas at th nd of ths nots, tim prmitting W now comput a coupl of asy xampls of xponntial matrics a 0 a Exampl 8 Suppos A is a 2 2 ral matrix Thn A 0 b k k 0 0 b k chck this and A a k! Ak k /k! 0 a 0 0 b k /k! 0 b 6
7 Similarly, if A is an n n diagonal matrix with diagonal ntris a,, a n, thn A is a diagonal matrix with diagonal ntris a,, an Exampl 9 An n n matrix A is calld nilpotnt if thr is a positiv intgr l such that A l 0 Hr 0 mans th n n matrix with all 0 ntris If A is nilpotnt, thn th infinit sum giving A has only finitly many nonzro trms and may b computd xplicitly by just adding up thos trms, i, A l k! Ak thn For instanc, if Thus A , A and A A I + A + 2 A SOLVING LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS WITH EXPONENTIAL MATRICES 3 Th main thorm Lt M n n R dnot th st of n n ral matrics For a fixd A M n n R, w may dfin a matrix-valud function γ A : R M n n R t At whr At is th scalar multipl of A by t Thus At k! Atk k! Ak t k Hr w ar using th fact that scalar multiplication commuts with matrix multiplication Idntifying M n n R with R n2 why can w do this?, w may viw γ A as a path As such, it maks 7
8 sns to diffrntiat: γ At d dt At d dt k A k! Ak t k k! Ak d dt tk k k! Ak t k k! Ak t k k k! Ak t k A At Svral stps hr rquir justification First and formost, w hav not justifid th third quality, which claims that diffrntiation commuts with infinit sris This is a dpr point that w will rturn to latr Th final stp also dsrvs attntion Obsrv that k rangs from to, but all of th apparancs of k in th sum ar in fact k s This rcovrs th xponntial function Givn this computation, w may prov th following thorm Thorm 20 Lt A M n n R and lt p 0 R n Thn th function x : R R n t At p 0 is a solution to th diffrntial quation x Ax with initial condition x0 p 0 Proof First not that by Exampl 8, 0 I whr 0 is th n n matrix of all 0 s Now obsrv that x0 A 0 p 0 Ip 0 p 0, so x satisfis th initial condition Finally, w may comput x d dt At p 0 A At p 0 Ax, as dsird Exampl 2 Considr th matrix A of Exampl 9 It is asy to comput At I + At + 3t 4t + 9t2 2 A2 t 2 0 6t 0 0 As such, w can us Thorm 20 to solv th diffrntial quation x 3y + 4z y 6z z 0 8
9 FIGURE 2 A vctor fild and flow lin givn by xponntiation of a nilpotnt matrix In particular, if our initial condition is γ0,,, thn γt At + 7t + 9t 2, + 6t, is a flow lin for this systm Figur 2 sktchs th vctor fild A x y and th flow lin γ z 32 Proprtis of th matrix xponntial W will us th following thorm to comput mor complicatd xponntial matrics and thus solv mor complicatd linar systms of diffrntial quations Thorm 22 If A, B M n n R and AB BA, thn A+B A B Proof Th ssntial obsrvation is that whn A and B commut, th binomial thorm applis so that k k k A + B k A l B l k k! l l!k l! Al B l k Hnc l0 A+B A + Bk k! k k! l0 l0 l! Al r+sk 9 l0 k! l!k l! Al B l k k l! Bl k r! Ar s! Bs
10 Obsrv that in this final doubl sum, th pairs r, s rang through all of N N xactly onc Thus laving som algbraic vrifications to th radr w may writ A+B r! Ar s! Bs r! Ar s! Bs A B, as dsird r0 s0 Rmark 23 Th abov thorm is gnrally fals if A and B do not commut! r0 Corollary 24 For all A M n n R, A is invrtibl with A A I Proof W hav A A AA, so Thorm 22 implis that I 0 A A A A Exampl 25 Lt s us this thorm to comput th xponntial of a 0 0 b + and 0 0 Thus a 0 0 b 0 0 a b b 0 0 a 0 0 b b s0 a b First not that a 0 a b a 0 Sinc is diagonal, w hav a a 0 0 a Sinc 2 0 b 0, w hav 0 0 Thus w may conclud that a b Mor gnrally, Thus th diffrntial quation 0 b 0 0 I + 0 b 0 0 b 0 a 0 b a 0 a a b 0 0 a a b t at at bt 0 at x a b x y y 0
11 FIGURE 3 Vctor fild and flow lin whn a b, p /4, and p 2 /4 with initial condition p, p 2 at tim 0 has solution a b t p p 2 at at bt p p 0 at at + p 2 at bt p 2 p 2 at x In Figur 3, w st a b Th plot dpicts th vctor fild along with a flow lin 0 y passing through /4, /4 at t 0 Explicitly, this solution is givn by t t /4 + t t/4, t /4 33 Priodic matrics Exampl 26 W will now considr a systm of diffrntial quations w hav sn bfor: x y y x W hav alrady sn that th solutions to this systm ar circls Dos th matrix xponntial rcovr ths solutions? W may r-xprss th abov systm as x y Obsrv that , 0 0 x 0 y , Sinc th fourth powr of this matrix is th idntity matrix, th pattrn now rpats, i, if A 0, thn A 0 4k+j A j whr k N and j is 0,, 2, or 3 Thus w may comput th xponntial At k k! Ak t k 2k! t2k k 2k+! t2k+ cos t sin t sin t cos t k 2k+! t2k+ k 2k! t2k
12 FIGURE 4 An llips as a solution to a homognous linar systm of diffrntial quations? Th sums hr look complicatd, but thy ar just rcording th various ± s, 0 s, factorials, and powrs of t which appar in th summation If at tim 0 w ar at th point r, 0, thn Thorm 20 tlls us that our systm has solution t At r 0 r cos t r sin t This prcisly dscribs a circl of radius r cntrd at th origin, which matchs our old solution of this problm Qustion 27 Do you think thr ar homognous systms of linar quations whos solutions ar llipss? If so, how might you form thm? If this sms mystrious right now, com back to this qustion aftr you hav larnd what a chang of basis is in Linar Algbra /2 0 Lt M which has invrs M /2 x Figur 4 plots th vctor fild MAM x y chck this! and considr th systm and a solution to this systm x y MAM y passing through, t t 0 givn by t cost + 2 sint, 5 sint This should sm quit tantalizing 34 A linar algbraic approach to th gnral 2 2 cas Lt A b an n n matrix and suppos A QBQ for Q, B n n matrics with Q invrtibl Thn A 2 QBQ QBQ QBQQ BQ QB 2 Q sinc QQ I Similarly, for any k 0, A k QB k Q 2
13 Thus At k! Ak t k k! QBk Q t k Q k! Bk t k Q Q Bt Q As such, if w can find B and Q such that Bt is computabl and A QBQ, thn w will b abl to comput At In th 2 2 cas, th following thorm prmits us to do just this a b Thorm 28 Lt A b a 2 2 matrix Thn A QBQ whr B is of th form c d λ 0 0 λ 2 or λ 0 λ Hr th ral or complx numbrs λ i ar th roots of th charactristic polynomial of A: x 2 a + dx + ad bc Proof This is th 2 2 cas of a powrful thorm from linar algbra about so-calld Jordan canonical form of matrics W will not commnt on th proof hr Rmark 29 As statd, th thorm dos not tll us anything about how to find Q In th first cas, λ 0 whr B this is rlativly simpl In this cas, it is possibl to find a basis of R 0 λ 2 of 2 ignvctors of B An ignvctor of B is a vctor v such that Bv λv for som scalar λ; th scalar λ 0 λ in such an quation is calld an ignvalu Th ignvalus of ar prcisly λ 0 λ and λ 2 2 chck this! If v and v 2 ar linarly indpndnt vctors such that Bv λ v and Bv 2 λ 2 v 2, thn w may tak Q to b th matrix with columns v and v 2 λ In th scond cas, in which B, th matrix Q is constructd in a diffrnt fashion 0 λ Th abov mthod dos not work bcaus B has only on ignvalu, λ, and is not alrady diagonal, and thus cannot hav a basis of ignvctors Whil it is not particularly complicatd, w will not study this mthod in ths nots 0 Exrcis Us ths mthods to comput th xponntial matrix of You will find λ 0 i, λ 2 i, so that Bt it it Aftr finding an invrtibl matrix Q such that 0 QBQ and computing At Q Bt Q you will b confrontd with a matrix of th form 2 it + it i 2 it it i 2 it + it 2 it + it Rcalling th idntity it cost + i sint you should s how this matrix simplifis to prvious computation of At Exampl 30 Considr th systm of diffrntial quations x 4x + 2y y 3x y 4 2 which is inducd by th matrix A This matrix has charactristic polynomial 3 x 2 3x 0 λ + 2λ 5 3
14 with roots λ 2, λ 2 5 Thus 2 0 B 0 5 W can comput Q and Q to b 2 Q 3 so At Q Bt Q 7 and Bt and Q 2t 0 0 5t /7 2/7 3/7 /7 2t + 6 5t 2 2t + 2 5t 3 2t + 3 5t 6 2t + 5t Th solution to th systm of diffrntial quations with initial conditions x0 7 and y0 0 is thus xt 2t + 6 5t 2 2t + 2 5t 7 yt 7 3 2t + 3 5t 6 2t + 5t 2t + 6 5t 0 3 2t + 3 5t It would b nic to additionally undrstand th curv dscribd by this solution To driv this, first not that th systm givn by B with initial condition p, p 2 has solution xt, yt p 2t, p 2 5t In this cas, x 5 y 2 p 5 p 2 2 which givs us a handl on th curv dscribd Th nxt thing to obsrv is that γt is a solution to th systm givn by B if and only if Qγt is a solution for th systm givn by A Why is this th cas? Hnc th solution curvs for A diffr from th ons for B via th linar chang of coordinats Q Now would b an opportun momnt to rturn to Qustion 27 and to considr what curvs ar dscribd whn a linar chang of coordinats is applid to a circl 4 THEORETICAL ODDS AND ENDS Givn th matrix xponntial s facility in solving diffrntial quations, w can now s why it was important to prov its convrgnc Rcall, though, that w only provd convrgnc with rspct to th oprator norm? Is this th sam thing as convrgnc with rspct to a mor traditional norm, lik th on inducd by considring n n matrics as points in R n2? W will sktch th proof of a thorm which, lik Voltair s Pangloss, assurs us that w liv in th bst of all possibl worlds Loosly spaking, it tlls us that all topologis on finit dimnsional ral vctor spacs which ar inducd by norms ar quivalnt Thus convrgnc with rspct to on norm is quivalnt to convrgnc with rspct to all othr norms In ordr to put mat on this statmnt, lt us rcall th dfinition of a norm Dfinition 3 A norm on R n is a function : R n R satisfying th following proprtis: for all x R n, x 0 with quality if and only if x 0, 2 for all x R n and λ R, λx λ x, and 3 for all x, y R n, x + y x + y Th Euclidan norm on R n is, of cours, a norm A priori, th oprator norm is dfind on th st of linar transformations R n R n Using our dictionary btwn linar transformations and matrics, though, w s that ach linar map R n R n corrsponds to a uniqu point in R n2 Thus it maks sns to think of th oprator norm as a norm on R n2 4
15 Whnvr w hav a norm, it inducs a distanc function d x, y x y, and w can dfin opn balls and a topology with rspct to this distanc function W can dfin limits of squncs with rspct to this norm via th usual ε-n formalism Dfinition 32 Two norms and ar quivalnt if thr ar positiv constants c and C such that c x x C x Rmark 33 Th apparnt asymmtry in this dfinition is an illusion Givn th chain of inqualitis c x x C x, w gt C x x c x for fr As such, quivalnc of norms is in fact an quivalnc rlation on th st of norms chck this! Th significanc of this dfinition is mad apparnt by th following thorm Thorm 34 If two norms and on R n ar quivalnt, thn a squnc a k is convrgnt with rspct to if and only if it is convrgnt with rspct to, and 2 a subst of R n is opn with rspct to if and only if it is opn with rspct to Th proof follows dirctly from th dfinitions chck this! and w will not prsnt it hr W can now proprly apprciat our Panglossian thorm: Thorm 35 For n 0, any two norms on R n ar quivalnt Corollary 36 Th matrix xponntial convrgs with rspct to all norms on R n2 Euclidan on including th usual Sinc it will b ssntial in our proof of th thorm, lt us rcall th Bolzano-Wirstrass thorm which you larnd in som for in Math 2 bfor procding Thorm 37 Bolzano-Wirstrass Evry boundd squnc in R has a convrgnt subsqunc Hr w man convrgnc with rspct to th standard absolut valu on R For x R n, w dfin x x + + x n ; it is calld th l norm or th taxi-cab norm think of th distanc your taxi must tak to travl btwn points in Manhattan Chck that is in fact a norm W will nd th following nhancd vrsion of th Bolzano-Wirstrass thorm Thorm 38 Bolzano-Wirstrass on R n with rspct to th norm In R n, any -boundd squnc has a -convrgnt subsqunc Proof Lt {x k } b a -boundd squnc in R n Lt x j k b th j-th coordinat of x k, j n Not that x k x k, so {x k } is a boundd squnc in R and thus has a convrgnt subsqunc {x lk } by Thorm 37 Now considr th squnc {x 2 l k } W again hav that x 2 l k x lk is boundd, so thr is a convrgnt subsqunc x hlk If n 2, w ar don with convrgnt subsqunc x h lk, x 2 h lk chck this! Othrwis, kp rpating th procss until it has bn don to all of th coordinats Rmark 39 It is a consqunc of Thorm 35 that th Bolzano-Wirstrass thorm on R n holds with rspct to any norm W ar now prpard to prov our main thorm 5
16 Proof Sinc quivalnc of norms is transitiv, it suffics to show that an arbitrary norm is quivalnt to W first chck that thr is a constant c such that c x x for all x R n Obsrv that x x i i, so th triangl inquality implis that x x i i If m max{ i }, thn w furthr hav x x i m m x Hnc w may tak c /m and driv that c x x It rmains to show that thr is a positiv constant C such that x C x for all x R n Suppos for contradiction that no such C xists In that cas, for ach k N w can find x k R n such that x k > k x k Lt y k x k / x k Thn y k so th squnc {y k } is boundd in th norm and thus has a subsqunc y lk convrging in to an lmnt y Sinc y lk, w also hav y chck this! and, in particular, y 0 Not, though, that y lk < /l k, so y 0, whnc y 0, a contradiction Thus thr must b som C > 0 such that x C x for all x, complting our proof 6
The Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationMA 262, Spring 2018, Final exam Version 01 (Green)
MA 262, Spring 218, Final xam Vrsion 1 (Grn) INSTRUCTIONS 1. Switch off your phon upon ntring th xam room. 2. Do not opn th xam booklt until you ar instructd to do so. 3. Bfor you opn th booklt, fill in
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #20 11/14/2013
18.782 Introduction to Arithmtic Gomtry Fall 2013 Lctur #20 11/14/2013 20.1 Dgr thorm for morphisms of curvs Lt us rstat th thorm givn at th nd of th last lctur, which w will now prov. Thorm 20.1. Lt φ:
More informationAddition of angular momentum
Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationQuasi-Classical States of the Simple Harmonic Oscillator
Quasi-Classical Stats of th Simpl Harmonic Oscillator (Draft Vrsion) Introduction: Why Look for Eignstats of th Annihilation Oprator? Excpt for th ground stat, th corrspondnc btwn th quantum nrgy ignstats
More informationMATH 319, WEEK 15: The Fundamental Matrix, Non-Homogeneous Systems of Differential Equations
MATH 39, WEEK 5: Th Fundamntal Matrix, Non-Homognous Systms of Diffrntial Equations Fundamntal Matrics Considr th problm of dtrmining th particular solution for an nsmbl of initial conditions For instanc,
More informationcycle that does not cross any edges (including its own), then it has at least
W prov th following thorm: Thorm If a K n is drawn in th plan in such a way that it has a hamiltonian cycl that dos not cross any dgs (including its own, thn it has at last n ( 4 48 π + O(n crossings Th
More informationRecall that by Theorems 10.3 and 10.4 together provide us the estimate o(n2 ), S(q) q 9, q=1
Chaptr 11 Th singular sris Rcall that by Thorms 10 and 104 togthr provid us th stimat 9 4 n 2 111 Rn = SnΓ 2 + on2, whr th singular sris Sn was dfind in Chaptr 10 as Sn = q=1 Sq q 9, with Sq = 1 a q gcda,q=1
More information2.3 Matrix Formulation
23 Matrix Formulation 43 A mor complicatd xampl ariss for a nonlinar systm of diffrntial quations Considr th following xampl Exampl 23 x y + x( x 2 y 2 y x + y( x 2 y 2 (233 Transforming to polar coordinats,
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationECE602 Exam 1 April 5, You must show ALL of your work for full credit.
ECE62 Exam April 5, 27 Nam: Solution Scor: / This xam is closd-book. You must show ALL of your work for full crdit. Plas rad th qustions carfully. Plas chck your answrs carfully. Calculators may NOT b
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationHydrogen Atom and One Electron Ions
Hydrogn Atom and On Elctron Ions Th Schrödingr quation for this two-body problm starts out th sam as th gnral two-body Schrödingr quation. First w sparat out th motion of th cntr of mass. Th intrnal potntial
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More informationCOMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.
C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH
More informationBrief Introduction to Statistical Mechanics
Brif Introduction to Statistical Mchanics. Purpos: Ths nots ar intndd to provid a vry quick introduction to Statistical Mchanics. Th fild is of cours far mor vast than could b containd in ths fw pags.
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationDerangements and Applications
2 3 47 6 23 Journal of Intgr Squncs, Vol. 6 (2003), Articl 03..2 Drangmnts and Applications Mhdi Hassani Dpartmnt of Mathmatics Institut for Advancd Studis in Basic Scincs Zanjan, Iran mhassani@iasbs.ac.ir
More informationExercise 1. Sketch the graph of the following function. (x 2
Writtn tst: Fbruary 9th, 06 Exrcis. Sktch th graph of th following function fx = x + x, spcifying: domain, possibl asymptots, monotonicity, continuity, local and global maxima or minima, and non-drivability
More informationSection 6.1. Question: 2. Let H be a subgroup of a group G. Then H operates on G by left multiplication. Describe the orbits for this operation.
MAT 444 H Barclo Spring 004 Homwork 6 Solutions Sction 6 Lt H b a subgroup of a group G Thn H oprats on G by lft multiplication Dscrib th orbits for this opration Th orbits of G ar th right costs of H
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More informationCombinatorial Networks Week 1, March 11-12
1 Nots on March 11 Combinatorial Ntwors W 1, March 11-1 11 Th Pigonhol Principl Th Pigonhol Principl If n objcts ar placd in hols, whr n >, thr xists a box with mor than on objcts 11 Thorm Givn a simpl
More informationu 3 = u 3 (x 1, x 2, x 3 )
Lctur 23: Curvilinar Coordinats (RHB 8.0 It is oftn convnint to work with variabls othr than th Cartsian coordinats x i ( = x, y, z. For xampl in Lctur 5 w mt sphrical polar and cylindrical polar coordinats.
More informationEinstein Equations for Tetrad Fields
Apiron, Vol 13, No, Octobr 006 6 Einstin Equations for Ttrad Filds Ali Rıza ŞAHİN, R T L Istanbul (Turky) Evry mtric tnsor can b xprssd by th innr product of ttrad filds W prov that Einstin quations for
More informationWeek 3: Connected Subgraphs
Wk 3: Connctd Subgraphs Sptmbr 19, 2016 1 Connctd Graphs Path, Distanc: A path from a vrtx x to a vrtx y in a graph G is rfrrd to an xy-path. Lt X, Y V (G). An (X, Y )-path is an xy-path with x X and y
More informationCPSC 665 : An Algorithmist s Toolkit Lecture 4 : 21 Jan Linear Programming
CPSC 665 : An Algorithmist s Toolkit Lctur 4 : 21 Jan 2015 Lcturr: Sushant Sachdva Linar Programming Scrib: Rasmus Kyng 1. Introduction An optimization problm rquirs us to find th minimum or maximum) of
More informationSCHUR S THEOREM REU SUMMER 2005
SCHUR S THEOREM REU SUMMER 2005 1. Combinatorial aroach Prhas th first rsult in th subjct blongs to I. Schur and dats back to 1916. On of his motivation was to study th local vrsion of th famous quation
More informationThus, because if either [G : H] or [H : K] is infinite, then [G : K] is infinite, then [G : K] = [G : H][H : K] for all infinite cases.
Homwork 5 M 373K Solutions Mark Lindbrg and Travis Schdlr 1. Prov that th ring Z/mZ (for m 0) is a fild if and only if m is prim. ( ) Proof by Contrapositiv: Hr, thr ar thr cass for m not prim. m 0: Whn
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More informationu x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula
7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting
More informationNEW APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA
NE APPLICATIONS OF THE ABEL-LIOUVILLE FORMULA Mirca I CÎRNU Ph Dp o Mathmatics III Faculty o Applid Scincs Univrsity Polithnica o Bucharst Cirnumirca @yahoocom Abstract In a rcnt papr [] 5 th indinit intgrals
More informationHomogeneous Constant Matrix Systems, Part I
39 Homognous Constant Matrix Systms, Part I Finally, w can start discussing mthods for solving a vry important class of diffrntial quation systms of diffrntial quations: homognous constant matrix systms
More informationCS 361 Meeting 12 10/3/18
CS 36 Mting 2 /3/8 Announcmnts. Homwork 4 is du Friday. If Friday is Mountain Day, homwork should b turnd in at my offic or th dpartmnt offic bfor 4. 2. Homwork 5 will b availabl ovr th wknd. 3. Our midtrm
More informationLecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields
Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration
More information1 Isoparametric Concept
UNIVERSITY OF CALIFORNIA BERKELEY Dpartmnt of Civil Enginring Spring 06 Structural Enginring, Mchanics and Matrials Profssor: S. Govindj Nots on D isoparamtric lmnts Isoparamtric Concpt Th isoparamtric
More informationDeift/Zhou Steepest descent, Part I
Lctur 9 Dift/Zhou Stpst dscnt, Part I W now focus on th cas of orthogonal polynomials for th wight w(x) = NV (x), V (x) = t x2 2 + x4 4. Sinc th wight dpnds on th paramtr N N w will writ π n,n, a n,n,
More informationFunction Spaces. a x 3. (Letting x = 1 =)) a(0) + b + c (1) = 0. Row reducing the matrix. b 1. e 4 3. e 9. >: (x = 1 =)) a(0) + b + c (1) = 0
unction Spacs Prrquisit: Sction 4.7, Coordinatization n this sction, w apply th tchniqus of Chaptr 4 to vctor spacs whos lmnts ar functions. Th vctor spacs P n and P ar familiar xampls of such spacs. Othr
More informationCOUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM
COUNTING TAMELY RAMIFIED EXTENSIONS OF LOCAL FIELDS UP TO ISOMORPHISM Jim Brown Dpartmnt of Mathmatical Scincs, Clmson Univrsity, Clmson, SC 9634, USA jimlb@g.clmson.du Robrt Cass Dpartmnt of Mathmatics,
More informationBasic Polyhedral theory
Basic Polyhdral thory Th st P = { A b} is calld a polyhdron. Lmma 1. Eithr th systm A = b, b 0, 0 has a solution or thr is a vctorπ such that π A 0, πb < 0 Thr cass, if solution in top row dos not ist
More informationSupplementary Materials
6 Supplmntary Matrials APPENDIX A PHYSICAL INTERPRETATION OF FUEL-RATE-SPEED FUNCTION A truck running on a road with grad/slop θ positiv if moving up and ngativ if moving down facs thr rsistancs: arodynamic
More informationChapter 10. The singular integral Introducing S(n) and J(n)
Chaptr Th singular intgral Our aim in this chaptr is to rplac th functions S (n) and J (n) by mor convnint xprssions; ths will b calld th singular sris S(n) and th singular intgral J(n). This will b don
More informationperm4 A cnt 0 for for if A i 1 A i cnt cnt 1 cnt i j. j k. k l. i k. j l. i l
h 4D, 4th Rank, Antisytric nsor and th 4D Equivalnt to th Cross Product or Mor Fun with nsors!!! Richard R Shiffan Digital Graphics Assoc 8 Dunkirk Av LA, Ca 95 rrs@isidu his docunt dscribs th four dinsional
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More information1 Minimum Cut Problem
CS 6 Lctur 6 Min Cut and argr s Algorithm Scribs: Png Hui How (05), Virginia Dat: May 4, 06 Minimum Cut Problm Today, w introduc th minimum cut problm. This problm has many motivations, on of which coms
More informationProblem Set 6 Solutions
6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr
More information(1) Then we could wave our hands over this and it would become:
MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and
More informationPropositional Logic. Combinatorial Problem Solving (CPS) Albert Oliveras Enric Rodríguez-Carbonell. May 17, 2018
Propositional Logic Combinatorial Problm Solving (CPS) Albrt Olivras Enric Rodríguz-Carbonll May 17, 2018 Ovrviw of th sssion Dfinition of Propositional Logic Gnral Concpts in Logic Rduction to SAT CNFs
More informationContent Skills Assessments Lessons. Identify, classify, and apply properties of negative and positive angles.
Tachr: CORE TRIGONOMETRY Yar: 2012-13 Cours: TRIGONOMETRY Month: All Months S p t m b r Angls Essntial Qustions Can I idntify draw ngativ positiv angls in stard position? Do I hav a working knowldg of
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim (implicit in notation and n a positiv intgr, lt ν(n dnot th xponnt of p in n, and U(n n/p ν(n, th unit
More information10. The Discrete-Time Fourier Transform (DTFT)
Th Discrt-Tim Fourir Transform (DTFT Dfinition of th discrt-tim Fourir transform Th Fourir rprsntation of signals plays an important rol in both continuous and discrt signal procssing In this sction w
More informationConstruction of asymmetric orthogonal arrays of strength three via a replacement method
isid/ms/26/2 Fbruary, 26 http://www.isid.ac.in/ statmath/indx.php?modul=prprint Construction of asymmtric orthogonal arrays of strngth thr via a rplacmnt mthod Tian-fang Zhang, Qiaoling Dng and Alok Dy
More informationBackground: We have discussed the PIB, HO, and the energy of the RR model. In this chapter, the H-atom, and atomic orbitals.
Chaptr 7 Th Hydrogn Atom Background: W hav discussd th PIB HO and th nrgy of th RR modl. In this chaptr th H-atom and atomic orbitals. * A singl particl moving undr a cntral forc adoptd from Scott Kirby
More information10. Limits involving infinity
. Limits involving infinity It is known from th it ruls for fundamntal arithmtic oprations (+,-,, ) that if two functions hav finit its at a (finit or infinit) point, that is, thy ar convrgnt, th it of
More informationThe graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the
Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th
More informationHomework #3. 1 x. dx. It therefore follows that a sum of the
Danil Cannon CS 62 / Luan March 5, 2009 Homwork # 1. Th natural logarithm is dfind by ln n = n 1 dx. It thrfor follows that a sum of th 1 x sam addnd ovr th sam intrval should b both asymptotically uppr-
More informationBifurcation Theory. , a stationary point, depends on the value of α. At certain values
Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local
More informationDifferential Equations
Prfac Hr ar m onlin nots for m diffrntial quations cours that I tach hr at Lamar Univrsit. Dspit th fact that ths ar m class nots, th should b accssibl to anon wanting to larn how to solv diffrntial quations
More informationBSc Engineering Sciences A. Y. 2017/18 Written exam of the course Mathematical Analysis 2 August 30, x n, ) n 2
BSc Enginring Scincs A. Y. 27/8 Writtn xam of th cours Mathmatical Analysis 2 August, 28. Givn th powr sris + n + n 2 x n, n n dtrmin its radius of convrgnc r, and study th convrgnc for x ±r. By th root
More informationBINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES. 1. Statement of results
BINOMIAL COEFFICIENTS INVOLVING INFINITE POWERS OF PRIMES DONALD M. DAVIS Abstract. If p is a prim and n a positiv intgr, lt ν p (n dnot th xponnt of p in n, and u p (n n/p νp(n th unit part of n. If α
More informationCalculus concepts derivatives
All rasonabl fforts hav bn mad to mak sur th nots ar accurat. Th author cannot b hld rsponsibl for any damags arising from th us of ths nots in any fashion. Calculus concpts drivativs Concpts involving
More informationWhat is a hereditary algebra?
What is a hrditary algbra? (On Ext 2 and th vanishing of Ext 2 ) Claus Michal Ringl At th Münstr workshop 2011, thr short lcturs wr arrangd in th styl of th rgular column in th Notics of th AMS: What is?
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More informationStrongly Connected Components
Strongly Connctd Componnts Lt G = (V, E) b a dirctd graph Writ if thr is a path from to in G Writ if and is an quivalnc rlation: implis and implis s quivalnc classs ar calld th strongly connctd componnts
More informationANALYSIS IN THE FREQUENCY DOMAIN
ANALYSIS IN THE FREQUENCY DOMAIN SPECTRAL DENSITY Dfinition Th spctral dnsit of a S.S.P. t also calld th spctrum of t is dfind as: + { γ }. jτ γ τ F τ τ In othr words, of th covarianc function. is dfind
More information(Upside-Down o Direct Rotation) β - Numbers
Amrican Journal of Mathmatics and Statistics 014, 4(): 58-64 DOI: 10593/jajms0140400 (Upsid-Down o Dirct Rotation) β - Numbrs Ammar Sddiq Mahmood 1, Shukriyah Sabir Ali,* 1 Dpartmnt of Mathmatics, Collg
More informationGeneral Notes About 2007 AP Physics Scoring Guidelines
AP PHYSICS C: ELECTRICITY AND MAGNETISM 2007 SCORING GUIDELINES Gnral Nots About 2007 AP Physics Scoring Guidlins 1. Th solutions contain th most common mthod of solving th fr-rspons qustions and th allocation
More informationSelf-Adjointness and Its Relationship to Quantum Mechanics. Ronald I. Frank 2016
Ronald I. Frank 06 Adjoint https://n.wikipdia.org/wiki/adjoint In gnral thr is an oprator and a procss that dfin its adjoint *. It is thn slf-adjoint if *. Innr product spac https://n.wikipdia.org/wiki/innr_product_spac
More informationMath 102. Rumbos Spring Solutions to Assignment #8. Solution: The matrix, A, corresponding to the system in (1) is
Math 12. Rumbos Spring 218 1 Solutions to Assignmnt #8 1. Construct a fundamntal matrix for th systm { ẋ 2y ẏ x + y. (1 Solution: Th matrix, A, corrsponding to th systm in (1 is 2 A. (2 1 1 Th charactristic
More informationA. Limits and Horizontal Asymptotes ( ) f x f x. f x. x "±# ( ).
A. Limits and Horizontal Asymptots What you ar finding: You can b askd to find lim x "a H.A.) problm is asking you find lim x "# and lim x "$#. or lim x "±#. Typically, a horizontal asymptot algbraically,
More informationThe van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012
Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor
More informationComputing and Communications -- Network Coding
89 90 98 00 Computing and Communications -- Ntwork Coding Dr. Zhiyong Chn Institut of Wirlss Communications Tchnology Shanghai Jiao Tong Univrsity China Lctur 5- Nov. 05 0 Classical Information Thory Sourc
More informationLimiting value of higher Mahler measure
Limiting valu of highr Mahlr masur Arunabha Biswas a, Chris Monico a, a Dpartmnt of Mathmatics & Statistics, Txas Tch Univrsity, Lubbock, TX 7949, USA Abstract W considr th k-highr Mahlr masur m k P )
More informationThere is an arbitrary overall complex phase that could be added to A, but since this makes no difference we set it to zero and choose A real.
Midtrm #, Physics 37A, Spring 07. Writ your rsponss blow or on xtra pags. Show your work, and tak car to xplain what you ar doing; partial crdit will b givn for incomplt answrs that dmonstrat som concptual
More informationMathematics. Complex Number rectangular form. Quadratic equation. Quadratic equation. Complex number Functions: sinusoids. Differentiation Integration
Mathmatics Compl numbr Functions: sinusoids Sin function, cosin function Diffrntiation Intgration Quadratic quation Quadratic quations: a b c 0 Solution: b b 4ac a Eampl: 1 0 a= b=- c=1 4 1 1or 1 1 Quadratic
More informationState-space behaviours 2 using eigenvalues
1 Stat-spac bhaviours 2 using ignvalus J A Rossitr Slids by Anthony Rossitr Introduction Th first vido dmonstratd that on can solv 2 x x( ( x(0) Th stat transition matrix Φ( can b computd using Laplac
More informationSec 2.3 Modeling with First Order Equations
Sc.3 Modling with First Ordr Equations Mathmatical modls charactriz physical systms, oftn using diffrntial quations. Modl Construction: Translating physical situation into mathmatical trms. Clarly stat
More informationINTEGRATION BY PARTS
Mathmatics Rvision Guids Intgration by Parts Pag of 7 MK HOME TUITION Mathmatics Rvision Guids Lvl: AS / A Lvl AQA : C Edcl: C OCR: C OCR MEI: C INTEGRATION BY PARTS Vrsion : Dat: --5 Eampls - 6 ar copyrightd
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr ot St #18 Introduction to DFT (via th DTFT) Rading Assignmnt: Sct. 7.1 of Proakis & Manolakis 1/24 Discrt Fourir Transform (DFT) W v sn that th DTFT is
More informationSuperposition. Thinning
Suprposition STAT253/317 Wintr 213 Lctur 11 Yibi Huang Fbruary 1, 213 5.3 Th Poisson Procsss 5.4 Gnralizations of th Poisson Procsss Th sum of two indpndnt Poisson procsss with rspctiv rats λ 1 and λ 2,
More informationMath 34A. Final Review
Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right
More informationSeparating principles below Ramsey s Theorem for Pairs
Sparating principls blow Ramsy s Thorm for Pairs Manul Lrman, Rd Solomon, Hnry Towsnr Fbruary 4, 2013 1 Introduction In rcnt yars, thr has bn a substantial amount of work in rvrs mathmatics concrning natural
More informationY 0. Standing Wave Interference between the incident & reflected waves Standing wave. A string with one end fixed on a wall
Staning Wav Intrfrnc btwn th incint & rflct wavs Staning wav A string with on n fix on a wall Incint: y, t) Y cos( t ) 1( Y 1 ( ) Y (St th incint wav s phas to b, i.., Y + ral & positiv.) Rflct: y, t)
More informationOn the irreducibility of some polynomials in two variables
ACTA ARITHMETICA LXXXII.3 (1997) On th irrducibility of som polynomials in two variabls by B. Brindza and Á. Pintér (Dbrcn) To th mmory of Paul Erdős Lt f(x) and g(y ) b polynomials with intgral cofficints
More informationAbstract Interpretation: concrete and abstract semantics
Abstract Intrprtation: concrt and abstract smantics Concrt smantics W considr a vry tiny languag that manags arithmtic oprations on intgrs valus. Th (concrt) smantics of th languags cab b dfind by th funzcion
More information1 General boundary conditions in diffusion
Gnral boundary conditions in diffusion Πρόβλημα 4.8 : Δίνεται μονοδιάτατη πλάκα πάχους, που το ένα άκρο της κρατιέται ε θερμοκραία T t και το άλλο ε θερμοκραία T 2 t. Αν η αρχική θερμοκραία της πλάκας
More information[1] (20 points) Find the general solutions of y y 2y = sin(t) + e t. Solution: y(t) = y c (t) + y p (t). Complementary Solutions: y
[] (2 points) Find th gnral solutions of y y 2y = sin(t) + t. y(t) = y c (t) + y p (t). Complmntary Solutions: y c y c 2y c =. = λ 2 λ 2 = (λ + )(λ 2), λ =, λ 2 = 2 y c = C t + C 2 2t. A Particular Solution
More informationLINEAR DELAY DIFFERENTIAL EQUATION WITH A POSITIVE AND A NEGATIVE TERM
Elctronic Journal of Diffrntial Equations, Vol. 2003(2003), No. 92, pp. 1 6. ISSN: 1072-6691. URL: http://jd.math.swt.du or http://jd.math.unt.du ftp jd.math.swt.du (login: ftp) LINEAR DELAY DIFFERENTIAL
More informationCramér-Rao Inequality: Let f(x; θ) be a probability density function with continuous parameter
WHEN THE CRAMÉR-RAO INEQUALITY PROVIDES NO INFORMATION STEVEN J. MILLER Abstract. W invstigat a on-paramtr family of probability dnsitis (rlatd to th Parto distribution, which dscribs many natural phnomna)
More informationDIFFERENTIAL EQUATION
MD DIFFERENTIAL EQUATION Sllabus : Ordinar diffrntial quations, thir ordr and dgr. Formation of diffrntial quations. Solution of diffrntial quations b th mthod of sparation of variabls, solution of homognous
More informationy = 2xe x + x 2 e x at (0, 3). solution: Since y is implicitly related to x we have to use implicit differentiation: 3 6y = 0 y = 1 2 x ln(b) ln(b)
4. y = y = + 5. Find th quation of th tangnt lin for th function y = ( + ) 3 whn = 0. solution: First not that whn = 0, y = (1 + 1) 3 = 8, so th lin gos through (0, 8) and thrfor its y-intrcpt is 8. y
More information4. (5a + b) 7 & x 1 = (3x 1)log 10 4 = log (M1) [4] d = 3 [4] T 2 = 5 + = 16 or or 16.
. 7 7 7... 7 7 (n )0 7 (M) 0(n ) 00 n (A) S ((7) 0(0)) (M) (7 00) 8897 (A). (5a b) 7 7... (5a)... (M) 7 5 5 (a b ) 5 5 a b (M)(A) So th cofficint is 75 (A) (C) [] S (7 7) (M) () 8897 (A) (C) [] 5. x.55
More informationNIL-BOHR SETS OF INTEGERS
NIL-BOHR SETS OF INTEGERS BERNARD HOST AND BRYNA KRA Abstract. W study rlations btwn substs of intgrs that ar larg, whr larg can b intrprtd in trms of siz (such as a st of positiv uppr dnsity or a st with
More informationData Assimilation 1. Alan O Neill National Centre for Earth Observation UK
Data Assimilation 1 Alan O Nill National Cntr for Earth Obsrvation UK Plan Motivation & basic idas Univariat (scalar) data assimilation Multivariat (vctor) data assimilation 3d-Variational Mthod (& optimal
More informationCalculus II (MAC )
Calculus II (MAC232-2) Tst 2 (25/6/25) Nam (PRINT): Plas show your work. An answr with no work rcivs no crdit. You may us th back of a pag if you nd mor spac for a problm. You may not us any calculators.
More informationA Uniform Approach to Three-Valued Semantics for µ-calculus on Abstractions of Hybrid Automata
A Uniform Approach to Thr-Valud Smantics for µ-calculus on Abstractions of Hybrid Automata (Haifa Vrification Confrnc 2008) Univrsity of Kaisrslautrn Octobr 28, 2008 Ovrviw 1. Prliminaris and 2. Gnric
More informationPartial Derivatives: Suppose that z = f(x, y) is a function of two variables.
Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv
More information