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1 Math Week 14 otes We will ot ecessarily fiish the material from a give day's otes o that day. We may also add or subtract some material as the week progresses, but these otes represet a i-depth outlie of what we pla to cover. These otes cover material i , Mo Nov Power laws ad least squares for log-log data; itroductio to ier product spaces. Aoucemets: Y EE'S today power law example forhw i gl.es 6 ier product spaces 6qei 2 Y Ex I Warm-up Exercise: Fid the liear regressio lie for the three poits to ,17 i e the least squares solutioto m Sketch µ lil fit wat m b l 1111 At I icosistet ATA I Atb least sgrs so fo i 2, c tai a.ie't m Yz b 46 sice ABIT BTAT

2 Review ad preview: O Wedesday before Thaksgivig we were discussig Sectio 6.6, Applicatios to liear models, which discusses how least squares solutios to liear systems, Sectio 6.5, are applied to liear models i experimetal scieces ad statistics. Today (o Moday) we'll apply these ideas to power laws, by lookig at a example related to Kepler's Laws. The Matlab scripts should be helpful for your homework problem this week about a height-weight power law for humas. The i the rest of Moday's class ad thru Wedesday we'll discuss Sectio , Ier product spaces ad applicatios. Ier product spaces are vector spaces which additioally pocess the algebraic equivalet of the dot product. As a result, ier product spaces have aalgous geometry related to orthogoality, projectio, agles. These ideas have amazig applicatios, for example to Fourier series (importat i applied mathematics ad physics) ad as a fu ad recet special case, image ad audio compressio algorithms. O Friday we'll begi Chapter 7, which is about "The Spectral Theorem" ad its applicatios. The spectral theorem is the fact that symmetric matrices A (oes for which A T = A) are always diagoalizable, ad with real eigevalues, ad furthermore so that the eigebasis ca always be chose to be ortho-ormal. There are may iterestig ad importat applicatios of the Spectral Theorem ad we should have time to discuss some of them i the remaig days of our course.

3 Example like Astroomical example As you may kow, Isaac Newto was motivated by Kepler's (observed) Laws of plaetary motio to discover the otios of velocity ad acceleratio, i.e. differetial calculus ad the itegral calculus, alog with the iverse square law of plaetary acceleratio aroud the su...from which he deduced the cocepts of mass ad force, ad that the uiversal iverse square law for gravitatoal attractio was the ONLY force law depedig oly o distace betwee objects, which was cosistet with Kepler's observatios! Kepler's three observatios were that (1) Plaets orbit the su i ellipses, with the su at oe of the ellipse foci. (2) A plaet sweeps out equal areas from the su, i equal time itervals, idepedetly of where it is i its orbit. (3) The square of the period of a plaetary orbit is directly proportioal to the cube of the orbit's semimajor axis. So, for roughly circular (elliptical) orbits, Keplers third law traslates to the statemet that the period P is related to the radius r, by the equatio P = b r 1.5, for some proportioality costat b. (Ad b = 1 i earthcetric uits below.) Let's see if that's cosistet with the followig data: Plaet mea distace r from su Orbital period t (i astroomical uits where 1=dist to earth) (i earth years) Mercury Earth Jupiter Uraus Pluto The Matlab script works with this data ad first fids the least squares fit to the log-log data. The it creates two figures, the first of which is the liear regressio lie plotted alog with a scatter plot of the log-log data. The secod plot is the graph of the power law for the period as a fuctio of radius, P = C r m alog with a scatter plot of the origial radius vs. period data. Step 1: Matlab is fidig the least squares solutio to the l-l data, Y = m X B (see ext page): log l p Crm m = b x = A T A A x = b A T A x = A T b 1 A T b HW wat to fid i power cost C log I fog 1 log P log C t tog r tog s 20T log I for us Matlab i logp log C t m logr Y B

4 Step 1: A I log periods

5 steps 2, 3: figure with liear regressio lie ad log-log data; ad figure with origial data ad power law graph. last portio of Matlab script: outputs:

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7 Examples of fuctio space ier products: f, g V = f : a, b s.t. f is cotiuous C a, b. a b f t g t dt (or some fixed positive multiple of this itegral). Exercise 1) Check the algebra requiremets a), b), c) for a ier product. a Sf g Cg f gabfitigltidt Sabgltifltidt b fifth Kf g 1 f h fabfitglgltythltildt fabfltigltldttfafltlhltlcf.cg7 ctf.gl study f cg Sabfittegltidt cost c f f 30 Cf f Jabfltidt 30 0 oly if f I O iff f o f is cot This ier product f, g is ot so differet from the dot product if you thik of Riema sums: Let t = b a ; t j = a j t, j = 1, 2,... The f, g = a b f t g t dt = lim j = 1 f t j g t j t f t 1 g t 1 = lim f t 2 : g t 2 : t. f t g t

8 Example For the ier product o C 1, 1 give by 1 I f, g f t g t dt 1 If oe applies Gram-Schmidt to the the set 1, t, t 2, t 3,... oe creates the (ormalized) Legedre polyomials which have a iterestig etry at Wikipedia. Projectig a cotiuous fuctio f oto W = spa 1, t, t 2,... t will create polyomical approximatios, that improve i the sese that p 2 lim f proj W f = 0. O Exercise 2 Fid the first three Legedre polyomials by usig Gram-Schmidt o the fuctios f 1 t = 1, f 2 t = t, f 3 t = t 2. (I your homework you will carry this oe step further.) Hf If Cf f vi utter fz cf ti L dt 2 Hf H E u utter sfz.fi tdt O Iff't Zz f z f the graphfr 5,53 Ez graphs to becotiued