Dimension of a Maximum Volume

Transcription

1 Dimesio of a Maximum Volume Robert Kreczer Deartmet of Mathematics ad Comutig Uiversity of Wiscosi-Steves Poit Steves Poit, WI Phoe: (715) rkrecze@uwsmail.uws.edu 1. INTRODUCTION. With mathematical comuter rograms becomig every day more owerful ad friedlier, we observe their growig ifluece o the teachig-learig rocess i the Mathematics. However, the adatatio of these techologies will a have ositive imact o educatio i Mathematics oly if we able to fid ew roblems whose level of difficulty ad origiality is aroriate to the sohisticated level of reset-day software used by the studets. While solvig such roblems, with the aid of comuters, we must ay a lot of attetio to how the studets utilize the emirical data obtaied from a comuter to suort their reasoig. I this rocess we should emhasize a critical attitude towards these results, ad thus itroduce the studets to idea of roof as a livig ad imortat art of the etire rocess ivolved i solvig roblems. I this article we will reset a examle of such a roblem, which is iterestig cotiuatio of the article [ 1] reseted at the last ICTCM. Let us deote by vol(b ) the volume of the uit ball i determied by the l orm, 1. More recisely, the - dimesioal measure of the set I [ 1] we showed that 1 1 B œ {(x,...,x ) : kx k...+ kx k Ÿ 1}. 2 1 ( ) : œ Š > > ( 1) vol(b ). (1.1) By usig Mathematica, we ca very easily sketch the grahs of vol(b ) for few fixed values of. I Figure 1, we show the Mathematica outut for œ 2, 3 ad 6.

2 Figure 1. The grahs of vol vs., with œ 1,2, ad 6. We observe that all these grahs have oe hum, which gets higher ad moves towards right as icreases. This aloe causes some roblems whe we try to sketch these grahs, because we must at the begiig assig large eough value for, for each searately, so that we ca see the hole hum. Secodly, there might be more tha oe hum, but they are simly ot dislayed by these grahs. Keeig this dilemma i our mid, we will roose the followig roblem: Fid / œ / (), /, such that volume of uit balls B attais its maximum, that is, for fixed 1, / œ / () is such a umber that /() vol(b ) Ÿ vol(b ), for all. Oe of the ossible aroach to solve this roblem would be to aly usual Calculus techiques, the derivatives, but ufortuately this aroach will ot work here. First, either the comutatio of derivative Dvol(B ) is ossible to carry out for most studets, or is its form obtaied from the Mathematica leasat. Secod, tryig to solve equatio Dvol(B ) œ 0, or eve aroximate its solutio, is harder ad messier yet. Third, the solutio of this equatio would be a real umber oly aroximatig /(). Therefore, we must abado this idea ad devise a ew aroach. 2. SHOWING THAT THERE IS ONLY ONE HUMP. Istead of aalyzig the derivative, we will cosider quotiet q() give by q(), œ vol(b ) vol(b -1) which remids us little bit of the formula for derivative. With the formula (1.1), ad the followig basic formulas for Beta (B) ad Gamma ( > ) fuctios:

3 _!-1 -x (i) >! ( ) œ ' x e dx. (iii) > (x 1) œ x > (x). 0 _ (ii) B(!", ) œ '!-1 "-1 >! ( ) >" ( ) x (1 x) dx. (iv) B(!", ) œ. 0 >! ( ") we will get that 2 q() œ ' 1 (1 x) x dx. (2.1) Formula (2.1) ca also be verified by the Mathematica. Sice it is easy to observe that (1-x) x Ÿ (1 1 x) x, for 0 Ÿ x Ÿ 1, we coclude that q( 1) q(), that is, the sequece q() is decreasig. By aalogy to the derivative, we may imagie that the grah of vol(b ), as a discrete set of oits, lies o the cocave dow curve. I additio, q(0) œ 2, ad sice the itegrad i (2.1) is less tha 1, by alyig Lebesgue's Theorem, we obtai that q() aroaches 0 as aroaches ositive ifiity. From these facts, therefore, we coclude that the fuctio vol(b ) ca have oly oe relative maximum, ad this relative maximum must be its absolute maximum at the same time. 3. Let Mathematica Fid /(). Sice we are ow cofidet that the first largest ecoutered umber o the list 1 2 {vol(b ), vol(b ),...,vol(b ),...} (3.1) is ideed the absolute maximum, we ca determie its ositio, for few chose values of, by emloyig the Mathematica to do it. For this comutatio we use little rogram show i Figure 2, ad the results of that comutatios were laced i the first colum of Table 1. As we try to comute /(), with this rogram, we discover quickly its two flaws. We must be able to set at frot a umber k, where k is ay

4 Figure 2. Mathematica's comutatio of /(1),...,v(10) Table 1.

5 umber greater tha /(). If k is smaller tha /(), list (3.1) is too short, it will ot iclude its absolute maximum, ad the obtaied outut will ot be correct. I oosed, if k is too large, the comutig time take by the Mathematica will become too large. As a matter of fact, with the values of k estimated from the grahs of vol(b ), the comutatio of /(), already for ž 10, becomes too log to be useful i ractice. 4. SO WHAT /() IS LIKE? Lookig at Table 1, first colum, we ca oly observe that /() becomes raidly large. I order to comrehed better these large umbers, we sketch the grah of l( /()) vs., usig data from Table 1. Figure 3. Grah of l( /()) Surrisigly, the oits lie almost o a straight lie, ad this idicates almost exoetial growth of /(). However, our observatio about /(), though imressive, is oly a qualitative statemet about /(), ad it still does ot solve our roblem to fid a exact formula for /(). Sice we do ot kow how to a achieve this goal, we will settle for less, for the time beig - we will try to aroximate /(). The mai idea, to fid a reasoable aroximatio of /(), is to select a coveiet subsequece form sequece vol(b ), ad cosider the maximum of this subsequece to be a aroximatio of /(). We decided, for the urose of simlicity, to select the subsequece r() œ vol(b ), whose formula, by usig (1.1), will become 8 x 1 vol(b ) œ, where x œ 2 > ( 1).!

6 8 Sice subsequece r() is the every th term of the sequece vol(b :) ad the grah of 8 vol(b :) is of the oe hum shae, the maximum term of the subsequece r() lies withi 8 terms to the left or to the right of the maximum term of vol(b :). Further, the th term i the subsequece r() is the ( )th term i the sequece vol(b ), so a idex of the maximum term of the subsequece r() multilied by should aroximate the maximum term of the sequece vol(b ), /(), with the error boud. O the other had, we ca easily determie the idex of the maximum term of r(), by realizig that the maximum of 8 x x x x! 1 2 r() œ œ... x occurs for the largest such that 1, that is x, that is, x, where x deotes Ÿ Ÿ œ g h g h the floor of x. Thus, we obtaied that : : 1 /() gxh œ ª 2 > (1 )«. (4.1) Now we would like to comare the accuracy of the aroximatio i (4.1). We already kow that the absolute error should be less tha. I Table 1 this error is dislayed i the third colum. I the forth colum, we comuted /() modulo. The umbers i these two colums are the same excet their sig. Accidetally, we discover that /() œ ª 1 2 > (1 )«mod (/()). (4.2) Obviously, the imressive formula (4.2) was discovered exerimetally, ad the the formal roof or disroof of it is still required. However, this seems to be oe roblem, left to a reader as a challege to tackle. REFERENCES 1. T.M. Aostol, Calculus,Volume 2, secod editio, Wiley W. Rudi, Real ad Comlex Aalysis, secod editio, McGraw Hill, S. Saks ad A. Zygmut, Aalytic Fuctios, secod editio elarged, Pastwowe Wydawictwo Naukowe, G. Schechtma ad M. Schmuckeschlager, Aother Remark O The Volume Of The Itersectio of Two L Balls, Lectures I Mathematics, Sriger Verlag, #1464, 1994, G. Smith ad M.K. Vamaamurthy, How Small Is A Uit Ball?, Mathematics Magazie, 62, 1988,

7

8