November 2003 - mathfun@mailman.xmission.com

[math-fun] odometer word game
by Richard Schroeppel 03 Jun '04

03 Jun '04

zimb zinc chromatic chromatid Rich

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[math-fun] Balls in Bags
by Jon Perry 09 Dec '03

09 Dec '03

Let there be n balls in a bag. Let k_i be a partition of n such that there are k_1 red balls, k_2 blue balls, etc..., and k_a yellow balls. Give each ball an unique number from 1 to n. What is the probability that a red ball has a number <=j. e.g. consider a bag with 20 balls numbered 1..20, and randomly coloured red (7) and green (13). What is the probability that a red ball is numbered either 1 or 2? Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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[math-fun] The spatial universe is finite !!!???
by asimovd＠aol.com 01 Dec '03

01 Dec '03

I just heard a report on an NPR station -- it was probably BBC news -- in which freelance mathematician Jeff Weeks (who wrote the book "The Shape of Space") was being interviewed after he had announced that the spatial universe is finite. There was not enough time allotted for him to give even a brief layperson's sketch of his reasoning. But it seemed to have something to do with the fluctuations in the background radiation from the Big Bang having bounded wavelengths. Does anyone know more about this research? Or where it might be written up? --Dan

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[math-fun] Permutation Pullze
by Jon Perry 26 Nov '03

26 Nov '03

If we are given p_i and p_j as permutations of P_n, i.e. all possible re-arrangements of 1..n, how is P(n,k) defined?, whereby(n,k) is the number of permutations of 1,..,n whereby the maximum absolute difference between any 2 elements of p_i and p_j is k. e.g. P(4,0)=1 1234->1234 P(4,1)=? 1234->1234 1234->2143 2143->1234 3142->4213 p(4,2)=? 1234->1234 1234->2143 2143->4321 P(n,0)=1 P(n,n-1)=n! Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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Fw: Fw: [math-fun] Fractile application: antennas
by Thane Plambeck 24 Nov '03

24 Nov '03

Here's an opinion from an (anonymous) friend on the antennas > > On Fri, Nov 21, Thane Plambeck writes ... > > Thane> Any comments... > > You're in trouble because I'm not really an antenna guy. Of course, > I do have an opinion ... > > There are a few key parameters of an antenna that determine its performance > (I'm going to describe them in terms of reception, but all of the > statements have a corresponding transmission formulation). The first > parameter is effective aperture (units of m^2). Effective aperture is not > the same as the physical area of an antenna, but it's very closely related. > The antenna's job is to collect RF energy in the form of a flux density > (Watts/m^2) from space. Physically small antennas have small effective > apertures; tiny antennas can't collect as much energy as big antennas > regardless of whether they're made from clever space-filling curves or coat > hangers. The second two parameters are radiation and circuit impedance, > and respectively describe how efficiently antennas couple RF energy into > free space and between their terminals and some other circuit. These are > quantities that are functions of antenna geometry and materials. If > fractal designs have any fundamental efficiency benefits to offer, it is in > the area of antenna geometry. > > One issue that antenna manufacturers face is that the coupling efficiency > and directivity (radiation pattern) of a particular antenna design is a > function of wavelength (among other things). For relatively broadband > applications, e.g., for UHF televsion which operates at frequencies from > 470-890 MHz in the US, special geometries are used to provide efficiency > across a broad range of operating frequencies. These antennas have > self-similar structure resulting in efficient reception across a wide range > of frequencies; the most familiar example is a log-periodic antenna which > you would recognize by sight if not by name (see, e.g., > http://www.ramayes.com/Log_Periodic_Antennas.htm) The self-similar > structure of various fractal curves might also have good properties in this > regard, but I can't immediately think of any physical reason why they would > have exceptional performance as compared to the alternatives. > > Another fractal antenna claim is that by "folding" the array geometry, an > antenna can be made small (retain its effective aperture) while > maintaining performance. I've been approached by other people working with > folded antennas, e.g., http://www.skycross.com, and the problem always > seems to be that these types of antennas are inefficient in coupling energy > to a transmitter or receiver. > > Finally, regarding the claims of enabling fast algorithms for beamforming, > there are probably some clever recursive algorithms that one can develop > for an array with fractal geometry. The real question is whether the > algorithms are less computationally intensive than standard techniques > (which typically require only the solution of a linear system of equations > of order equal to the number of elements in the array) for arrays of > realistic size. > > So, I'm suspicious. Without giving the subject the thought that it no > doubt deserves, my intuition is that there are some nice > mathematical/algorithmic properties of arrays with fractal geometry but > that the arrays being proposed by these people have no significant > performance advantage relative to the other self-similar and folded designs > that are well known in the industry. > >

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[math-fun] Fractile application: antennas
by Richard Schroeppel 22 Nov '03

22 Nov '03

Source: Penn State Date Posted: 2003-10-20 http://www.sciencedaily.com/releases/2003/10/031020054315.htm FRACTAL-SHAPED TILES DEVELOPED FOR NEW BROADBAND ANTENNA CLASS Penn State engineers have developed innovative design methods for a new class of antennas composed of an array of fractal-shaped tiles that offer anywhere from a 4:1 to 8:1 improvement in bandwidth compared to their conventional counterparts. Many natural objects, such as tree branches and their root systems, peaks and valleys in a landscape and rivers and their tributaries are versions of mathematical fractals which appear pleasingly irregular to the eye but are actually made of self-similar, repeated units. The new broadband antennas are composed of irregular but self-similar, repeated fractal-shaped unit tiles or "fractiles" which cover an entire plane without any gaps or overlaps. The outer boundary contour of an array built of fractiles follows a fractal distribution. Dr. Douglas H. Werner, professor of electrical engineering and senior scientist in Penn State's Applied Research Laboratory, will describe the new antennas and their generation at the 2003 IEEE AP-S Topical Conference on Wireless Communication Technology, Oct. 16, in Honolulu, Hawaii. His paper is "A New Design Methodology for Modular Broadband Arrays Based on Fractal Tilings." His co-authors are Waroth Kuhirun, graduate student, and Dr. Pingjuan Werner, associate professor of electrical engineering. While fractal concepts have been used previously in antenna design, Werner and his research team are the first to introduce a design approach for broadband phased array antenna systems that combines aspects of tiling theory with fractal geometry. Once the specific fractile array has been designed, the Penn State team exploits the fact that fractal arrays are generated recursively or via successive stages of growth starting from a simple initial unit, to develop fast recursive algorithms for calculating radiation patterns. Using the recursive property, they have also developed rapid algorithms for adaptive beam forming, especially for arrays with multiple stages of growth that contain a relatively large number of elements. Werner says, "The availability of fast beam forming algorithms is especially advantageous for designing smart antenna systems." The Penn State team has also shown that a fractile array made of unit tiles based on the Peano-Gosper curve, for example, offers performance advantages over a similar-sized array with conventional square boundaries. The Peano-Gosper fractile array produces no grating lobes over a much wider frequency band than conventional periodic planar square arrays. Werner explains that "Grating lobes are sidelobes with the same intensity as the mainbeam. They are undesirable because they take energy away from the main beam and focus it in unintended directions, causing a reduction in the gain of an antenna array." The University is patenting the team's approach to Peano-Gosper and related fractile arrays. The team has also been awarded a grant through the Applied Research Laboratory to build and test a prototype.

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[math-fun] CA Whites may be nastier than NH Whites
by Eugene Salamin 21 Nov '03

21 Nov '03

An automated weather station has been operating since September on 14250 ft White Mountain Summit, the high point of the White Mountain Range, which is the next range east of the California Sierras. Already it has recorded a wind gust of 154 MPH. I expect it won't be long before they break the current record of 231 MPH held by New Hampshire's Mt. Washington. This can be a very pleasant place to visit in the summer: most scenic drive in California, 4700 year old bristlecone pines, 7.5 mi 3000 ft hike to summit, early July wildflowers. It can also be nasty. On a backpack to Pellisier Flats, I woke up in a well below freezing whiteout at 13000 ft and had to navigate out by compass. http://www.wrcc.dri.edu/weather/wmtn.html http://www.wmrs.edu/ http://www.sonic.net/bristlecone/ Gene __________________________________ Do you Yahoo!? Free Pop-Up Blocker - Get it now http://companion.yahoo.com/

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[math-fun] Method for magic cubes
by Christian Boyer 18 Nov '03

18 Nov '03

Some words about the method used, by Walter and me, for our perfect magic cube of order 5. We have first computed a large amount of auxiliary cubes of order 3. These auxiliary cubes are central "symmetrical", meaning that all their 13 alignments of 3 numbers going through the center have x+63+y = 189. These auxiliary cubes have some other partial magic characteristics: 29/49 alignments (including 14/18 diagonals and 4/4 triagonals) have already the same sum = 189. You can check these characteristics, looking at the central cube of order 3 included in our magic cube of order 5. And after that, using these auxiliary cubes, we have tried to fill always by computer the 5^3 - 3^3 = 98 missing numbers, mainly using complementary numbers x+189+y = 315. That's why you can seen a lot of various symmetries in our cube. Running several computers during several weeks before to find the first solution. With this method, it is very easy to find cubes of order 5 with 26 magic diagonals. And more than 1500 cubes with 28 magic diagonals were found before to find our first cube with 30 magic diagonals. We used more than 80,000 different auxiliary cubes of order 3 before to find the first solution. The method used previously by Walter for his cube of order 6 is similar, using a central auxiliary cube of order 4. Christian Boyer. www.multimagie.com/indexengl.htm > -----Message d'origine----- > De: Richard Schroeppel [mailto:rcs@CS.Arizona.EDU] > Date: lundi 17 novembre 2003 21:24 > À: cboyer(a)club-internet.fr > Objet: magic cubes > > Feel free to send more details about the construction/search methods for > the 5^3 and 6^3 to the math-fun list. > > Rich Schroeppel rcs(a)cs.arizona.edu >

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[math-fun] Re: powers of PseudoAntisymmetric (-1,0,1)- Matrices
by Dan Hoey 17 Nov '03

17 Nov '03

"wouter meeussen" <wouter.meeussen(a)pandora.be> writes: > One counter-example to spoil things a bit: What is this a counterexample of? > for n=7, a small sample of (-1,0,1)-matrices whose powers are > 'same', shows that not all are pseudo-Antisymmetric (= AntiSymmetric > + Diagonal) What about [ 0 1 ] [ 1 0 ] ? Isn't that a non-PA (-1,0,1)-matrix whose powers are PA (-1,0,1)-matrices? But the (powerlength = divisor of 12) remains valid. That's not true of the permutation matrix [ 0 1 0 0 0 ] [ 0 0 1 0 0 ] [ 0 0 0 1 0 ] [ 0 0 0 0 1 ] [ 1 0 0 0 0 ]. Earlier, you wrote > > Consider > > the (-1,0,1)-matrices T with properties : Det[T] not zero > > (invertible), all powers T^k are also invertible (-1,0,1) matrices. > > Properties: > > powerlength of T divides 12, > > Det[t] is 1 or -1, > > T is pseudoAntisymmetric, What are you asserting or conjecturing about those matrices and those properties? By the way, a more standard term for "powerlength" would be "order", or "multiplicative order" if you use the word "order" for some other property. Certainly any matrix of finite multiplicative order must have unit determinant, but I can't see any other likely conjecture. Dan

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[math-fun] Re: powers of PseudoAntisymmetric (-1,0,1)- Matrices
by wouter meeussen 15 Nov '03

15 Nov '03

One counter-example to spoil things a bit: for n=7, a small sample of (-1,0,1)-matrices whose powers are 'same', shows that not all are pseudo-Antisymmetric (= AntiSymmetric + Diagonal) try: { {-1, 0, 0, 1, 0, 0, 0}, { 0, -1, 0, 0, 0, 0, 0}, { 0, 0, 0, 0, 0, 1, -1}, {-1, 0, 0, 0, 0, 0, 0}, { 0, 0, 0, 0, 1, 0, 0}, { 0, 0, -1, 0, 0, -1, 1}, { 0, 0, -1, 0, 0, -1, 0} } so my program might have missed some for n=4 or n=5, since I examined only the pseudo-antisymmetric ones. But the (powerlength = divisor of 12) remains valid. When interpreted as adjacency-matrices of (oriented?) graphs, do their matrix-powers have a 'graph-theoretical meaning'? Wouter. ----- Original Message ----- From: "wouter meeussen" <wouter.meeussen(a)pandora.be> To: "math-fun" <math-fun(a)mailman.xmission.com>; "Seqfan (E-mail)" <seqfan(a)ext.jussieu.fr> Cc: "Marc LeBrun" <mlb(a)fxpt.com> Sent: Saturday, November 08, 2003 7:27 PM Subject: moRe: powers of PseudoAntisymmetric (-1,0,1)- Matrices > rehash of thread 15/08/2003, with some news. > OEIS?Anum=A072148 Sequence: 2,14,92,796,7672 > > Definitions: > pseudoAntisymmetric : T(i,j)= -T(j,i) for j<i , so T = diagonal+Antisymmetric. > (my definition, forgive..) > > powerlength: minimal p>0 so that T^p = Identity > > > > Consider > the (-1,0,1)-matrices T with properties : Det[T] not zero (invertible), > all powers T^k are also invertible (-1,0,1) matrices. > > > Properties: > powerlength of T divides 12, > Det[t] is 1 or -1, > T is pseudoAntisymmetric, > > the powers T^k need not be all pseudoAntisymmetric: > > for 4x4 matrices, > all those with 8 non-zero elements have > powerlength 4, and their powers 2 and 3 are not pseudoAntisymmetric; > > for the 5x5 matrices, > all those with 9 non-zero elements have > powerlength 4, and their powers 2 and 3 are not pseudoAntisymmetric; > all those with 10 non-zero elements have > powerlength 12, and their powers 2,3,6,7,10 and 11 are not pseudoAntisymmetric; > > There is a system in this madness, > but this margin is too small... > > > W. > > (I owe Marc LeBrun <mlb(a)fxpt.com> for help, > partial insight & lots inspiration, thanx Marc) > > > I put the 796 4-by4 and the 7672 5-by-5 on > http://users.pandora.be/Wouter.Meeussen/pseudoAntisymmMatrixPowers_4.txt > http://users.pandora.be/Wouter.Meeussen/pseudoAntisymmMatrixPowers_5.txt > > > >

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