July 2015 - mathfun@mailman.xmission.com

[math-fun] LambertW; and Geometric Discriminant
by rcs＠xmission.com 01 Jul '15

01 Jul '15

Why don't we name the functions X e^X and X logX, 'xex' and 'xlogx'? The inverses could be 'axex' and 'axlogx'. This would simplify life for those of us who only make occasional use of LambertW, and need to consult the definition each time. ----- Geometric Interpretation of the Discriminant Following up Gene's note on Discriminant = square of Vandermonde determinant ... The determinant is +- the area/volume/measure of a parallelogram/piped/? with origin 0 and each row (or each column) as a generating edge. The determinant of the matrix [ 1 2 ] [ 3 4 ] is -2, which is also the area of the parallelogram 00 - 12 - 46 - 34. (The +- depends on your sign convention, and is extinguished if you square the Vandermonde determinant.) The determinant of a V matrix is 0 whenever two roots of the related polynomial are equal. Rich

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Re: [math-fun] Geometric significance of algebraic discriminant ?
by Axel Vogt 01 Jul '15

01 Jul '15

> From: Henry Baker <hbaker1(a)pipeline.com> > To: math-fun(a)mailman.xmission.com > Subject: [math-fun] Geometric significance of algebraic discriminant ? > > I recall learning a bunch of algebra about the discriminant, which becomes zero when there are coincident roots. > > https://en.wikipedia.org/wiki/Discriminant > > Has someone come up with geometric insights about this particular formula ? > > In the case of a quadratic, the formula is (x1-x2)^2, but this isn't the real number |x1-x2|^2. Perhaps the norm of the discriminant (DD*) is more important? > > What about the discriminant of the cubic ? Shouldn't this say something interesting about the triangle in the complex plane formed by the roots? Perhaps one might get some inspiration through the book Gelfand, Kapranov, Zelevinsky: Discriminants, resultants, and multidimensional determinants (1994), may be already in Chap 1 Though I admit that I missed the energy to 'read' it

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Re: [math-fun] Fano Plane puzzle
by Bill Gosper 01 Jul '15

01 Jul '15

The Sierra Club has you beaten. They've quoted the volume of nuclear waste in cubic liters. -- Gene Why do only bad things come in cubic liters? I saw a GreenPeace flyer celebrating the sabotage of a drainpipe dumping "thousands of cubic liters of pharmaceutical waste" into the ocean. Unfortunately, the photo showed only three-dimensional damage, leaving hundreds of square gallons unspilled, resulting in attacks by fifteen pound purple jellyfish: http://sanfrancisco.cbslocal.com/2015/06/12/big-purple-sea-slugs-wash-ashor… --rwg From: rwg <rwg(a)sdf.org> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Wednesday, June 24, 2015 6:28 PM Subject: Re: [math-fun] Fano Plane puzzle On 2015-06-24 17:07, Dan Asimov wrote: On Jun 24, 2015, at 3:22 PM, David Wilson <davidwwilson(a)comcast.net> wrote: Sadly, it would difficult to make an office toy in R^6. Not at all. In R^6 they have even better machine shops than we do. —Dan And the wealth to operate them. Like I was just telling the kids, On Wed, Jun 24, 2015 at 7:19 AM, Bill Gosper <billgosper(a)gmail.com> wrote: Subj: Scrooge McDuck was said to have three cubic acres of money. How much is that in square gallons? In[169]:= UnitConvert[Quantity[3, "Acres"]^3, Quantity["Gallons"]^2] Out[169]= Quantity[95610810544128000000000000000000000000000000000000000/ 765615812545937377500183749853000049, ("Gallons")^2] Oops, that's 27 cubic acres! Should have been In[170]:= UnitConvert[Quantity[3, "Acres"^3], Quantity["Gallons"]^2] Out[170]= Quantity[10623423393792000000000000000000000000000000000000000/ 765615812545937377500183749853000049, ("Gallons")^2] I hate it when ducks exaggerate. --Bill -----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com <mailto:math-fun-bounces@mailman.xmission.com>] Dan Asimov Sent: Wednesday, June 24, 2015 3:30 PM But there could be beautiful configurations for the Fano plane in higher dimensions. Maybe in 6-space, where the vertices of the unit simplex s_6 form 140 unit equilateral triangles.

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