[math-fun] Fwd: numberphile.com
It looks like MSRI is getting into serious outreach. Bob Baillie passed along the url http://www.numberphile.com I haven't looked at the insides yet, but the front page looks tasty. Rich
It appears that MSRI is listed as one of the “sponsors” of the website, and apparently that means the website can use the MSRI logo, but the website claims that it is independent of MSRI. The website looks very nice — at first glance. But I don’t think MSRI would intentionally support a video that claims 1+2+3+ . . . = -1/12, since the research institute seems mainly interested in mathematical truth. —Dan On Feb 28, 2014, at 1:33 PM, rcs@xmission.com wrote:
It looks like MSRI is getting into serious outreach. Bob Baillie passed along the url http://www.numberphile.com I haven't looked at the insides yet, but the front page looks tasty.
Rich
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It's saying that ζ(-1) = -1/12. -- Gene
________________________________ From: Dan Asimov <dasimov@earthlink.net> To: math-fun <math-fun@mailman.xmission.com> Sent: Friday, February 28, 2014 2:29 PM Subject: Re: [math-fun] Fwd: numberphile.com
It appears that MSRI is listed as one of the “sponsors” of the website, and apparently that means the website can use the MSRI logo, but the website claims that it is independent of MSRI.
The website looks very nice — at first glance. But I don’t think MSRI would intentionally support a video that claims 1+2+3+ . . . = -1/12, since the research institute seems mainly interested in mathematical truth.
—Dan
On Feb 28, 2014, at 1:33 PM, rcs@xmission.com wrote:
It looks like MSRI is getting into serious outreach. Bob Baillie passed along the url http://www.numberphile.com I haven't looked at the insides yet, but the front page looks tasty.
Rich
That I can agree with! (But the URL sent by Bob Baillie showed the equation 1+2+3+ . . . = -1/12 as a partial visual description of the video, something I can’t agree with.) —Dan On Feb 28, 2014, at 2:38 PM, Eugene Salamin <gene_salamin@yahoo.com> wrote:
It's saying that ζ(-1) = -1/12.
The purpose of the graphics is to grab attention. Sounds like this one did its job. Charles Greathouse Analyst/Programmer Case Western Reserve University On Fri, Feb 28, 2014 at 5:45 PM, Dan Asimov <dasimov@earthlink.net> wrote:
That I can agree with! (But the URL sent by Bob Baillie showed the equation 1+2+3+ . . . = -1/12 as a partial visual description of the video, something I can't agree with.)
--Dan
On Feb 28, 2014, at 2:38 PM, Eugene Salamin <gene_salamin@yahoo.com> wrote:
It's saying that ζ(-1) = -1/12.
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On Fri, Feb 28, 2014 at 3:45 PM, Dan Asimov <dasimov@earthlink.net> wrote:
That I can agree with! (But the URL sent by Bob Baillie showed the equation 1+2+3+ . . . = -1/12 as a partial visual description of the video, something I can't agree with.)
Does that equation make sense p-adically for p coprime to 12? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
No. For p coprime to 12, the p-adic norm of -1/12 is 1. But the partial sum n(n+1) has p-adic norm at most 1/p when n or n+1 is a multiple of p. -- Gene
________________________________ From: Mike Stay <metaweta@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Cc: Eugene Salamin <gene_salamin@yahoo.com> Sent: Friday, February 28, 2014 3:03 PM Subject: Re: [math-fun] Fwd: numberphile.com
On Fri, Feb 28, 2014 at 3:45 PM, Dan Asimov <dasimov@earthlink.net> wrote:
That I can agree with! (But the URL sent by Bob Baillie showed the equation 1+2+3+ . . . = -1/12 as a partial visual description of the video, something I can't agree with.)
Does that equation make sense p-adically for p coprime to 12? -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Numberphile started out making videos for fun, became fairly popular, then went around looking for grant money to be able to continue making them and found some in MSRI. On Fri, Feb 28, 2014 at 3:29 PM, Dan Asimov <dasimov@earthlink.net> wrote:
It appears that MSRI is listed as one of the "sponsors" of the website, and apparently that means the website can use the MSRI logo, but the website claims that it is independent of MSRI.
The website looks very nice -- at first glance. But I don't think MSRI would intentionally support a video that claims 1+2+3+ . . . = -1/12, since the research institute seems mainly interested in mathematical truth.
--Dan
On Feb 28, 2014, at 1:33 PM, rcs@xmission.com wrote:
It looks like MSRI is getting into serious outreach. Bob Baillie passed along the url http://www.numberphile.com I haven't looked at the insides yet, but the front page looks tasty.
Rich
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Dan, can you say more about why you object to using the equation 1+2+3+4+...=-1/12 in this way? Is this equation, for you, exactly as objectionable as (say) 1+2+4+8+...=-1, and for the exact same reason? And is that reason the conviction that "1+2+3+4+..." must (in the absence of side explanations) connote the limit (in Cauchy's sense) of 1+2+...+n as n goes to infinity? Jim Propp On Friday, February 28, 2014, Dan Asimov <dasimov@earthlink.net> wrote:
It appears that MSRI is listed as one of the "sponsors" of the website, and apparently that means the website can use the MSRI logo, but the website claims that it is independent of MSRI.
The website looks very nice -- at first glance. But I don't think MSRI would intentionally support a video that claims 1+2+3+ . . . = -1/12, since the research institute seems mainly interested in mathematical truth.
--Dan
On Feb 28, 2014, at 1:33 PM, rcs@xmission.com <javascript:;> wrote:
It looks like MSRI is getting into serious outreach. Bob Baillie passed along the url http://www.numberphile.com I haven't looked at the insides yet, but the front page looks tasty.
Rich
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Oops, I meant "denote", not "connote". Jim On Friday, February 28, 2014, James Propp <jamespropp@gmail.com> wrote:
Dan, can you say more about why you object to using the equation 1+2+3+4+...=-1/12 in this way? Is this equation, for you, exactly as objectionable as (say) 1+2+4+8+...=-1, and for the exact same reason?
And is that reason the conviction that "1+2+3+4+..." must (in the absence of side explanations) connote the limit (in Cauchy's sense) of 1+2+...+n as n goes to infinity?
Jim Propp
On Friday, February 28, 2014, Dan Asimov <dasimov@earthlink.net<javascript:_e(%7B%7D,'cvml','dasimov@earthlink.net');>> wrote:
It appears that MSRI is listed as one of the "sponsors" of the website, and apparently that means the website can use the MSRI logo, but the website claims that it is independent of MSRI.
The website looks very nice -- at first glance. But I don't think MSRI would intentionally support a video that claims 1+2+3+ . . . = -1/12, since the research institute seems mainly interested in mathematical truth.
--Dan
On Feb 28, 2014, at 1:33 PM, rcs@xmission.com wrote:
It looks like MSRI is getting into serious outreach. Bob Baillie passed along the url http://www.numberphile.com I haven't looked at the insides yet, but the front page looks tasty.
Rich
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I agree with Dan. Properly speaking, the only correct limit for these two series is infinity. But we all understand how the incorrect sums arise, and I am not going to make a fuss about it. -- Gene
________________________________ From: James Propp <jamespropp@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Friday, February 28, 2014 6:03 PM Subject: Re: [math-fun] Fwd: numberphile.com
Dan, can you say more about why you object to using the equation 1+2+3+4+...=-1/12 in this way? Is this equation, for you, exactly as objectionable as (say) 1+2+4+8+...=-1, and for the exact same reason?
And is that reason the conviction that "1+2+3+4+..." must (in the absence of side explanations) connote the limit (in Cauchy's sense) of 1+2+...+n as n goes to infinity?
Jim Propp
On Friday, February 28, 2014, Dan Asimov <dasimov@earthlink.net> wrote:
It appears that MSRI is listed as one of the "sponsors" of the website, and apparently that means the website can use the MSRI logo, but the website claims that it is independent of MSRI.
The website looks very nice -- at first glance. But I don't think MSRI would intentionally support a video that claims 1+2+3+ . . . = -1/12, since the research institute seems mainly interested in mathematical truth.
--Dan
On Feb 28, 2014, at 1:33 PM, rcs@xmission.com <javascript:;> wrote:
It looks like MSRI is getting into serious outreach. Bob Baillie passed along the url http://www.numberphile.com I haven't looked at the insides yet, but the front page looks tasty.
Rich
On Fri, Feb 28, 2014 at 7:03 PM, James Propp <jamespropp@gmail.com> wrote:
Dan, can you say more about why you object to using the equation 1+2+3+4+...=-1/12 in this way? Is this equation, for you, exactly as objectionable as (say) 1+2+4+8+...=-1, and for the exact same reason?
And is that reason the conviction that "1+2+3+4+..." must (in the absence of side explanations) connote the limit (in Cauchy's sense) of 1+2+...+n as n goes to infinity?
I would guess his objection has to do with abuse of notation, in that one is implicitly assuming a nontrivial summation method. Are there series whose sum gives different results if you use Abel, Borel, Cesaro, Euler, or Lambert summation? If not, then perhaps the objection is unfounded. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
Here's a short path from divergent series summation mumbo-jumbo to total nonsense: If you sum 1+1+2+5+14+42+... (the Catalan numbers) by plugging x=1 into the formula 1+x+2x^2+5x^3+14x^4+...=(1-sqrt(1-4x))/2, you'll get a non-real complex number a+bi with b non-zero. Now take the complex conjugate of the equation 1+1+2+5+...=a+bi. The left hand side (being a sum of ordinary integers) stays the same, but the right hand side becomes a-bi. So a+bi=a-bi, so 2bi=0, so i=0, which gives -1=i^2=0^2=0. Adding 2 to both sides gives 1=2. Therefore (for the usual reason) I am the Pope. Jim Propp On Friday, February 28, 2014, Mike Stay <metaweta@gmail.com> wrote:
On Fri, Feb 28, 2014 at 7:03 PM, James Propp <jamespropp@gmail.com<javascript:;>> wrote:
Dan, can you say more about why you object to using the equation 1+2+3+4+...=-1/12 in this way? Is this equation, for you, exactly as objectionable as (say) 1+2+4+8+...=-1, and for the exact same reason?
And is that reason the conviction that "1+2+3+4+..." must (in the absence of side explanations) connote the limit (in Cauchy's sense) of 1+2+...+n as n goes to infinity?
I would guess his objection has to do with abuse of notation, in that one is implicitly assuming a nontrivial summation method. Are there series whose sum gives different results if you use Abel, Borel, Cesaro, Euler, or Lambert summation? If not, then perhaps the objection is unfounded. -- Mike Stay - metaweta@gmail.com <javascript:;> http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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I like the Catalan example --- but it doesn't quite address Mike's question. When a sum is assigned to a divergent series via analytical extension, the value obtained will vary according to the analytical representation involved. The original example apparently implicitly interpreted terms as coefficients in a Dirichlet series; utilising Taylor series yields instead 1 x^0 + 2 x^1 + 3 x^2 + ... = 1/(1 - x)^2 at x = 1 , yielding (some flavour of) infinity. Anyway, Jim isn't the pope --- I am. WFL On 3/1/14, James Propp <jamespropp@gmail.com> wrote:
Here's a short path from divergent series summation mumbo-jumbo to total nonsense:
If you sum 1+1+2+5+14+42+... (the Catalan numbers) by plugging x=1 into the formula 1+x+2x^2+5x^3+14x^4+...=(1-sqrt(1-4x))/2, you'll get a non-real complex number a+bi with b non-zero.
Now take the complex conjugate of the equation 1+1+2+5+...=a+bi. The left hand side (being a sum of ordinary integers) stays the same, but the right hand side becomes a-bi.
So a+bi=a-bi, so 2bi=0, so i=0, which gives -1=i^2=0^2=0. Adding 2 to both sides gives 1=2. Therefore (for the usual reason) I am the Pope.
Jim Propp
On Friday, February 28, 2014, Mike Stay <metaweta@gmail.com> wrote:
On Fri, Feb 28, 2014 at 7:03 PM, James Propp <jamespropp@gmail.com<javascript:;>> wrote:
Dan, can you say more about why you object to using the equation 1+2+3+4+...=-1/12 in this way? Is this equation, for you, exactly as objectionable as (say) 1+2+4+8+...=-1, and for the exact same reason?
And is that reason the conviction that "1+2+3+4+..." must (in the absence of side explanations) connote the limit (in Cauchy's sense) of 1+2+...+n as n goes to infinity?
I would guess his objection has to do with abuse of notation, in that one is implicitly assuming a nontrivial summation method. Are there series whose sum gives different results if you use Abel, Borel, Cesaro, Euler, or Lambert summation? If not, then perhaps the objection is unfounded. -- Mike Stay - metaweta@gmail.com <javascript:;> http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
="James Propp" <jamespropp@gmail.com> Here's a short path from divergent series summation mumbo-jumbo to total nonsense:
Very nice! A fast cutoff from mumbo-jumbo to total nonsense will save me lots of time! (Are we there yet?<;-)
... you'll get a non-real complex number a+bi with b non-zero.
Assuming of course you legitimize imaginary quantities! But of course that was also once controversial. One could instead take the position that this is an argument against this evaluation of the sum, much like:
="Fred Lunnon" <fred.lunnon@gmail.com> ... = 1/(1 - x)^2 at x = 1 , yielding (some flavour of) infinity.
If we can disallow expressions like 1/0 why can't we also disallow a+bi?
="James Propp" <jamespropp@gmail.com> Now take the complex conjugate of the equation 1+1+2+5+...=a+bi.
This assumes that you can take the complex conjugate of this infinite aggregate object by conjugating its components. Since this led to a contradictory result, one could instead argue that you've just proved that this is an inadmissible operation. Can you construct a contradiction from a divergent sum whose generating function at unity is finite and real-valued?
="Dan Asimov" <dasimov@earthlink.net> Yeah, it¹s as simple as that. The definition of the existence and, if so, of the value of an infinite sum is very well-established in math. If you mean something different, how hard is it to say as much?
Well, well-established for some t > Euler anyway. But I agree, math is a game and if you are going to play by unorthodox rules it's more sporting to say so at the outset. Actually I don't mind these textbook arguments as demonstrations that inconsistent rule sets lead to contradictions, thereby highlighting the underlying inconsistencies. I only object to dismissing out-of-hand anything but the most rigidly orthodox interpretation of what the symbol "1+2+3+4+..." means as The One Truth without likewise some acknowledgement of their actually being an underlying adopted canon. Mostly in practice even this doesn't matter, except when the pedagogic purpose is to delight and astonish with unexpected consequences, as it was here, in order to motivate deeper engagement. It's kill-joy to prohibit magicians from positing that rabbits may be extracted from hats.
="James Propp" <jamespropp@gmail.com> Therefore (for the usual reason) I am the Pope.
="Fred Lunnon" <fred.lunnon@gmail.com> Anyway, Jim isn't the pope --- I am.
No need to fight guys, let's just get rid of this pesky "excluded middle"! "To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true; so that he who says of anything that it is, or that it is not, will say either what is true or what is false." --Aristotle, Metaphysics, Book IV, Part 7 (per Wikipedia) Clear as mud! --MLB
Yeah, it’s as simple as that. The definition of the existence and, if so, of the value of an infinite sum is very well-established in math. If you mean something different, how hard is it to say as much? —Dan On Feb 28, 2014, at 6:03 PM, James Propp <jamespropp@gmail.com> wrote:
Dan, can you say more about why you object to using the equation 1+2+3+4+...=-1/12 in this way? Is this equation, for you, exactly as objectionable as (say) 1+2+4+8+...=-1, and for the exact same reason?
And is that reason the conviction that "1+2+3+4+..." must (in the absence of side explanations) connote the limit (in Cauchy's sense) of 1+2+...+n as n goes to infinity?
On Feb 28, 2014, at 5:29 PM, Dan Asimov <dasimov@earthlink.net> wrote:
The website looks very nice — at first glance. But I don’t think MSRI would intentionally support a video that claims 1+2+3+ . . . = -1/12, since the research institute seems mainly interested in mathematical truth.
This series comes up in the Casimir effect. In class, before I try to make sense of it, I remind my students of the story, where Gauss's 3rd grade teacher asks the class to sum 1+2+ … +100. Okay, they all know that one. What is less well known, I then go on, is that Riemann's 3rd grade teacher, not to be outdone, had the class sum 1+2+ … -Veit
Question: Let Q(n) := exp(pi*sqrt(n)) for n = 1,2,3,…. What can be said about the distribution of the sequence frac(Q(n)) = Q(n) (mod 1) ??? Since my next birthday, #n, satisfies exp(pi*sqrt(n)) is close to an integer, I was wondering what the sequence (letting f(x) := min{frac(x),1-frac(x)} be the distance of x to its nearest integer): n_1 = 1, and n_(k+1) := min{n | f(Q(n))) < f(Q(n_k))} looks like. I.e., n for which f(Q(x)) is a record low. —Dan
On 3/1/2014 4:02 PM, Dan Asimov wrote:
Question:
Let Q(n) := exp(pi*sqrt(n)) for n = 1,2,3,Â….
What can be said about the distribution of the sequence frac(Q(n)) = Q(n) (mod 1) ???
Since my next birthday, #n, satisfies exp(pi*sqrt(n)) is close to an integer, I was wondering what the sequence (letting f(x) := min{frac(x),1-frac(x)} be the distance of x to its nearest integer):
n_1 = 1, and
n_(k+1) := min{n | f(Q(n))) < f(Q(n_k))}
looks like. I.e., n for which f(Q(x)) is a record low.
The sequence of integers yielding record lows is oeis.org/A069014 . Apparently no further value is known that beats 163, so the sequence shows only nine terms: 1, 2, 6, 17, 22, 25, 37, 58, 163. -- Fred W. Helenius fredh@ix.netcom.com
Hmm, interesting — I’d heard of 22 and 58, but not of 6, 25, 37. It was especially the famous case of 163 that got me thinking: If the sequence of frac(Q(n)), n=1,2,3, is uniformly distributed in [0,1] then the sequence of recordholders n_k must be infinite. —Dan On Mar 1, 2014, at 7:40 PM, Fred W. Helenius <fredh@ix.netcom.com> wrote:
On 3/1/2014 4:02 PM, Dan Asimov wrote:
Question:
Let Q(n) := exp(pi*sqrt(n)) for n = 1,2,3,….
What can be said about the distribution of the sequence frac(Q(n)) = Q(n) (mod 1) ???
Since my next birthday, #n, satisfies exp(pi*sqrt(n)) is close to an integer, I was wondering what the sequence (letting f(x) := min{frac(x),1-frac(x)} be the distance of x to its nearest integer):
n_1 = 1, and
n_(k+1) := min{n | f(Q(n))) < f(Q(n_k))}
looks like. I.e., n for which f(Q(x)) is a record low.
The sequence of integers yielding record lows is oeis.org/A069014 . Apparently no further value is known that beats 163, so the sequence shows only nine terms: 1, 2, 6, 17, 22, 25, 37, 58, 163.
I think that the claim that the "truth" of [1] 1 + 2 + 3 + 4 + ... = -1/12 like that of any other mathematical statement, depends on the theoretical context. Is it true that 1 + 1 = 2, 1, or 0? It all depends if we are working in integer, Boolean, or Z(2) arithmetic. Using standard convergent series, the series 1 + 2 + 3 + 4 + ... Is clearly divergent and has no sum. However, Numberphile derives [1] via formal series manipulations (all of which appear legitimate for convergent series) arriving at the series 1 - 1 + 1 - 1 + ... = 1/2 evaluated using Cesaro summation as opposed to partial sum limits. This suggests that [1] is perhaps a legitimate result in a broader theory of infinite series based on Cesaro sums as opposed to the standard theory of convergent partial sums. My question would be, is this Cesaro-based theory of infinite series coherent? I suspect it is. If physicists routinely use formal manipulations and Cesaro sums such as those shown in the Numberphile video to evaluate classically divergent series, it would be an embarrassment if they had no mathematical foundation to assure them that their evaluations were unique in general or correct in a specific instance. If some particular method evaluating divergent series yields values that appear to work in some particular physics application, this is perhaps evidence, but certainly not proof, that the methods are legitimate and reliable. What is needed is a mathematical theory built on solid mathematical foundations in which [1] and like statements are theorems on a par with the rest of mathematics. Seriously, do physicists routinely use these types of formal manipulations without any mathematical foundation? I'm guessing not. If physicists are using equation like [1] in actual physics, I suspect they must have some theory to justify their methods.
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of Dan Asimov Sent: Friday, February 28, 2014 5:30 PM To: math-fun Subject: Re: [math-fun] Fwd: numberphile.com
It appears that MSRI is listed as one of the "sponsors" of the website, and apparently that means the website can use the MSRI logo, but the website claims that it is independent of MSRI.
The website looks very nice - at first glance. But I don't think MSRI would intentionally support a video that claims 1+2+3+ . . . = -1/12, since the research institute seems mainly interested in mathematical truth.
-Dan
On Feb 28, 2014, at 1:33 PM, rcs@xmission.com wrote:
It looks like MSRI is getting into serious outreach. Bob Baillie passed along the url http://www.numberphile.com I haven't looked at the insides yet, but the front page looks tasty.
Rich
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I don't think the use of zeta(-1) = -1/12 in physics has anything to do with these formal manipulations --- they have to do with extending zeta(s) analytically in the complex plane. What's curious is that there is a set of formal manipulations that leads to the same answer. But this may be true for _every_ possible answer. Cris On Mar 1, 2014, at 2:07 PM, David Wilson <davidwwilson@comcast.net> wrote:
Seriously, do physicists routinely use these types of formal manipulations without any mathematical foundation? I'm guessing not. If physicists are using equation like [1] in actual physics, I suspect they must have some theory to justify their methods.
Cristopher Moore Professor, Santa Fe Institute The Nature of Computation Cristopher Moore and Stephan Mertens Available now at all good bookstores, or through Oxford University Press http://www.nature-of-computation.org/
I guess that's my question. There are clearly some no-no's with formal manipulation of infinite series, for instance, if a series includes an infinite number of both positive and negative terms, you can't arbitrarily reorder the elements . However, there are likely some "safe" manipulations, like termwise sum and product, scalar multiplication, zero term elision, &c, that do not alter the value of the sum. The question is, if we start with Cesaro-summable sequences and apply only safe manipulations, do we end up with unique values for sequences like 1 + 2 + 3 + 4 + ...?
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of Cris Moore Sent: Saturday, March 01, 2014 5:03 PM To: math-fun Subject: Re: [math-fun] numberphile.com
I don't think the use of zeta(-1) = -1/12 in physics has anything to do with these formal manipulations --- they have to do with extending zeta(s) analytically in the complex plane.
What's curious is that there is a set of formal manipulations that leads to the same answer. But this may be true for _every_ possible answer.
Cris
On Mar 1, 2014, at 2:07 PM, David Wilson <davidwwilson@comcast.net> wrote:
Seriously, do physicists routinely use these types of formal manipulations without any mathematical foundation? I'm guessing not. If physicists are using equation like [1] in actual physics, I suspect they must have some theory to justify their methods.
Cristopher Moore Professor, Santa Fe Institute
The Nature of Computation Cristopher Moore and Stephan Mertens Available now at all good bookstores, or through Oxford University Press http://www.nature-of-computation.org/
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On 3/1/14, David Wilson <davidwwilson@comcast.net> wrote:
... Seriously, do physicists routinely use these types of formal manipulations without any mathematical foundation? I'm guessing not. If physicists are using equation like [1] in actual physics, I suspect they must have some theory to justify their methods.
Take a look at a relevant, well-written and interesting general article at http://en.wikipedia.org/wiki/Casimir_effect WFL
i prefer this version, which LISTS all the videos: http://www.numberphile.com/text_index.html and as far as 1+2+3+ . . . = -1/12 is concerned, this is a standard way to "sum" a divergent series. bob --- rcs@xmission.com wrote:
It looks like MSRI is getting into serious outreach. Bob Baillie passed along the url http://www.numberphile.com I haven't looked at the insides yet, but the front page looks tasty.
Rich
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participants (13)
-
Charles Greathouse -
Cris Moore -
Dan Asimov -
David Wilson -
Eugene Salamin -
Fred Lunnon -
Fred W. Helenius -
James Propp -
Marc LeBrun -
Mike Stay -
rcs＠xmission.com -
Robert Baillie -
Veit Elser