February 2013 - mathfun@mailman.xmission.com

Re: [math-fun] Weird leap-year rule
by Keith F. Lynch 11 Feb '13

11 Feb '13

apgoucher(a)gmx.com (Adam P. Goucher) wrote: > Year 2^n has 29 - n days, and year 3^n has 29 + n days. > Hence, February can be of any integer length ... Correct. gareth.mccaughan(a)pobox.com (Gareth McCaughan) wrote: > ... which I think equals 365 + log 2. I didn't follow your reasoning, but that's the correct answer. The way I figured it: All numbers are divisible by 1, so add 1. Half the numbers are divisible by 2, so subtract 1/2. A third of the numbers are divisible by 3, so add 1/3. A fourth of the numbers are divisible by 4, so subtract 1/4. A fifth of the numbers are divisible by 5, so add 1/5. etc... And it's well known that 1 - 1/2 + 1/3 - 1/4 + ... converges to the natural log of 2. To make it really rigorous I'd probably have to deal with the fact that that series isn't absolutely convergent, i.e. by rearranging the terms you can make it converge to something else; in fact you can get it to converge to any real number whatsoever. Log 2 comes from the most natural (npi) ordering, and as year numbers ascend, the integers are introduced in their natural order, for whatever that's worth. Anyhow, like Hans Havermann, I added up the first few terms (for sufficiently large values of "few") and saw that it does indeed seem to converge to that value. "I have only proved it correct, not tried it." -- Donald Knuth "Trust, but verify." -- Ronald Reagan gladhobo(a)teksavvy.com (Hans Havermann) wrote: > The February days-added numbers are < https://oeis.org/A048272 >. > Checking the 1 0000-term link that is provided there for numbers < > -28 tells us that this first happens in the year 2880. Correct, that is indeed the first year in which February has a negative length. I ran a program to test every number until it found it. A friend of mine to whom I gave the problem in person worked it out in front of me with pencil and paper. >> What then will be the first year in which a whole year has a >> negative length? > ... 17297280. Correct. One curiosity about this calendar is that, like ours, it's on a four-year cycle, even though the number four isn't explicitly in the rule. Years are successively: * long (>365 days) * normal length (365 days) * long (>365 days) * short (<365 days) and this pattern repeats forever. I also came up with simple rules for calendars which gave pi in the average year length. Can anyone else find such a rule? Or rules that give e, Euler's constant, or other interesting numbers in the average year length? Has anyone else done this sort of thing before, or is it original with me?

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Re: [math-fun] Weird leap-year rule
by Adam P. Goucher 11 Feb '13

11 Feb '13

> > And it's well known that 1 - 1/2 + 1/3 - 1/4 + ... converges to > > the natural log of 2. > > Oh, that's sooooo much simpler than what I did! And I bet > it's possible to make it rigorous enough despite the fact > that the series isn't absolutely convergent. Yes, just take the first n! terms, and let n tend to infinity. Sincerely, Adam P. Goucher

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[math-fun] Attesting to Atoms
by James Propp 10 Feb '13

10 Feb '13

I just watched George Hart's video https://simonsfoundation.org/multimedia/attesting-to-atoms/ and was left with a vexing disquiet about the fact that the macroscopic structure of crystals seems to imply the existence of atoms and yet gives us no information about how big atoms are. If the observed structure of macroscopic crystals is compatible with an infinite range of models of reality, each positing the existence of atoms but at ever-smaller scales, could there be some sort of projective limit of these theories, with "cubes all the way down" but no bottom level? I'm not saying it's a believable physical theory, but it seems like it would give an example of a universe with crystals but without atoms. Or is this idea incoherent in some way? Jim Propp

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[math-fun] Attesting to atoms with crystals
by Warren Smith 10 Feb '13

10 Feb '13

Molecular beam epitaxy grows crystals 1 atom layer at a time.

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[math-fun] Weird leap-year rule
by Keith F. Lynch 10 Feb '13

10 Feb '13

In our calendar, February has 28 days unless the year is divisible by 4, in which case it has 29, unless the year is divisible by 100, in which case it has 28, unless the year is divisible by 400, in which case it has 29. But there's a world about a trillion light years away that by a remarkable coincidence has a calendar almost identical to ours. The one difference is their leap year rules. As with us, their February starts with 28 days. But it gains a day for every odd number that the year is divisible by, and loses a day for every even number that the year is divisible by. Divisors include 1 and the number itself. So in the year 1, February has 29 days, since 1 is divisible by just one odd number, namely itself, and is not divisible by any even numbers. So year 1 has 366 days. February of the year 2 has 28 days, since 2 is divisible by one odd number (1) and one even number (2). So year 1 has 365 days. Like us, they never had a year zero, but unlike us they have a good excuse for the lack: Since zero is divisible by infinitely many odd numbers and infinitely many even numbers, it would be impossible to decide on February's length. Eventually, they have a crisis. For the first time, February will have a negative length, and there's no provision for dealing with that. Borrow a day from January or March? The puzzle is, in what year does this first happen? Or will this actually never happen? Suppose they do decide to borrow days from other months when February has a negative length. What then will be the first year in which a whole year has a negative length? Or will this actually never happen? Is there any (integer number of days) length, positive, negative, or zero, that a year can never have? Finally, and most interestingly, what is the average length of a year in their calendar? If possible, express it in closed form rather than as a numerical estimate.

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[math-fun] Lissajous Figure
by dupre＠math.rutgers.edu 10 Feb '13

10 Feb '13

The Lissajous figure also is ambiguous in that it can be seen as rotating either clockwise or counterclockwise around a vertical axis. I am not as successful in being able to see it as rotating in either direction about a horizontal axis, but maybe some people are able to do this. Art DuPre'

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[math-fun] Lissajous Figure
by dupre＠math.rutgers.edu 10 Feb '13

10 Feb '13

The Lissajous figure also is ambiguous in that it can be seed as rotating either clockwise or counterclockwise around a vertical axis. I am not as successful in being able to see it as rotating in either direction about a horizontal axis, but maybe some people are able to do this.

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Re: [math-fun] Weird leap-year rule
by Adam P. Goucher 10 Feb '13

10 Feb '13

Year 2^n has 29 - n days, and year 3^n has 29 + n days. Hence, February can be of any integer length on Planet Keith. Sincerely, Adam P. Goucher http://cp4space.wordpress.com > ----- Original Message ----- > From: Keith F. Lynch > Sent: 02/09/13 10:33 PM > To: math-fun(a)mailman.xmission.com > Subject: [math-fun] Weird leap-year rule > > In our calendar, February has 28 days unless the year is divisible by > 4, in which case it has 29, unless the year is divisible by 100, in > which case it has 28, unless the year is divisible by 400, in which > case it has 29. > > But there's a world about a trillion light years away that by a > remarkable coincidence has a calendar almost identical to ours. The > one difference is their leap year rules. As with us, their February > starts with 28 days. But it gains a day for every odd number that the > year is divisible by, and loses a day for every even number that the > year is divisible by. Divisors include 1 and the number itself. > > So in the year 1, February has 29 days, since 1 is divisible by just > one odd number, namely itself, and is not divisible by any even > numbers. So year 1 has 366 days. February of the year 2 has 28 days, > since 2 is divisible by one odd number (1) and one even number (2). > So year 1 has 365 days. > > Like us, they never had a year zero, but unlike us they have a good > excuse for the lack: Since zero is divisible by infinitely many odd > numbers and infinitely many even numbers, it would be impossible to > decide on February's length. > > Eventually, they have a crisis. For the first time, February will > have a negative length, and there's no provision for dealing with > that. Borrow a day from January or March? The puzzle is, in what > year does this first happen? Or will this actually never happen? > > Suppose they do decide to borrow days from other months when February > has a negative length. What then will be the first year in which a > whole year has a negative length? Or will this actually never happen? > > Is there any (integer number of days) length, positive, negative, or > zero, that a year can never have? > > Finally, and most interestingly, what is the average length of a year > in their calendar? If possible, express it in closed form rather than > as a numerical estimate. > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] Revolution in physics coming?
by Warren Smith 09 Feb '13

09 Feb '13

>H.Baker: Re quantum computation: I'm still extremely uneasy with Shor's quantum factoring algorithm. The problem isn't with his algorithm, per se, but with the basic quantum model on which it is based. Leaving aside the potentially unsolvable problem of achieving the amount of isolation from the environment required to factor a large number, I'm not so sure that the underlying physics can represent the quantum phases to the amount of precision required to factor very large numbers. --see this 2003 paper by me, which argues that "quantum computers won't work." More precisely, it argues that in an computer-sciency asymptotic limit they will be no more powerful (up to polynomial factor) than old-fashioned computers. I personally find my counterargument quite convincing, despite fact few paid attention to it. However, the asymptopia in which my argument is convincing, leaves a lot of room below it, i.e. you might be able to build a quantum computer to factor 100000-digit numbers, that might not be asymptopia yet! So my argument is not so great in that sense. http://rangevoting.org/WarrenSmithPages/homepage/works.html paper #64 Argument against quantum computers (or against certain decoherence models) http://rangevoting.org/WarrenSmithPages/homepage/whywontwork2.ps

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Re: [math-fun] Revolution in physics coming?
by Henry Baker 09 Feb '13

09 Feb '13

If I knew how to make my ranting more coherent, I'd be getting crank late night phone calls from Sweden ! IMHO, there is a necessary connection between quantum states & information. Quantum entanglement is "merely" the reduction in bits necessary to describe the world. The only really solid (nailed down) connection between information theory and physics which exists today is entropy and thermodynamics. Therefore, the only really persuasive argument for a physicist will be to start with physical entropy (taking cues from Maxwell, Boltzmann, Planck, Bekenstein) and derive quantum computation directly, bypassing all of "elementary" physics completely. I suspect that quantumness is a necessary property of everything in the universe, so worrying about the quantumness of every one of the distinct particles makes the problem needlessly complicated. I had hoped that Verlinde's approach would provide some progress along these lines, but I haven't heard anything more. There's a video of him discussing it which is worth while seeing http://homer.kitp.ucsb.edu/online/joint98/verlinde2/snd/Verlinde2_TheorySem… 354MBytes, 93 minutes. The paper: http://arxiv.org/abs/1001.0785 Re quantum computation: I'm still extremely uneasy with Shor's quantum factoring algorithm. The problem isn't with his algorithm, per se, but with the basic quantum model on which it is based. Leaving aside the potentially unsolvable problem of achieving the amount of isolation from the environment required to factor a large number, I'm not so sure that the underlying physics can represent the quantum phases to the amount of precision required to factor very large numbers. I think that quantum theory has something very new to add to computer science, but I'm not sure that the current concept of a "qubit" is that item. Qubits are still stuck in the completely unrealistic Aleph_1 (=2^(Aleph_0) by the continuum hypothesis) world, and so can't really be said to be part of _computer science/information theory_ at all. We also need a new type of quantum gate that doesn't depend upon a "collapse of the wave function" in order to operate. Given the lack of progress in quantum computation/qubits in the last decade, I'd say it's way past time to move on and draw some new cards from the deck. At 08:47 AM 2/9/2013, Warren Smith wrote: >I'd be interested to here more ranting on the topic from H.Baker. (A bit more coherent >preferably, he sounds like he's getting at something interesting, but what?)

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