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eppur_se_muova

(36,227 posts)
Thu Mar 31, 2016, 07:20 PM Mar 2016

Mathematician finds his 'new' solution to Poisson formula problem buried in 1959 paper (phys.org)

(Phys.org)—As Yves Meyer was getting ready to publish a detailed mathematical proof that he had spent months working on, he decided do a final search of the existing literature. In the reference list of one of the papers he had just peer-reviewed, he noticed what he describes as a "bizarre" paper published in 1959 by Andrew Paul Guinand. Upon further investigation, he was shocked to discover that Guinand had formulated the exact same proof to solve the same problem that Meyer had been working on, though the solution had remained deeply buried and completely forgotten.

Meyer, a Professor Emeritus at the École Normale Supérieure de Cachan, accordingly revised and published his paper, which appeared just a few weeks ago in the Proceedings of the National Academy of Sciences. In his work, he proves that there is not just one, but many Poisson summation formulas, using a simpler solution than was previously known.

Meyer—who has spent his career making fundamental contributions to wavelet theory and number theory, and recently won the Gauss Prize—explains that at first he was somewhat embarrassed that someone else had made the same discovery many decades earlier. However, he also interprets the experience as an example of a more universal pattern: that all of human discovery builds on what comes before.

"Suddenly I understood what I have been steadily doing in my scientific life," Meyer told Phys.org. "I was transmitting a heritage. Today I can express my gratitude to Guinand, who was a great person, both as a human being and as a mathematician."
***
more: http://phys.org/news/2016-03-mathematician-solution-poisson-formula-problem.html#jCp




Worth a read; it's a little more complicated that what's in the first four paras.

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Mathematician finds his 'new' solution to Poisson formula problem buried in 1959 paper (phys.org) (Original Post) eppur_se_muova Mar 2016 OP
One man's fish is another man's poisson. immoderate Mar 2016 #1
Terrible Lucky Luciano Mar 2016 #5
This is an elegant solution MariaThinks Mar 2016 #2
I'm not a maths person, enlightenment Mar 2016 #3
Yikes ! Don't have 55 min right at the moment, but if it's Sophie Germain ... eppur_se_muova Mar 2016 #4
It is Sophie Germain. enlightenment Mar 2016 #6
Shuffling cards and modulo arithmetic. Jim__ Apr 2016 #7

enlightenment

(8,830 posts)
3. I'm not a maths person,
Thu Mar 31, 2016, 07:44 PM
Mar 2016

and can't make an informed comment about the math - but I can appreciate the honesty of Dr. Meyer. Kudos.

eppur_se_muova, are you familiar with the Gresham Lectures (London, UK)? I was there in February and enjoyed listening to Dr. Raymond Flood give a talk on Gauss and Germain. He's a very good lecturer - even I grasped (for that moment, don't ask me now) the basics of the math!

Here's the lecture, uploaded by Gresham College:

eppur_se_muova

(36,227 posts)
4. Yikes ! Don't have 55 min right at the moment, but if it's Sophie Germain ...
Thu Mar 31, 2016, 07:53 PM
Mar 2016

... it's probably worth a listen right there. Thanks !

https://primes.utm.edu/top20/page.php?id=2

Of course, Prof Meyer's proof is probably in Gauss's notebooks somewhere as well.

enlightenment

(8,830 posts)
6. It is Sophie Germain.
Thu Mar 31, 2016, 11:55 PM
Mar 2016

Prof. Flood examines the relationship (correspondence) between the two - and discusses some of the mathematical models.

It was a great lecture to listen to, but if you would rather read it, Gresham College offers all their lectures as text, as well.
http://www.gresham.ac.uk/lectures-and-events/gauss-and-germain

He also has a great talk on Babbage and Lovelace - and there are more . . .
http://www.gresham.ac.uk/watch/?subject=mathematics


on edit: Thanks for the link to the notebooks!

Jim__

(14,045 posts)
7. Shuffling cards and modulo arithmetic.
Sat Apr 2, 2016, 10:07 AM
Apr 2016

Based on that video, modulo arithmetic is an important tool in analysing some aspects of theoretical math. I find it interesting because it is also a simple practical tool that can be used to explain things like how shuffling cards works.

A riffle shuffle of a card deck is when you hold half the deck in one hand, half in the other and then interleave the cards from each half deck. This type of shuffle:
[center][/center]
A perfect riffle shuffle is when the deck is split exactly in half (for a normal deck, 26 cards in each hand) and interleave the cards one from one hand, then one from the other. An out-shuffle is when the cards that are on the top and bottom of the deck remain on the top and bottom after the shuffle.

A perfect riffle out-shuffle mixes the cards from the top and bottom of the deck, something like this:

[center]1 --> 1 ......
....... 2 <-- 27
2 --> 3 ......
....... 4 <-- 28
.
.
.
13 --> 25 ......
........ 26 <-- 39
14 --> 27 ......
........ 28 <-- 40
.
.
.
25 --> 49 ......
........ 50 <-- 51
26 --> 51 ......
.........52 <-- 52[/center]

8 consecutive perfect riffle out-shuffles returns all the cards to their original position, as in:

Deck size = 52 Shuffle type - outShuffle
Cycle 1: 1 -> 1
Cycle 2: 2 -> 3 -> 5 -> 9 -> 17 -> 33 -> 14 -> 27 -> 2
Cycle 3: 4 -> 7 -> 13 -> 25 -> 49 -> 46 -> 40 -> 28 -> 4
Cycle 4: 6 -> 11 -> 21 -> 41 -> 30 -> 8 -> 15 -> 29 -> 6
Cycle 5: 10 -> 19 -> 37 -> 22 -> 43 -> 34 -> 16 -> 31 -> 10
Cycle 6: 12 -> 23 -> 45 -> 38 -> 24 -> 47 -> 42 -> 32 -> 12
Cycle 7: 18 -> 35 -> 18
Cycle 8: 20 -> 39 -> 26 -> 51 -> 50 -> 48 -> 44 -> 36 -> 20
Cycle 9: 52 -> 52

There is a detailed mathematical description of what's going on here; but, the gist of it is that 28 = 256 and 51 x 5 = 255, so 28 ? 1 (mod 51) or 28 k ? k (mod 51).

A perfect riffle shuffle using an in-shuffle would take 52 shuffles before restoring the cards to their original order. An in-shuffle differs from an out-shuffle in how it handles the top and bottom cards. With an in-shuffle, the top card from the bottom half of the deck becomes the top card in the shuffled deck and the bottom card from the top half of the deck becomes the bottom card of the shuffled deck.
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