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Philippe Beaudoin
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Philippe Beaudoin

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Paranormal phenomenon : normal person :: Ulam sequence : mathematician.
 
A hidden signal in the Ulam sequence

The Ulam sequence is a sequence of positive integers a_n, where a_1=1, a_2=2, and where each a_n for n > 2 is defined to be the smallest integer that can be expressed as the sum of two distinct earlier terms in a unique way.

The first few terms of the sequence are shown in the picture: 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47. The third term is 3, because 3=1+2. The fourth term is 4, because although 4 can be expressed in two ways as the sum of two earlier terms (1+3, or 2+2) the sum “2+2” is not a sum of distinct earlier terms. The fifth term is not 5, because 5=1+4=2+3; rather, it is 6, which is 2+4. The sixth term is not 7, because 7=1+6=3+4, but rather 8, which is 2+6. And so on.

The Ulam sequence is named after the Polish-American mathematician Stanisław Ulam (1909–1984) who introduced it in 1964 in a survey on unsolved problems. Ulam remarked that it can be notoriously difficult to answer questions about the properties of sequences like this one, even if they are defined by a simple rule. 

In particular, Ulam was interested in whether one could find the asymptotic density of this particular sequence. Roughly speaking, this is asking what proportion of the integers in between 1 and N are members of the Ulam sequence if N is very large. Empirically, the answer seems to be about 7.4%, but there is not even a proof that the density is bigger than Ulam's predicted value of zero. Ulam's sequence has been described as “quite erratic” and it has been said that it “does not appear to follow any recognizable pattern”. 

Even though the Ulam sequence has been around for 50 years, very little is known about it. However, the recent paper A Hidden Signal in the Ulam sequence by Stefan Steinerberger (http://arxiv.org/abs/1507.00267) unearths some surprisingly rigid structure in the sequence. In the context of Fourier series, it is natural to study the function f_N(x) obtained by summing the values of cos(a_n x) from n=1 to N. The expected result here is that whenever x is not zero, the absolute value of f_N(x) should be approximately equal to the square root of N. However, this is not what happens, because there is a mysterious constant α, equal to about 2.571, such that f_N(α) is approximately equal to c times N, for a negative constant c around –0.8.

The existence of such a constant can be thought of (metaphorically) as a signal embedded in the sequence. Steinerberger shows that the constant α is given by 2.5714474995...; there is another such constant at 2π–α by symmetry. The signal can be demonstrated in an elementary way by considering the values of cos(α a_n), measuring angles in radians: remarkably, with a very small number of exceptions, the values are all negative.

The paper also considers versions of the Ulam sequence in which the first two terms are replaced with other numbers, and similar phenomena occur. The case where a_1=2 and a_2=3 seems to be especially striking.

Relevant links

Stanisław Ulam was the bomb: he participated in the Manhattan Project and originated the Teller–Ulam design of thermonuclear weapons. He was also the chair of my department in Boulder in the 1960s. More information on Ulam can be found here: https://en.wikipedia.org/wiki/Stanislaw_Ulam

The On-Line Encyclopedia of Integer Sequences has more about the Ulam sequence: https://oeis.org/A002858

#mathematics #scienceeveryday #spnetwork arXiv:1507.00267
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I'm totally geeking out over my newly built lego computer, from +Chris McVeigh's gorgeous collection. (http://chrismcveigh.com)
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Wow :-)
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I'm very happy to work for a company that takes the threat of climate change seriously.
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No army can stop an idea whose time has come.
 
On the consequences of making books freely available online

Earlier today I posted the final beta chapter of my book about neural networks at http://neuralnetworksanddeeplearning.com.

The book is online and freely available.   

One pleasing consequence of this availability is that many people get to see the book - the book will pass 400,000 readers later today or tomorrow, according to Google Analytics.   Of course, that number is not a true representation of impact, since many of those people no doubt glanced briefly at the book, and decided to move on to other things.

But there is a closely related number that gives me particular pleasure: according to Google Analytics the book has been downloaded in 208 countries.  That list includes places such as Djibouti, Comoros, Burkina Faso, and many others.  

As in other countries, no doubt some of the readers in those countries only looked briefly at the book.  But I'm truly delighted that putting the book freely online gives people all over the world the opportunity to look at the book, and decide if it might be useful to them.

If I'd gone the conventional publishing route, my book would just now be entering production, and would have been seen by perhaps a few dozen people, probably in 2 or 3 countries.  Online availability really makes a big difference!
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Where was this parenting book when I needed it?

/cc +Robin Sacks​​ +Robert Bridson
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Tell her magic is French and she has to say the spell in French. ;) (Although at that point she's probably too experienced in French. Try Spanish or Mandarin or whatever language she wants to learn next. ;))
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We're only able to tell that we're not performing at our best for a day or two. So if you get the Sunday blues and stay up watching Netflix or reading on your iPhone, and then you have a long day on Monday and an early meeting on Tuesday, you may have already settled into a sub-optimal performance routine for the week, compounding the negative effects of sleep deprivation every subsequent night.

Your brain just loses the ability to tell the difference.

People "who are chronically sleep deprived may no longer be capable of reliably appraising their own sleepiness — or they simply don't experience levels of sleepiness in any way commensurate with their actual deprivation," writes Mark Wolverton in Psychology Today, describing the research David Dinges of the University of Pennsylvania. You may feel a little bit sleepy — nothing more — but your body and your brain are just struggling to get by on such a severe deficit.
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Oh very interesting to know!
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Do you ever find yourself trying to avoid long lines or wondering when is the best time to go grocery shopping, pick up coffee or hit the gym (hint: avoid Monday after work)? You’re in luck!

Now, you can avoid the wait and see the busiest times of the week at millions of places and businesses around the world directly from Google Search. For example, just search for "Blue Bottle Williamsburg", tap on the title and see how busy it gets throughout the day. Enjoy your extra time!
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With the information, I probably will end up not doing anything... ;)
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This is awesome. Also: pepperonis are now banned from airplanes.
 
Playing with your food...
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That tube attached to it is probably bringing a healthy dose of pure oxygen to the mix. IIUC a thermal lance is just fuel+oxygen+fire. Pepperoni sounds like a pretty good fuel. :)
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I wish more textbooks were written like +Michael Nielsen's excellent (and free) Neural Networks and Deep Learning, which goes beyond proofs and explains what may have led someone to have such clever ideas in the first place.

The second mystery is how someone could ever have discovered backpropagation in the first place? It's one thing to follow the steps in an algorithm, or even to follow the proof that the algorithm works. But that doesn't mean you understand the problem so well that you could have discovered the algorithm in the first place. Is there a plausible line of reasoning that could have led you to discover the backpropagation algorithm? In this section I'll address both these mysteries.

/cc +Robert Bridson, +Michiel van de Panne.
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Just watched Kieślowski's masterpiece: Trois couleurs: Bleu and it prompted me to learn more about the poet behind these images. I found this citation quite inspiring, and it may explain why I'm so moved by his movies:

It comes from a deep-rooted conviction that if there is anything worthwhile doing for the sake of culture, then it is touching on subject matters and situations which link people, and not those that divide people. There are too many things in the world which divide people, such as religion, politics, history, and nationalism. If culture is capable of anything, then it is finding that which unites us all. And there are so many things which unite people. It doesn't matter who you are or who I am, if your tooth aches or mine, it's still the same pain. Feelings are what link people together, because the word 'love' has the same meaning for everybody. Or 'fear', or 'suffering'. We all fear the same way and the same things. And we all love in the same way. That's why I tell about these things, because in all other things I immediately find division.

/cc +Seb Paquet 
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And now for some food myth debunking...
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Education
  • University of British Columbia
    2011
  • Université de Montréal
  • École Polytechnique
Basic Information
Gender
Male
Relationship
In a relationship
Apps with Google+ Sign-in
  • Bonza Word Puzzle
  • Crossy Road
Story
Tagline
Code + Boardgames
Introduction
I stayed in school too long, they gave me a PhD for it. Then I created two popular open source projects, gwt-platform and jukito. In June 2011 I joined Google as a software developer on Chrome.

I also play and design boardgames, Quebec is my first published design.

Moderation style
I like Google+ more than any other social network because the discussions I have here are more substantial, more polite, and more interesting than anywhere else on the Net. However, to maintain that level of quality, I will sometimes (very rarely) delete comments that I feel throw the conversation in undesirable directions. It does not mean I do not welcome opinions that differ from mine. However, if:
  • you attack someone else personally,
  • you propose an outrageous claim without any backing evidence,
  • you are too agressive,
it is very likely that I will delete your comment.

Fortunately, this happens so seldom that I can take the time to contact people whose comments I delete. You're always welcome to come back and voice your opinion in a more polite way, or with stronger evidence. Naturally, you're also welcome to reshare my post and comment in any way you like on your own stream.

Disclaimer! Although I work for Google, I use Google+ to publish my personal opinions. Any views I express here are mine alone and do not necessarily reflect those of my employer.

Bragging rights
Gave a talk at Google I/O before I was a googler.
Work
Occupation
Software developer
Skills
programming, boardgame design
Employment
  • Google
    Software developer, 2011 - present
  • BookedIn
    Lead developer, 2010 - 2011
  • ArcBees
    CTO, 2010 - 2011
  • University of British Columbia
    Postdoc, 2008 - 2010
  • Digital Fiction
    Software developer, 2001 - 2003
  • Matrox
    ASIC Architect, 2000 - 2001
  • Neo Dimensions
    CEO, 1999 - 2000
  • Université de Montréal
    Research and teching assistant, 1997 - 1998
  • Nortel
    Intern, 1996 - 1996
  • NetCorp
    Intern, 1995 - 1995
  • Teleglobe Canada
    Intern, 1993 - 1993
Places
Map of the places this user has livedMap of the places this user has livedMap of the places this user has lived
Currently
Ville Mont-Royal
Previously
Trois-Rivières - Montréal - Vancouver
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