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Seb Paquet
Worked at Sceneverse, Inc.
Attended Université de Montréal
Lives in Montreal
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Seb Paquet

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#Math  
 
The Lonely Runner Conjecture

Suppose that six runners are running around and around a circular track, each running at a different constant speed, but in the same direction. Will each runner eventually be in a situation where each of the other five runners is at least 1/6 of a lap away? According to the Lonely Runner Conjecture, the answer is yes.

This animation from Wikipedia by Claudio Rocchini illustrates the situation for six runners. The runners are initially represented by black dots, which turn to yellow as soon as the corresponding runner has 1/6 of a lap of empty track both ahead of and behind them. Eventually, all the runners have had the experience of being in this state.

In general if the number of runners on the circular track is k, a runner is said to be lonely if each of the other runners is at least 1/k of a lap away. The conjecture is that if k is at least 2 and each runner runs at a different speed, then each runner will be “lonely” at some point.

The conjecture was originally made by J.M. Wills in 1967. Although it is easy to state, the conjecture is even now only known to be true in general for small values of k. The case k=2 is trivial, the case k=3 is not too hard, and the case k=4 was proved in 1972. A proof of the case k=5 involving a computer check was found in 1984, and a more elegant proof was found in 1998. The case k=6 was proved in 2001, with a shorter proof being found in 2004. The case k=7 was proved by Barajas and Serra in 2008, but the problem is still open if k is 8 or more.

The Lonely Runner Conjecture has points of contact with other areas of mathematics. As Barajas and Serra explain in the introduction to their paper (http://arxiv.org/abs/0710.4495), the positive solution of the conjecture would prove a theorem about nowhere zero flows in regular matroids. The known results about the conjecture have also been used to compute chromatic numbers of distance graphs.

If all the runners' speeds are irrational multiples of each other, the problem becomes easy and the conjecture is known to be true. For this reason, attacks on the problem focus on the most difficult case, where all the speeds are rational multiples of each other. This case quickly reduces to the case where all the runners' speeds are integers with no common factor greater than one. We may also assume that one of the speeds is zero, meaning that one of the “runners” is stationary. 

In the case of six runners, the stationary runner will be lonely when the other five runners are bunched in the 4/6 of the track centred at the point diametrically opposite the stationary runner. This point of view can be used to recast the conjecture in terms of a line of sight problem in Euclidean space. My previous post on this topic goes into more detail about this aspect of the problem (https://plus.google.com/101584889282878921052/posts/JMtPiZeVVnT) and provides a link to another relevant paper.

Picture source: http://en.wikipedia.org/wiki/Lonely_runner_conjecture

#mathematics #scienceeveryday #spnetwork arXiv:0710.4495
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Bob Haugen's profile photo
 
I need to mute all animated gifs, regardless of how fascinating they may be...for my mental health...
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Seb Paquet

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Yes baby
 
Methodology has by now become so crowded with empty sophistication that it is extremely difficult to perceive the simple errors at the basis. It is like fighting the hydra - cut off one ugly head, and eight formalizations take its place. In this situation the only answer is superficiality: when sophistication loses content then the only way of keeping in touch with reality is to be crude and superficial. This is what I intend to be.

--- Paul Feyerabend
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Seb Paquet

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Beautiful. Yann Tiersen's music doesn't hurt.
 
Nice super creative "juggling".
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"The mathematical modeling of society is made possible, according to Pentland, by the innate tractability of human beings. We may think of ourselves as rational actors, in conscious control of our choices, but most of what we do is reflexive. Our behavior is determined by our subliminal reactions to the influence of other people, particularly those in the various peer groups we belong to. “The power of social physics,” he writes, “comes from the fact that almost all of our day-to-day actions are habitual, based mostly on what we have learned from observing the behavior of others.” Once you map and measure all of a person’s social influences, you can develop a statistical model that predicts that person’s behavior, just as you can model the path a billiard ball will take after it strikes other balls."
Tapping into big data, researchers and planners are building mathematical models of personal and civic behavior. But the models may hide rather than reveal the deepest sources of social ills.
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Edward Vielmetti's profile photoGuido Stevens's profile photoSeb Paquet's profile photoPhilippe Beaudoin's profile photo
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Reductionist model that trivializes human behavior into simple interacting finite-state machines.
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Have him in circles
4,698 people
Stéphane Brodu's profile photo

Seb Paquet

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American middle class becoming an oxymoron
 
I reckon it doesn't make much sense to directly compare USD incomes in different countries because the prices are quite different everywhere.
After three decades of slow growth, middle-class incomes in the U.S. appear to trail those of Canada. Poor Americans now make less than the poor in several other countries.
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Do not miss.
 
This is awesome and hilarious.
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John hd Dewey's profile photoAlexander Nikitin's profile photo
 
Fascinating. The inherent time delay in neural communication seems to contribute to the an apparent smoothness in gate. Takeaway the delay and motion becomes sharper. 
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Seb Paquet

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In a way...yes. #JustSaying  
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easy
In his latest video, CrazyRussianHacker has a tip for taking off your t-shirt very quickly using one hand. Just tuck your elbow in, grab the underside of the sleeve with the opposite hand and yank ...
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Have him in circles
4,698 people
Stéphane Brodu's profile photo
Work
Occupation
Culture Hacker
Employment
  • Sceneverse, Inc.
    VP & Social Alchemist
  • National Research Council of Canada
    Research Officer
  • Socialtext, Inc.
  • Téluq-UQAM
    Professor
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Map of the places this user has livedMap of the places this user has livedMap of the places this user has lived
Currently
Montreal
Previously
moncton - hull, qc - vancouver, bc
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Bragging rights
I coined the phrase "Ridiculously Easy Group-Forming".
Education
  • Université de Montréal
    Ph.D. Computer Science, 2003
  • Université de Montréal
    B.Sc. Physics, 1997
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Male
Birthday
October 1