The Lonely Runner with Seven Runners
نویسندگان
چکیده
منابع مشابه
The Lonely Runner with Seven Runners
Suppose k + 1 runners having nonzero constant speeds run laps on a unit-length circular track starting at the same time and place. A runner is said to be lonely if she is at distance at least 1/(k + 1) along the track to every other runner. The lonely runner conjecture states that every runner gets lonely. The conjecture has been proved up to six runners (k ≤ 5). A formulation of the problem is...
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The Lonely Runner Conjecture, posed independently by Wills and by Cusick, states that for any set of runners running along the unit circle with constant different speeds and starting at the same point, there is a time where all of them are far enough from the origin. We study the correlation among the time that runners spend close to the origin. By means of these correlations, we improve a resu...
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For x real, let {x} be the fractional part of x (i.e. {x} = x − bxc). In this paper we prove the k = 5 case of the following conjecture (the lonely runner conjecture): for any k positive reals v1, . . . , vk there exists a real number t such that 1/(k + 1) ≤ {vit} ≤ k/(k + 1) for i = 1, . . . , k.
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We prove the following result. Let G be an undirected graph. If G has a nowhere zero ow with at most k di erent values, then it also has one with values from the set f1; : : : ; kg. When k 5, this is a trivial consequence of Seymour's \sixow theorem". When k 4 our proof is based on a lovely number theoretic problem which we call the \Lonely Runner Conjecture". Suppose k runners having nonzero c...
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The Lonely Runner Conjecture of J. Wills asserts the following. For any vector v ∈ (R−{0})n−1 there exists t ∈ R such that every component of tv has distance at least 1/n to the nearest integer. We study those vectors v for which the bound of this conjecture is attained. In particular, we construct an infinite family of such tight vectors. This family, plus three sporadic examples, constitute a...
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2008
ISSN: 1077-8926
DOI: 10.37236/772