Related papers: A proof of the lonely runner conjecture
We show that the shifted Lonely Runner Conjecture (sLRC) holds for 5 runners. We also determine that there are exactly 3 primitive tight instances of the conjecture, only two of which are tight for the non-shifted conjecture (LRC). Our…
At a first glance, the problem of illuminating the boundary of a convex body by external light sources and the problem of covering a convex body by its smaller positive homothetic copies appear to be quite different. They are in fact two…
We give a definition for Obstacle Problems with measure data and general obstacles. For such problems we prove existence and uniqueness of solutions and consistency with the classical theory of Variational Inequalities. Continuous…
We are concerned with the computational problem of determining the covering radius of a rational polytope. This parameter is defined as the minimal dilation factor that is needed for the lattice translates of the correspondingly dilated…
Wills conjectured that, for any non-zero integers $u_1,\ldots,u_k$, there is a real number $t$ such that, for all $i=1,\ldots,k$, \[\lVert tu_i\rVert\geq\frac{1}{k+1},\] where $\lVert x\rVert$ is the distance from $x$ to the closest…
Linear-width is a well-known and highly regarded graph parameter. The concept of Single Ideal and Linear obstacle serves as an obstruction to linear-width on a connectivity sysem. In this concise paper, we present an alternative proof for…
We consider (n+1) runners with given constant unique integer speeds running along the circumference of a circle whose circumferential length is one, and all runners starting from the same point. We define and give lower bounds to a first…
We survey recent developments on the Restriction conjecture.
The purpose of this note is to give a self contained description of Walls finiteness obstruction.
All experiments attempting to verify the invariance of speed of light directly are based on two-way speed measurement. The challenge in one-way speed measurement, the requirement of spatially separated synchronised clocks, can be possibly…
In this work we present a general introduction to the Signorini problem (or thin obstacle problem). It is a self-contained survey that aims to cover the main currently known results regarding the thin obstacle problem. We present the theory…
In this paper, we study the distribution of the boundary points of expansion. As an application, we say something about the lonely runner problem. We show that given $k$ runners $\mathcal{S}_i$ round a unit circular track with the condition…
The Halting Problem is ill-conceived and ill-defined.
Based on the results people have obtained, we try to prove the Jacobian conjecture, but there is a gap in the proof.
It is proved that one can choose a control function on an arbitrary small open subset of the boundary of an obstacle so that the total radiation from this obstacle for a fixed direction of the incident plane wave and for a fixed wave number…
Nous refutons, sous une certaine hypothese combinatoire, la "nonrevisiting path conjecture". Abstract: In this article, we give, under some hypothesis, a couterexample to the nonrevisiting path conjecture.
We survey the state of the union-closed sets conjecture.
We show that the conclusion of a recent experiment [arXiv:0912.3934] that claims to have discovered that "the speed of light seems to depend on the motion of the observer" is wrong.
We consider the classical problem of making mobile processes gather or converge at a same position (as performed by swarms of animals in Nature). Existing works assume that each process can see all other processes, or all processes within a…
We prove Union-Closed sets conjecture.