Related papers: The Lonely Runner Conjecture turns 60
A fixed number of passengers independently board one of several buses uniformly at random. The lonely passenger problem is to prove that the probability of at least one passenger being the only one in their bus is increasing in the number…
Many probability measures in the multiverse depend exponentially on some observable parameters, giving rise to potential problems such as youngness bias, Q-catastrophe etc. In this paper we explore a possibility that the exponential runaway…
We consider the problem of locating the source (starting vertex) of a simple random walk, given a snapshot of the set of edges (or vertices) visited in the first $n$ steps. Considering lattices $\mathbb{Z}^d$, in dimensions $d \geq 5$, we…
A thought experiment is described and the probability of a particular type of results is predicted according to the quantum formalism. Then, the assumption is made that there exists a particle that travels from the source to one of the…
We prove an estimate for the probability that a simple random walk in a simply connected subset A of Z^2 starting on the boundary exits A at another specified boundary point. The estimates are uniform over all domains of a given inradius.…
We review some representative results for first-passage problems involving so-called mortal or evanescent walkers, i.e., walkers with a finite lifetime. The mortality constraint plays a key role in the modeling of many real scenarios, as it…
As quantum theory celebrates its 100th birthday, spectacular successes are mixed with outstanding puzzles and promises of new technologies. This article reviews both the successes of quantum theory and the ongoing debate about its…
The results summarized here are intended as rigorous mathematical statements on various physical models coming from condensed matter physics, statistical mechanics (classical and quantum), quantum field theory and cold atoms physics. The…
The development of inventions is theorized as a process of searching and recombining existing knowledge components. Previous studies under this theory have examined myriad characteristics of recombined knowledge and their performance…
Random walk is one of the most classical and well-studied model in probability theory. For two correlated random walks on lattice, every step of the random walks has only two states, moving in the same direction or moving in the opposite…
We consider a walker that at each step keeps the same direction with a probabilitythat depends on the time already spent in the direction the walker is currently moving. In this paper, we study some asymptotic properties of this persistent…
In this issue we demonstrate the very inspiring role of the continuous-time random walk (CTRW) formalism and its numerous modifications thanks to their flexibility and various applications as well its promising perspectives in different…
Injuries to the knee joint are very common for long-distance and frequent runners, an issue which is often attributed to fatigue. We address the problem of fatigue detection from biomechanical data from different sources, consisting of…
In honor of Daniel Greenberger's 65th birthday I record for posterity two superb examples of his wit, offer a proof of an important theorem on quantum correlations that even those of us over 60 can understand, and suggest, by trying to make…
The Polya number of a classical random walk on a regular lattice is known to depend solely on the dimension of the lattice. For one and two dimensions it equals one, meaning unit probability to return to the origin. This result is extremely…
Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are…
We prove a conjecture due to Nicolas Chevallier concerning unimodular matrices related to simultaneous Diophantine approximation to real numbers.
Autoformalisation, the task of expressing informal mathematical statements in formal language, is often viewed as a direct translation process. This, however, disregards a critical preceding step: conjecturing. Many mathematical problems…
A time-dependent model of space-time is used to suggest simple explanations for the increasing of the Astronomical unit, as well as the increasing of the distance from the Earth to the Moon. The Pioneer anomaly is also considered.
Using the discrepancy metric, we analyze the rate of convergence of a random walk on the circle generated by d rotations, and establish sharp rates that show that badly approximable d-tuples in R^d give rise to walks with the fastest…