Related papers: On the Gaussian Moat Problem
We study the motion of two non-interacting quantum particles performing a random walk on a line and analyze the probability that the two particles are detected at a particular position after a certain number of steps (meeting problem). The…
Gaussian processes are a powerful framework for quantifying uncertainty and for sequential decision-making but are limited by the requirement of solving linear systems. In general, this has a cubic cost in dataset size and is sensitive to…
We consider in this article an Elephant Random Walk evolving in the plane. Specifically, this is a reinforced stochastic process in which the $n$th step is given by a random rotation of one of the previous steps chosen uniformly at random.…
A. Tarski uses in his system for the elementary geometry only the primitive concept of point, and the two primitive relations betweenness and equidistance. Another approach is the relations to be on lines instead of points. W.…
Mixture models are one of the most widely used statistical tools when dealing with data from heterogeneous populations. This paper considers the long-standing debate over finite mixture and infinite mixtures and brings the two modelling…
We consider the motion of a particle on a Galton Watson tree, when the probabilities of jumping from a vertex to any one of its neighbours is determined by a random process. Given the tree, positive weights are assigned to the edges in such…
In this paper a new class of modified-Hessian equations, closely related to the Optimal Transportation Equation, will be introduced and studied. In particular, the existence of globally smooth, classical solutions of these equations…
We consider a new model of a branching random walk on a multidimensional lattice with continuous time and one source of particle reproduction and death, as well as an infinite number of sources in which, in addition to the walk, only…
We show that the particle motion in Bohmian mechanics, given by the solution of an ordinary differential equation, exists globally: For a large class of potentials the singularities of the velocity field and infinity will not be reached in…
This work is a numerical experiment of stochastic motion of conservative Hamiltonian system or weakly damped Brownian particles. The objective is to prove the existence of path probability and to compute its values. By observing a large…
We prove a limit theorem on the convergence of the distributions of the scaled last exit time over a slowly moving nonlinear boundary for a class of Gaussian stationary processes. The limit is a double exponential (Gumbel) distribution.
A unitarily invariant projective framework is introduced to analyze the complexity of path-following methods for the eigenvalue problem. A condition number, and its relation to the distance to ill-posedness, is given. A Newton map…
The first-passage-time problem for a Brownian motion with alternating infinitesimal moments through a constant boundary is considered under the assumption that the time intervals between consecutive changes of these moments are described by…
We investigate necessary and sufficient conditions for the extendibility and boundedness of Gaussian curvature, Mean curvature and principal curvatures near all types of singularities on fronts. We also study the convergence to infinite…
Complex metrics are a double-edged sword: they allow one to replace singular spacetimes, such as those containing a big bang, with regular metrics, yet they can also describe unphysical solutions in which quantum transitions may be more…
In this study, the problem of overcoming the infinite energy barrier separating the bradyonic and tachyonic realms is investigated. Making use of the Majorana equation for particles with arbitrary spin and the Heisenberg uncertainty…
Motivated by Tweedie's formula for the Compound Decision problem, we examine the theoretical foundations of empirical Bayes estimators that directly model the marginal density $m(y)$. Our main result shows that polynomial log-marginals of…
Gaussian process (GP) modulated Cox processes are widely used to model point patterns. Existing approaches require a mapping (link function) between the unconstrained GP and the positive intensity function. This commonly yields solutions…
We prove local well-posedness for the Muskat problem on the half-plane, which models motion of an interface between two fluids of distinct densities (e.g., oil and water) in a porous medium (e.g., an aquifer) that sits atop an impermeable…
A vast literature on convergence guarantees for gradient descent and derived methods exists at the moment. However, a simple practical situation remains unexplored: when a fixed step size is used, can we expect gradient descent to converge…