Related papers: A Fire Fighter's Problem
A nonlinear equation describing curved stationary flames with arbitrary gas expansion $\theta = \rho_{{\rm fuel}}/\rho_{{\rm burnt}}$, subject to the Landau-Darrieus instability, is obtained in a closed form without an assumption of weak…
We consider a family of controlled reaction-diffusion equations, describing the spatial spreading of an invasive biological species. For a given propagation speed $c\in{I\!\!R}$, we seek a control with minimum cost, which achieves a…
We consider a perimeter surveillance problem introduced by Kingston, Beard, and Holt in 2008 and studied by Davis, Humphrey, and Kingston in 2019. In this problem, $n$ drones surveil a finite interval, moving at uniform speed and exchanging…
We consider a mean curvature flow in a cone, that is, a hypersurface in a cone which moves toward the opening with normal velocity equaling to the mean curvature, and the contact angle between the hypersurface and the cone boundary being…
We consider the setup employed in a recent experiment (Ramanathan et al 2011 Phys. Rev. Lett. 106 130401) devoted to the study of the instability of the superfluid flow of a toroidal Bose-Einstein condensate in presence of a repulsive…
Long range flight by fixed-wing aircraft without propulsion systems can be accomplished by "soaring" -- exploiting randomly located updrafts to gain altitude which is expended in gliding flight. As the location of updrafts is uncertain and…
This work considers the distance constrained formation control problem with an additional constraint requiring that the formation exhibits a specified spatial symmetry. We employ recent results from the theory of symmetry-forced rigidity to…
The collapse of a vapor bubble near a flat solid boundary results in the formation of a jet that is directed towards the boundary. In more complex geometries such as corners, predictions of the collapse cannot be made in a straightforward…
We consider a perimeter defense problem in a planar conical environment comprising a single turret that has a finite range and non-zero service time. The turret seeks to defend a concentric perimeter against $N\geq 2$ intruders. Upon…
The finite colliding bullets problem is the following simple problem: consider a gun, whose barrel remains in a fixed direction; let $(V_i)_{1\le i\le n}$ be an i.i.d.\ family of random variables with uniform distribution on $[0,1]$; shoot…
A flashover occurs when a fire spreads very rapidly through crevices due to intense heat. Flashovers present one of the most frightening and challenging fire phenomena to those who regularly encounter them: firefighters. Firefighters'…
We consider convection in an internally heated layer of fluid that is bounded below by a perfect insulator and above by a poor conductor. The poorly conducting boundary is modelled by a fixed heat flux. Using solely analytical methods, we…
We determine the asymptotic spreading speed of an invasive species, which invades the territory of a native competitor, governed by a diffusive competition model with a free boundary in a spherically symmetric setting. This free boundary…
Analysis of the speed of propagation in parabolic operators is frequently carried out considering the minimal speed at which its traveling waves move. This value depends on the solution concept being considered. We analyze an extensive…
This study investigates the critical conditions for flame propagation in channels with cold walls. We analyze the impact of the Lewis number and flow amplitude ($A$) on the minimum channel width required to sustain a premixed flame. Our…
We prove a blow-up criterion in terms of an $L_2$-bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve…
Forest fires pose a natural threat with devastating social, environmental, and economic implications. The rapid and highly uncertain rate of spread of wildfires necessitates a trustworthy digital tool capable of providing real-time…
The containment game is a full information game for two players, initialised with a set of occupied vertices in an infinite connected graph $G$. On the $t$-th turn, the first player, called Spreader, extends the occupied set to $g(t)$…
Recently Andrews and Bryan [3] discovered a comparison function which allows them to shorten the classical proof of the well-known fact that the curve shortening flow shrinks embedded closed curves in the plane to a round point. Using this…
We study the contraction of a convex immersed plane curve with speed (1/{\alpha})k^{{\alpha}}, where {\alpha}in(0,1] is a constant and show that, if the blow-up rate of the curvature is of type one, it will converge to a homothetic…