Eternal and m-eternal domination are concerned with using mobile guards to protect a graph against infinite sequences of attacks at vertices. Eternal domination allows one guard to move per attack, whereas more than one guard may move per attack in the m-eternal domination model. Inequality chains consisting of the domination, eternal domination, m-eternal domination, independence, and clique c...

Abstract. Eternal solutions of parabolic equations (those which are defined for all time) are typically rather rare. For example, the heat equation has exactly one eternal solution – the trivial solution. While solutions to the heat equation exist for all forward time, they cannot be extended backwards in time. Nonlinearities exasperate the situation somewhat, in that solutions may form singula...

Inflation is known to be generically eternal to the future: the false vacuum is thermalized in some regions of space, while inflation continues in other regions. Here, we address the question of whether inflation can also be eternal to the past. We argue that such a steady-state picture is impossible and, therefore, that inflation must have had a beginning. First, it is shown that the old infla...

We investigate the condition for eternal inflation to take place in the noncommutative spacetime. We find that the possibility for eternal inflation’s happening is greatly suppressed in this case. If eternal inflation can not happen in the low energy region where the noncommutativity is very weak (the UV region), it will never happen along the whole inflationary history. Based on these conclusi...

In the eternal dominating set problem, guards form a dominating set on a graph and at each step, a vertex is attacked. After each attack, if the guards can “move” to form a dominating set that contains the attacked vertex, then the guards have successfully defended against the attack. We wish to determine the minimum number of guards required to successfully defend against any possible sequence...

Models of eternal inflation predict a stochastic self-similar geometry of the universe at very large scales and allow existence of points that never thermalize. I explore the fractal geometry of the resulting spacetime, using coordinate-independent quantities. The formalism of stochastic inflation can be used to obtain the fractal dimension of the set of eternally inflating points (the “eternal...

An eternal $m$-secure set of a graph $G = (V,E)$ is aset $S_0subseteq V$ that can defend against any sequence ofsingle-vertex attacks by means of multiple-guard shifts along theedges of $G$. A suitable placement of the guards is called aneternal $m$-secure set. The eternal $m$-security number$sigma_m(G)$ is the minimum cardinality among all eternal$m$-secure sets in $G$. An edge $uvin E(G)$ is ...

Motivated by the lessons of black hole complementarity, we develop a causal patch description of eternal inflation. We argue that an observer cannot ascribe a semiclassical geometry to regions outside his horizon, because the large-scale metric is governed by the fluctuations of quantum fields. In order to identify what is within the horizon, it is necessary to understand the late time asymptot...