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...

a roman dominating function (rdf) on a graph g = (v,e) is defined to be a function satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. a set s v is a restrained dominating set if every vertex not in s is adjacent to a vertex in s and to a vertex in . we define a restrained roman dominating function on a graph g = (v,e) to be ...

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 ...

In the random geometric graph G(n,rn), n vertices are placed randomly in Euclidean d-space and edges added between any pair of distant at most rn from each other. We establish strong laws large numbers (LLNs) for a class parameters, evaluated G(n,rn) thermodynamic limit with nrnd= const., also dense nrnd→∞, rn→0. Examples include domination number, independence clique-covering eternal number tr...

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...