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...

Starting with values {f 0 j }j∈Z assigned to the integers, a binary subdivision scheme defines recursively values {fk j }j∈Z, respectively assigned to the binary points {2−kj}j∈Z. The purpose of subdivision analysis is to study the convergence of such processes and to establish the existence of a limit function on R and its smoothness class. A general treatment of uniform subdivision can be fou...

We study the influence of edge subdivision on the convex domination number. We show that in general an edge subdivision can arbitrarily increase and arbitrarily decrease the convex domination number. We also find some bounds for unicyclic graphs and we investigate graphs G for which the convex domination number changes after subdivision of any edge in G.

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...

The eternal inflation scenario predicts that our observable Universe resides inside a single bubble embedded in a vast inflating multiverse. We present the first observational tests of eternal inflation, performing a search for cosmological signatures of collisions with other bubble universes in cosmic microwave background data from the WMAP satellite. We conclude that the WMAP 7-year data do n...

The Catmull-Clark subdivision algorithm consists of an operator that can be decomposed into a refinement operator and a successively executed smoothing operator, where the refinement operator splits each face with m vertices into m quadrilateral subfaces and the smoothing operator replaces each internal vertex with an affine combination of its neighboring vertices and itself. Over regular meshe...

Given a polygonal subdivision of the plane and some query point, we would like to know which region (or edge or vertex) contains it. In one dimension, the problem reduces to locating a query point in a subdivision of the line into intervals, and is easily solved by binary search. We will look for a method to extend the idea of binary search into two dimensions. Given a subdivision with m edges,...

We show that Sobolev norm convergence of a stationary subdivision scheme is equivalent to standard norm convergence of the stationary subdivision scheme to a limit function in the Sobolev space. Stationary subdivision is an iterative method for the generation of reenable functions which are the main building blocks for the construction of wavelets. For this reason and also because of its use in...

let g(v,e) be a graph with p vertices and q edges. a graph g is said to have an odd mean labeling if there exists a function f : v (g) → {0, 1, 2,...,2q - 1} satisfying f is 1 - 1 and the induced map f* : e(g) → {1, 3, 5,...,2q - 1} defi ned by f*(uv) = (f(u) + f(v))/2 if f(u) + f(v) is evenf*(uv) = (f(u) + f(v) + 1)/2 if f(u) + f(v) is odd is a bijection. a graph that admits an odd mean lab...