An Improved Upper Bound for Bootstrap Percolation in All Dimensions

In r-neighbour bootstrap percolation on the vertex set of a graph G, a set A of initially infected vertices spreads by infecting, at each time step, all uninfected vertices with at least r previously infected neighbours. When the elements of A are chosen independently with some probability p, it is natural to study the critical probability pc(G, r) at which it becomes likely that all of V(G) will eventually become infected. Improving a result of Balogh, Bollobás and Morris, we give a bound on the second term in the expansion of the critical probability when G = [n]d and d ⩾ r ⩾ 2. We show that for all d ⩾ r ⩾ 2 there exists a constant cd,r > 0 such that if n is sufficiently large, then $$p_c (\left[ n \right]^d ,{\rm{ }}r){\rm{\le }}\left( {\frac{{\lambda (d,r)}}{{\log _{(r - 1)} (n)}} - \frac{{c_{d,r} }}{{(\log _{(r - 1)} (n))^{3/2} }}} \right)^{d - r + 1} ,$$where λ(d, r) is an exact constant and log(k) (n) denotes the k-times iterated natural logarithm of n.

Frames and subspaces

This thesis will consist of three parts. In the first part we find the closest probabilistic Parseval frame to a given probabilistic frame in the 2 Wasserstein Distance. It is known that in the traditional [symbol]2 distance the closest Parseval frame to a frame [phi] = {[symbol]i} N[i=1] [symbol] R[d] is [phi[�] = {[symbol] � i }N[i]=1 = {S [-1/2][[symbol]i]} N[i=1] where S is the frame operator of [phi]. We use this fact to prove a similar statement about probabilistic frames in the 2 Wasserstein metric. In the second part, we will associate a complex vector with a rank 2 real projection. Using this association we will answer many open questions in frame theory. In particular we will prove Edidin's theorem in phase retrieval in the complex case, answer a question on mutually unbiased bases, a question on equiangular lines, and a question on fusion frames. In the last part we will give a way to calculate the exact constant for the [symbol]1 � [symbol]2 inequality and use this method to prove a couple of interesting theorems

Railroad Capital Stock Changes in the Post-Deregulation Period

This paper involves the estimation of a model of railway road and equipment capital stocks and the changes in their levels that have occurred since 1983. The model is based on balancing the level of investment and the level of degradation of the capital stocks to create a data series for roadway capital and for equipment capital. A two-stage least squares errors in variables model is applied. This is appropriate as degradation and exact constant dollar investment are not directly observable. Results obtained from the model indicate that Class I railroads have increased their absolute capital stock levels over the period examined. This holds for both roadway and equipment capital although roadway capital has increased at a somewhat faster rate.