Mustazee Rahman

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We consider the problem of finding an induced subgraph in a random dregular graph such that its components have bounded size as the size of the graph gets arbitrarily large. We show that for any threshold τ , the largest size density of such an induced subgraph with component sizes bounded by τ is at most 2(log d)/d for asymptotically large d. A matching(More)
We study the recursions A(n) = A(n−a−A(n−b))+A(A(n−b)) where a ≥ 0, b ≥ 1 are integers and the superscript k denotes a k-fold composition, and also the recursion C(n) = C(n − s − C(n − 1)) + C(n − s − 2 − C(n − 3)) where s ≥ 0 is an interger. We prove that under suitable initial conditions the sequences A(n) and C(n) will be defined for all positive(More)
The optimal power flow problem is concerned with finding a proper operating point for a power network while attempting to minimize some cost function and satisfy several network constraints. In this report we analyze the optimal power flow problem subject to contingency constraints, which demands that there be enough power to meet demands in the network in(More)
We study invariant percolation processes on the d-regular tree that are obtained as a factor of an iid process. We show that the density of any factor of iid site percolation process with finite clusters is asymptotically at most (log d)/d as d → ∞. This bound is asymptotically optimal as it can be realized by independent sets. One implication of the result(More)