Low-complexity computations for nilpotent subgroup problems

Low complexity differential geometric computations

In this thesis, we consider the problem of fast and efficient indexing techniques for time sequences which evolve on manifold-valued spaces. Using manifolds is a convenient way to work with complex features that often do not live in Euclidean spaces. However, computing standard notions of geodesic distance, mean etc. can get very involved due to the underlying non-linearity associated with the ...

Quantum measurements for hidden subgroup problems with optimal sample complexity

One of the central issues in the hidden subgroup problem is to bound the sample complexity, i.e., the number of identical samples of coset states sufficient and necessary to solve the problem. In this paper, we present general bounds for the sample complexity of the identification and decision versions of the hidden subgroup problem. As a consequence of the bounds, we show that the sample compl...

Stability Computations for Nilpotent Hopf bifurcations in Coupled Cell Systems

Vanderbauwhede and van Gils, Krupa, and Langford studied unfoldings of bifurcations with purely imaginary eigenvalues and a nonsemisimple linearization, which generically occurs in codimension three. In networks of identical coupled ODE these nilpotent Hopf bifurcations can occur in codimension one. Elmhirst and Golubitsky showed that these bifurcations can lead to surprising branching patterns...

Improving computations by using low complexity matrix algebras

(the proof of the fact that Bn is uniquely defined is left to the reader, it could be a further exercise). First observe that any Bn is monic, in fact, if Bn(x) = anx n + an−1xn−1 + . . . then Bn(x+ 1)−Bn(x) = (an(x + 1) + an−1(x+ 1)n−1 + . . .)− (anx + an−1x + . . .) = [an(x n + nxn−1 + . . .) + an−1(x n−1 + (n− 1)xn−2 + . . .) + . . .]− (anx + an−1x + . . .) = annx n−1 + (·)xn−2 + . . . and t...

Finite Groups With a Certain Number of Elements Pairwise Generating a Non-Nilpotent Subgroup