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This paper proposes a privacy-preserving algorithm to solve the average consensus problem based on Shamir's secret sharing scheme, in which a network of agents reach an agreement on their states without exposing their individual state until an agreement is reached. Unlike other methods, the proposed algorithm renders the network resistant to the collusion of any given number of neighbors (even with all neighbors' colluding). Another virtue of this work is that such a method can protect the network consensus procedure from eavesdropping.

We consider a similar type of scenario for the disappearance of uniform of hyperbolicity as in Bjerklöv and Saprykina (2008 Nonlinearity 21), where it was proved that the minimum distance between invariant stable and unstable bundles has a linear power law dependence on parameters. In this scenario we prove that the Lyapunov exponent is sharp 1 2 -Hölder continuous. In particular, we show that the Lyapunov exponent of Schrödinger cocycles with a potential having a unique non-degenerate minimum, is sharp 1 2 -Hölder continuous below the lowest energy of the spectrum, in the large coupling regime.

Let Φ be a quasi-periodically forced quadratic map, where the rotation constant ω is a Diophantine irrational. A strange non-chaotic attractor (SNA) is an invariant (under Φ) attracting graph of a nowhere continuous measurable function ψ from the circle T to [0, 1].This paper investigates how a smooth attractor degenerates into a strange one, as a parameter β approaches a critical value β 0 , and the asymptotics behind the bifurcation of the attractor from smooth to strange. In our model, the cause of the strange attractor is a so-called torus collision, whereby an attractor collides with a repeller.Our results show that the asymptotic minimum distance between the two colliding invariant curves decreases linearly in the parameter β , as β approaches the critical parameter value β 0 from below.Furthermore, we have been able to show that the asymptotic growth of the supremum of the derivative of the attracting graph is asymptotically bounded from both sides by a constant times the reciprocal of the square root of the minimum distance above.