Crypto-currencies like Bitcoin have recently attracted a lot of interest. A crucial ingredient into such systems is the "mining" of a Nakamoto blockchain. We model mining as a Poisson process with time-dependent intensity and use this model to derive predictions about block times for various hashrate scenarios (exponentially rising hash rate being the most important). We also analyse Bitcoin's method to update the "network difficulty" as a mechanism to keep block times stable. Since it yields systematically too fast blocks for exponential hash-rate growth, we propose a new method to update difficulty. Our proposed method performs much better at ensuring stable average block times over longer periods of time, which we verify both in simulations of artificial growth scenarios and with real-world data. Besides Bitcoin itself, this has practical benefits particularly for systems like Namecoin. It can be used to make name expiration times more predictable, preventing accidental loss of names.
Level-sets are a flexible method to describe geometries and their changes according to a speed field. This can be used in a wide variety of applications. We will present a Hopf-Lax formula that can be used to represent the solution of the level-set equation as well as the described geometries directly. This formula is a generalisation of existing results to the case of speed fields without a uniform, positive lower bound. The corresponding equation is of Hamilton-Jacobi type with a nonconvex Hamiltonian. Our representation formula can be used both for theoretical and numerical purposes. In the latter case, the Fast Marching Method can be applied, leading to very efficient and robust numerical calculations of the geometry evolutions. We will also apply the level-set framework to an illustrative problem in PDEconstrained shape optimisation, and present numerical results.
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