On Approximations of the Curve Shortening Flow and of the Mean Curvature Flow based on the DeTurck trick

In this paper we discuss novel numerical schemes for the computation of the curve shortening and mean curvature flows that are based on special reparametrizations. The main idea is to use special solutions to the harmonic map heat flow in order to reparametrize the equations of motion. This idea is widely known from the Ricci flow as the DeTurck trick. By introducing a variable time scale for the harmonic map heat flow, we obtain families of numerical schemes for the reparametrized flows. For the curve shortening flow this family unveils a surprising geometric connection between the numerical schemes in [5] and [9]. For the mean curvature flow we obtain families of schemes with good mesh properties similar to those in [3]. We prove error estimates for the semi-discrete scheme of the curve shortening flow. The behaviour of the fully-discrete schemes with respect to the redistribution of mesh points is studied in numerical experiments. We also discuss possible generalizations of our ideas to other extrinsic flows

Transition probabilities and measurement statistics of postselected ensembles

It is well-known that a quantum measurement can enhance the transition probability between two quantum states. Such a measurement operates after preparation of the initial state and before postselecting for the final state. Here we analyze this kind of scenario in detail and determine which probability distributions on a finite number of outcomes can occur for an intermediate measurement with postselection, for given values of the following two quantities: (i) the transition probability without measurement, (ii) the transition probability with measurement. This is done for both the cases of projective measurements and of generalized measurements. Among other constraints, this quantifies a trade-off between high randomness in a projective measurement and high measurement-modified transition probability. An intermediate projective measurement can enhance a transition probability such that the failure probability decreases by a factor of up to 2, but not by more.Comment: 23 pages, 5 figures, minor updat

Nonprofit Organizations as Ideal Type of Socially Responsible and Impact Investors

Nonprofit organizations (NPOs) as mission-driven organizations could profit from investing in stocks diametrically opposed to their mission, as they serve as a perfect hedge. Earning more income from oil or tobacco companies when there is a greater need for ecological interventions or cancer research might help effectively fighting the cause. We show the flaw in this logic as in its optimal state, this strategy is at most a financial zero-sum game. However, as NPOs strive at creating net value by aiming at a most effective mission-accomplishment, socially responsible and impact investments may offer a better way of doing so. We present NPOs as an ideal type of a socially responsible and impact investor and give the corresponding formal economic reasoning. For mission-driven organizations only the combination of financial and mission-based goals allows for an effective, goal-oriented financial decision-making. The full application of this logic is what is broadly understood under the term of mission investing (MI). Based on a theoretic introduction, we present a formalized way of analyzing multidimensional tradeoffs in the case of NPOs being mission-driven investors. This formalization will supply NPOs with a tool that enables them to capture their investments’ financial and mission-based impact and therefore the full benefit of responsible and impact-driven investments