The vortex blob method as a second-grade non-Newtonian fluid

We show that a certain class of vortex blob approximations for ideal hydrodynamics in two dimensions can be rigorously understood as solutions to the equations of second-grade non-Newtonian fluids with zero viscosity, and initial data in the space of Radon measures ${\mathcal M}({\mathbb R}^2)$. The solutions of this regularized PDE, also known as the averaged Euler or Euler-$\alpha$ equations, are geodesics on the volume preserving diffeomorphism group with respect to a new weak right invariant metric. We prove global existence of unique weak solutions (geodesics) for initial vorticity in ${\mathcal M}({\mathbb R}^2)$ such as point-vortex data, and show that the associated coadjoint orbit is preserved by the flow. Moreover, solutions of this particular vortex blob method converge to solutions of the Euler equations with bounded initial vorticity, provided that the initial data is approximated weakly in measure, and the total variation of the approximation also converges. In particular, this includes grid-based approximation schemes of the type that are usually used for vortex methods

Stability under Galerkin truncation of A-stable Runge--Kutta discretizations in time

We consider semilinear evolution equations for which the linear part is normal and generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of Hilbert spaces. We approximate their semiflow by an implicit, A-stable Runge--Kutta discretization in time and a spectral Galerkin truncation in space. We show regularity of the Galerkin-truncated semiflow and its time-discretization on open sets of initial values with bounds that are uniform in the spatial resolution and the initial value. We also prove convergence of the space-time discretization without any condition that couples the time step to the spatial resolution. Then we estimate the Galerkin truncation error for the semiflow of the evolution equation, its Runge--Kutta discretization, and their respective derivatives, showing how the order of the Galerkin truncation error depends on the smoothness of the initial data. Our results apply, in particular, to the semilinear wave equation and to the nonlinear Schr\"odinger equation

Exponentially accurate Hamiltonian embeddings of symplectic A-stable Runge--Kutta methods for Hamiltonian semilinear evolution equations

We prove that a class of A-stable symplectic Runge--Kutta time semidiscretizations (including the Gauss--Legendre methods) applied to a class of semilinear Hamiltonian PDEs which are well-posed on spaces of analytic functions with analytic initial data can be embedded into a modified Hamiltonian flow up to an exponentially small error. As a consequence, such time-semidiscretizations conserve the modified Hamiltonian up to an exponentially small error. The modified Hamiltonian is $O(h^p)$-close to the original energy where $p$ is the order of the method and $h$ the time step-size. Examples of such systems are the semilinear wave equation or the nonlinear Schr\"odinger equation with analytic nonlinearity and periodic boundary conditions. Standard Hamiltonian interpolation results do not apply here because of the occurrence of unbounded operators in the construction of the modified vector field. This loss of regularity in the construction can be taken care of by projecting the PDE to a subspace where the operators occurring in the evolution equation are bounded and by coupling the number of excited modes as well as the number of terms in the expansion of the modified vector field with the step size. This way we obtain exponential estimates of the form $O(\exp(-c/h^{1/(1+q)}))$ with $c>0$ and $q \geq 0$; for the semilinear wave equation, $q=1$, and for the nonlinear Schr\"odinger equation, $q=2$. We give an example which shows that analyticity of the initial data is necessary to obtain exponential estimates

Belief in free will affects causal attributions when judging others' behavior

Free will is a cornerstone of our society, and psychological research demonstrates that questioning its existence impacts social behavior. ;In six studies, we tested whether believing in free will is related to the correspondence bias, which reflects people's automatic tendency to overestimate the influence of internal as compared to external factors when interpreting others' behavior. All studies demonstrate a positive relationship between the strength of the belief in free will and the correspondence bias. Moreover, in two experimental studies, we showed that weakening participants' belief in free will leads to a reduction of the correspondence bias. Finally, the last study demonstrates that believing in free will predicts prescribed punishment and reward behavior, and that this relation is mediated by the correspondence bias. Overall, these studies show that believing in free will impacts fundamental social-cognitive processes that are involved in the understanding of others' behavior

Graphical functions in parametric space

Graphical functions are positive functions on the punctured complex plane $\mathbb{C}\setminus\{0,1\}$ which arise in quantum field theory. We generalize a parametric integral representation for graphical functions due to Lam, Lebrun and Nakanishi, which implies the real analyticity of graphical functions. Moreover we prove a formula that relates graphical functions of planar dual graphs.Comment: v2: extended introduction, minor changes in notation and correction of misprint

An empirical comparison of different implicit measures to predict consumer choice

While past research has found that implicit measures are good predictors of affectively driven, but not cognitively driven, behavior it has not yet been tested which implicit measures best predict behavior. By implementing a consumer context, in the present experiment, we assessed two explicit measures (i.e. self-reported habit and tastiness) and three implicit measures (i.e. manikin task, affective priming, ID-EAST) in order to test the predictive validity of affectively versus cognitively driven choices. The results indicate that irrespective of whether participants chose affectively or cognitively, both explicit measures, but not the implicit measures, predicted consumer choice very strongly. Likewise, when comparing the predictive validity among all measures, the explicit measures were the best predictors of consumer choice. Theoretical implications and limitations of the study are discussed

The effect of money priming on self-focus in the imitation-inhibition task : a registered report

The self-sufficiency hypothesis suggests that priming individuals with money makes them focus more strongly on themselves than on others. However, recently, research supporting this claim has been heavily criticized and some attempts to replicate have failed. A reason for the inconsistent findings in the field may lay in the common use of explicit measures, because they tend to rely on one or just a few items and are thus prone to demand effects and low reliability. In the present research, we administered, in two experiments, the imitation-inhibition task – a robust, unobtrusive, and reliable paradigm that is sensitive to self-other focus on a trial-by-trial basis. A pilot study found an increased focus on the self as compared to others when primed with money. Building on this finding, a preregistered high-powered experiment replicated this effect, suggesting that money primes may indeed increase a focus on the self. An additionally carried out meta-analysis indicates that automatic imitation is modulated by self-other focus and that money primes lead to a smaller focus on the self than conventional methods. Overall, the found effects are rather small and several limitations, such as order effects, call for a cautious interpretation of the findings

BCS theory in the weak magnetic field regime for systems with nonzero flux and exponential estimates on the adiabatic theorem in extended quantum lattice systems

In the main part of this PhD thesis, we consider a periodically realized microscopic superconductor described by BCS theory, which is subject to external electromagnetic fields. We show that the superconductor is properly described by Ginzburg--Landau theory in the macroscopic and weak magnetic field limit. The main novelty of our results is to allow for a non-vanishing magnetic flux through the unit cell of the lattice of periodicity. These main results are supplemented by various unpublished notes in the field of BCS theory. Furthermore, we preface the presentation of these results with a comprehensive introduction suitable for master's or PhD students. Thereby, we hope to contribute to filling the gap of missing introductory literature in the field. The thesis comprises a second topic, in which we provide ideas for setting up quantum lattice systems in order to prove exponential estimates for the adiabatic theorem. These notes are the result of studies in this field, which have been conducted during a research stay at the University of British Columbia (UBC) in Vancouver, Canada.Comment: PhD thesi