On Iterated Dominance, Matrix Elimination, and Matched Paths

We study computational problems arising from the iterated removal of weakly dominated actions in anonymous games. Our main result shows that it is NP-complete to decide whether an anonymous game with three actions can be solved via iterated weak dominance. The two-action case can be reformulated as a natural elimination problem on a matrix, the complexity of which turns out to be surprisingly difficult to characterize and ultimately remains open. We however establish connections to a matching problem along paths in a directed graph, which is computationally hard in general but can also be used to identify tractable cases of matrix elimination. We finally identify different classes of anonymous games where iterated dominance is in P and NP-complete, respectively.Comment: 12 pages, 3 figures, 27th International Symposium on Theoretical Aspects of Computer Science (STACS

Simplicial blowups and discrete normal surfaces in simpcomp

simpcomp is an extension to GAP, the well known system for computational discrete algebra. It allows the user to work with simplicial complexes. In the latest version, support for simplicial blowups and discrete normal surfaces was added, both features unique to simpcomp. Furthermore, new functions for constructing certain infinite series of triangulations have been implemented and interfaces to other software packages have been improved to previous versions.Comment: 10 page

On annealed elliptic Green function estimates

We consider a random, uniformly elliptic coefficient field $a$ on the lattice $\mathbb{Z}^d$. The distribution $\langle \cdot \rangle$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ resp. $L^1$ in probability, i.e. they obtained bounds on $\langle |\nabla_x G(t,x,y)|^2 \rangle^{\frac{1}{2}}$ and $\langle |\nabla_x \nabla_y G(t,x,y)| \rangle$, see T. Delmotte and J.-D. Deuschel: On estimating the derivatives of symmetric diffusions in stationary random environments, with applications to the $\nabla\phi$ interface model, Probab. Theory Relat. Fields 133 (2005), 358--390. In particular, the elliptic Green function $G(x,y)$ satisfies optimal annealed bounds. In a recent work, the authors extended these elliptic bounds to higher moments, i.e. $L^p$ in probability for all $p<\infty$, see D. Marahrens and F. Otto: {Annealed estimates on the Green function}, arXiv:1304.4408 (2013). In this note, we present a new argument that relies purely on elliptic theory to derive the elliptic estimates (see Proposition 1.2 below) for $\langle |\nabla_x G(x,y)|^2 \rangle^{\frac{1}{2}}$ and $\langle |\nabla_x \nabla_y G(x,y)| \rangle$.Comment: 15 page

Stacked polytopes and tight triangulations of manifolds

Tightness of a triangulated manifold is a topological condition, roughly meaning that any simplexwise linear embedding of the triangulation into euclidean space is "as convex as possible". It can thus be understood as a generalization of the concept of convexity. In even dimensions, super-neighborliness is known to be a purely combinatorial condition which implies the tightness of a triangulation. Here we present other sufficient and purely combinatorial conditions which can be applied to the odd-dimensional case as well. One of the conditions is that all vertex links are stacked spheres, which implies that the triangulation is in Walkup's class $\mathcal{K}(d)$. We show that in any dimension $d\geq 4$ \emph{tight-neighborly} triangulations as defined by Lutz, Sulanke and Swartz are tight. Furthermore, triangulations with $k$-stacked vertex links and the centrally symmetric case are discussed.Comment: 28 pages, 2 figure

Time Quasilattices in Dissipative Dynamical Systems

We establish the existence of `time quasilattices' as stable trajectories in dissipative dynamical systems. These tilings of the time axis, with two unit cells of different durations, can be generated as cuts through a periodic lattice spanned by two orthogonal directions of time. We show that there are precisely two admissible time quasilattices, which we term the infinite Pell and Clapeyron words, reached by a generalization of the period-doubling cascade. Finite Pell and Clapeyron words of increasing length provide systematic periodic approximations to time quasilattices which can be verified experimentally. The results apply to all systems featuring the universal sequence of periodic windows. We provide examples of discrete-time maps, and periodically-driven continuous-time dynamical systems. We identify quantum many-body systems in which time quasilattices develop rigidity via the interaction of many degrees of freedom, thus constituting dissipative discrete `time quasicrystals'.Comment: 38 pages, 14 figures. This version incorporates "Pell and Clapeyron Words as Stable Trajectories in Dynamical Systems", arXiv:1707.09333. Submission to SciPos

The photon polarization tensor in a homogeneous magnetic or electric field

We revisit the photon polarization tensor in a homogeneous external magnetic or electric field. The starting point of our considerations is the momentum space representation of the one-loop photon polarization tensor in the presence of a homogeneous electromagnetic field, known in terms of a double parameter integral. Our focus is on explicit analytical insights for both on- and off-the-light-cone dynamics in a wide range of well-specified physical parameter regimes, ranging from the perturbative to the manifestly nonperturbative strong field regime. The basic ideas underlying well-established approximations to the photon polarization tensor are carefully examined and critically reviewed. In particular, we systematically keep track of all contributions, both the ones to be neglected and those to be taken into account explicitly, to all orders. This allows us to study their ranges of applicability in a much more systematic and rigorous way. We point out the limitations of such approximations and manage to go beyond at several instances.Comment: 43 pages, 2 figures; two misprints in Eqs. (118) and (142) corrected (a factor 2^(-2/3) was missing