Manhattan orbifolds

We investigate a class of metrics for 2-manifolds in which, except for a discrete set of singular points, the metric is locally isometric to an L_1 (or equivalently L_infinity) metric, and show that with certain additional conditions such metrics are injective. We use this construction to find the tight span of squaregraphs and related graphs, and we find an injective metric that approximates the distances in the hyperbolic plane analogously to the way the rectilinear metrics approximate the Euclidean distance.Comment: 17 pages, 15 figures. Some definitions and proofs have been revised since the previous version, and a new example has been adde

Numerical treatment of a generalized Vandermonde system of equations

AbstractA stable method is proposed for the numerical solution of a linear system of equations having a generalized Vandermonde matrix. The method is based on Gaussian elimination and establishes explicit expressions for the elements of the resulting upper triangular matrix. These elements can be computed by means of sums of exclusively positive terms. In an important special case these sums can be reduced to simple recursions. Finally the method is retraced for the case of a confluent type of generalized Vandermonde matrix

The simplicial boundary of a CAT(0) cube complex

For a CAT(0) cube complex $\mathbf X$, we define a simplicial flag complex $\partial_\Delta\mathbf X$, called the \emph{simplicial boundary}, which is a natural setting for studying non-hyperbolic behavior of $\mathbf X$. We compare $\partial_\Delta\mathbf X$ to the Roller, visual, and Tits boundaries of $\mathbf X$ and give conditions under which the natural CAT(1) metric on $\partial_\Delta\mathbf X$ makes it (quasi)isometric to the Tits boundary. $\partial_\Delta\mathbf X$ allows us to interpolate between studying geodesic rays in $\mathbf X$ and the geometry of its \emph{contact graph} $\Gamma\mathbf X$, which is known to be quasi-isometric to a tree, and we characterize essential cube complexes for which the contact graph is bounded. Using related techniques, we study divergence of combinatorial geodesics in $\mathbf X$ using $\partial_\Delta\mathbf X$. Finally, we rephrase the rank-rigidity theorem of Caprace-Sageev in terms of group actions on $\Gamma\mathbf X$ and $\partial_\Delta\mathbf X$ and state characterizations of cubulated groups with linear divergence in terms of $\Gamma\mathbf X$ and $\partial_\Delta\mathbf X$.Comment: Lemma 3.18 was not stated correctly. This is fixed, and a minor adjustment to the beginning of the proof of Theorem 3.19 has been made as a result. Statements other than 3.18 do not need to change. I thank Abdul Zalloum for the correction. See also: arXiv:2004.01182 (this version differs from previous only by addition of the preceding link, at administrators' request

Making sense of punishment: Transgressors' interpretation of punishment motives determines the effects of sanctions

Punishment is expected to have an educative, behaviour-controlling effect on the transgressor. Yet, this effect often remains unattained. Here, we test the hypothesis that transgressors' inferences about punisher motives crucially shape transgressors' post-punishment attitudes and behaviour. As such, we give primacy to the social and relational dimensions of punishment in explicating how sanctions affect outcomes. Across four studies using different methodologies (N = 1189), our findings suggest that (a) communicating punishment respectfully increases transgressor perceptions that the punisher is trying to repair the relationship between the transgressor and their group (relationship-oriented motive) and reduces perceptions of harm-oriented and self-serving motives, and that (b) attributing punishment to relationship-oriented (vs. harm/self-oriented, or even victim-oriented) motives increases prosocial attitudes and behaviour. This research consolidates and extends various theoretical perspectives on interactions in justice settings, providing suggestions for how best to deliver sanctions to transgressors