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In this article we consider the application of Euler's homogeneous function theorem together with Stokes' theorem to exactly integrate families of polynomial spaces over general polygonal and polyhedral (polytopic) domains in two and three dimensions, respectively. This approach allows for the integrals to be evaluated based on only computing the values of the integrand and its derivatives at the vertices of the polytopic domain, without the need to construct a sub-tessellation of the underlying domain of interest. Here, we present a detailed analysis of the computational complexity of the proposed algorithm and show that this depends on three key factors: the ambient dimension of the underlying polytopic domain; the size of the requested polynomial space to be integrated; and the size of a directed graph related to the polytopic domain. This general approach is then employed to compute the volume integrals arising within the discontinuous Galerkin finite element approximation of the linear transport equation. Numerical experiments are presented which highlight the efficiency of the proposed algorithm when compared to standard quadrature approaches defined on a sub-tessellation of the polytopic elements.

This article is about the death of a graduate school for spiritual formation and theological education. I ask what the death of this school can teach us about teaching theology and religion for the sake of student transformation and more loving pedagogies. I also ask how, when an institution dies, it can die beautifully. The underlying thesis of this article is that there is, alive in our classrooms (and ourselves, perhaps), a spiritual wound that can be named an experiential death of divine love--and that this wound is manifesting as unshepherded fear, rage, and discontent. I suggest that this spiritual wound be tended through skillful pedagogical tenderness, flexibility, and liberative co-creation with students to cultivate a presence of love and feed the spiritual hungers of our time. I also suggest that a school (and a teacher) dies beautifully when its death is allowed and free to become a scene of beautiful instruction.

Planar cell polarity (PCP)--the coordinated polarisation of a whole field of cells within the plane of a tissue-relies on the interaction of three modules: a global module that couples individual cellular polarity to the tissue axis, a local module that aligns the axis of polarisation of neighbouring cells, and a readout module that directs the correct outgrowth of PCP-regulated structures such as hairs and bristles. While much is known about the molecular components that are required for PCP, the functional details of--and interactions between--the modules remain unclear. In this work, we perform a mathematical and computational analysis of two previously proposed computational models of the local module (Amonlirdviman et al., Science, 307, 2005; Le Garrec et al., Dev. Dyn., 235, 2006). Both models can reproduce wild-type and mutant phenotypes of PCP observed in the Drosophila wing under the assumption that a tissue-wide polarity cue from the global module persists throughout the development of PCP. We demonstrate that both models can also generate tissue-level PCP when provided with only a transient initial polarity cue. However, in these models such transient cues are not sufficient to ensure robustness of the resulting cellular polarisation.

This article is about the death of a graduate school for spiritual formation and theological education. I ask what the death of this school can teach us about teaching theology and religion for the sake of student transformation and more loving pedagogies. I also ask how, when an institution dies, it can die beautifully. The underlying thesis of this article is that there is, alive in our classrooms (and ourselves, perhaps), a spiritual wound that can be named an experiential death of divine love—and that this wound is manifesting as unshepherded fear, rage, and discontent. I suggest that this spiritual wound be tended through skillful pedagogical tenderness, flexibility, and liberative co‐creation with students to cultivate a presence of love and feed the spiritual hungers of our time. I also suggest that a school (and a teacher) dies beautifully when its death is allowed and free to become a scene of beautiful instruction. [ABSTRACT FROM AUTHOR]