We define and compute the energy of higher curvature gravity theories in arbitrary dimensions. Generically, these theories admit constant curvature vacua (even in the absence of an explicit cosmological constant), and asymptotically constant curvature solutions with non-trivial energy properties. For concreteness, we study quadratic curvature models in detail. Among them, the one whose action i...

We present a systematic discussion of algorithms to multiply a vector by a matrix expressed as the Kronecker product of sparse matrices, extending previous work in a unified notational framework. Then, we use our results to define new algorithms for the solution of large structured Markov models. In addition to a comprehensive overview of existing approaches, we give new results with respect to...

We construct a concrete example of constant Gauss curvature $K = 1$ on the 2-sphere having all geodesics closed and same length.

We give an integral-geometric proof of the Gauss-Bonnet theorem for hypersurfaces in constant curvature spaces. As a tool, we obtain variation formulas in integral geometry with interest in its own.

On the basis of the integral theorems about the mean curvature and Gauss curvature, geometric conservation laws for cells or vesicles are proved. These conservation laws may depict various special bionano structures discovered in experiments, such as the membrane nanotubes and singular points grown from the surfaces of cells or vesicles. Potential applications of the conservation laws to lipid ...

We propose a novel but natural definition of conserved quantities for gravity models quadratic and higher in curvature. Based on the spatial asymptotics of curvature rather than of metric, it avoids the GR energy machinery’s more egregious problems–such as zero energy “theorems” and failure in flat backgrounds – in this fourth-derivative realm. In D > 4, the present expression indeed correctly ...

The study of constant mean curvature surfaces in a space-form has been an active field since the work of H. Hopf in the 1920’s and H.Liebmann in the years around 1900. The questions which are generally of interest are global questions of existence and uniqueness in complete 3-manifolds. We deal in this short paper on a question of existence and uniqueness with respect to the complex structure a...

We leverage recent insights from second-order optimisation for neural networks to construct a Kronecker factored Laplace approximation to the posterior over the weights of a trained network. Our approximation requires no modification of the training procedure, enabling practitioners to estimate the uncertainty of their models currently used in production without having to retrain them. We exten...

The dried raisin, the crushed soda can, and the collapsed bicycle inner tube exemplify the nonlinear mechanical response of naturally curved elastic surfaces with different intrinsic curvatures to a variety of different external loads. To understand the formation and evolution of these features in a minimal setting, we consider a simple assay: the response of curved surfaces to point indentatio...