Cartan Ribbonization of Surfaces and a Topological Inspection

We develop the concept of Cartan ribbons and a method by which they can be used to ribbonize any given surface in space by intrinsically flat ribbons. The geodesic curvature along the center curve on the surface agrees with the geodesic curvature of the corresponding Cartan development curve, and this makes a rolling strategy successful. Essentially, it follows from the orientational alignment of the two co-moving Darboux frames during the rolling. Using closed center curves we obtain closed approximating Cartan ribbons that contribute zero to the total curvature integral of the ribbonization. This paves the way for a particular simple topological inspection -- it is reduced to the question of how the ribbons organize their edges relative to each other. The Gauss-Bonnet theorem leads to this topological inspection of the vertices. Finally, we display two examples of ribbonizations of surfaces, namely of a torus using two ribbons, and of an ellipsoid using its closed curvature lines as center curves for the ribbons. The topological inspection of the torus ribbonization is particularly simple as it has no vertex points, giving directly the Euler characteristic $0$. The ellipsoid has $4$ vertices -- corresponding to the $4$ umbilical points -- each of degree one and each therefore contributing one-half to the Euler characteristic

Non-existence of an invariant measure for a homogeneous ellipsoid rolling on the plane

It is known that the reduced equations for an axially symmetric homogeneous ellipsoid that rolls without slipping on the plane possess a smooth invariant measure. We show that such an invariant measure does not exist in the case when all of the semi-axes of the ellipsoid have different length.Comment: v2: Minor changes after journal review. This text uses the theory developed in arXiv:1304.1788 for the specific example of a homogeneous ellipsoid rolling on the plan

The evolution of pebble size and shape in space and time

We propose a mathematical model which suggests that the two main geological observations about shingle beaches, i.e. the emergence of predominant pebble size ratios and strong segregation by size are interrelated. Our model is a based on a system of ODEs called the box equations, describing the evolution of pebble ratios. We derive these ODEs as a heuristic approximation of Bloore's PDE describing collisional abrasion. While representing a radical simplification of the latter, our system admits the inclusion of additional terms related to frictional abrasion. We show that nontrivial attractors (corresponding to predominant pebble size ratios) only exist in the presence of friction. By interpreting our equations as a Markov process, we illustrate by direct simulation that these attractors may only stabilized by the ongoing segregation process.Comment: 22 pages, 8 figure

Ellipsoidal Prediction Regions for Multivariate Uncertainty Characterization

While substantial advances are observed in probabilistic forecasting for power system operation and electricity market applications, most approaches are still developed in a univariate framework. This prevents from informing about the interdependence structure among locations, lead times and variables of interest. Such dependencies are key in a large share of operational problems involving renewable power generation, load and electricity prices for instance. The few methods that account for dependencies translate to sampling scenarios based on given marginals and dependence structures. However, for classes of decision-making problems based on robust, interval chance-constrained optimization, necessary inputs take the form of polyhedra or ellipsoids. Consequently, we propose a systematic framework to readily generate and evaluate ellipsoidal prediction regions, with predefined probability and minimum volume. A skill score is proposed for quantitative assessment of the quality of prediction ellipsoids. A set of experiments is used to illustrate the discrimination ability of the proposed scoring rule for misspecification of ellipsoidal prediction regions. Application results based on three datasets with wind, PV power and electricity prices, allow us to assess the skill of the resulting ellipsoidal prediction regions, in terms of calibration, sharpness and overall skill.Comment: 8 pages, 7 Figures, Submitted to IEEE Transactions on Power System

Topological monodromy as an obstruction to Hamiltonization of nonholonomic systems: pro or contra?

The phenomenon of a topological monodromy in integrable Hamiltonian and nonholonomic systems is discussed. An efficient method for computing and visualizing the monodromy is developed. The comparative analysis of the topological monodromy is given for the rolling ellipsoid of revolution problem in two cases, namely, on a smooth and on a rough plane. The first of these systems is Hamiltonian, the second is nonholonomic. We show that, from the viewpoint of monodromy, there is no difference between the two systems, and thus disprove the conjecture by Cushman and Duistermaat stating that the topological monodromy gives a topological obstruction for Hamiltonization of the rolling ellipsoid of revolution on a rough plane.Comment: 31 pages, 11 figure

New LpL_p Affine Isoperimetric Inequalities

We prove new $L_p$ affine isoperimetric inequalities for all $p \in [-\infty,1)$. We establish, for all $p\neq -n$, a duality formula which shows that $L_p$ affine surface area of a convex body $K$ equals $L_\frac{n^2}{p}$ affine surface area of the polar body $K^\circ$

Human sperm accumulation near surfaces: a simulation study

A hybrid boundary integral/slender body algorithm for modelling flagellar cell motility is presented. The algorithm uses the boundary element method to represent the ‘wedge-shaped’ head of the human sperm cell and a slender body theory representation of the flagellum. The head morphology is specified carefully due to its significant effect on the force and torque balance and hence movement of the free-swimming cell. The technique is used to investigate the mechanisms for the accumulation of human spermatozoa near surfaces. Sperm swimming in an infinite fluid, and near a plane boundary, with prescribed planar and three-dimensional flagellar waveforms are simulated. Both planar and ‘elliptical helicoid’ beating cells are predicted to accumulate at distances of approximately 8.5–22 μm from surfaces, for flagellar beating with angular wavenumber of 3π to 4π. Planar beating cells with wavenumber of approximately 2.4π or greater are predicted to accumulate at a finite distance, while cells with wavenumber of approximately 2π or less are predicted to escape from the surface, likely due to the breakdown of the stable swimming configuration. In the stable swimming trajectory the cell has a small angle of inclination away from the surface, no greater than approximately 0.5°. The trapping effect need not depend on specialized non-planar components of the flagellar beat but rather is a consequence of force and torque balance and the physical effect of the image systems in a no-slip plane boundary. The effect is relatively weak, so that a cell initially one body length from the surface and inclined at an angle of 4°–6° towards the surface will not be trapped but will rather be deflected from the surface. Cells performing rolling motility, where the flagellum sweeps out a ‘conical envelope’, are predicted to align with the surface provided that they approach with sufficiently steep angle. However simulation of cells swimming against a surface in such a configuration is not possible in the present framework. Simulated human sperm cells performing a planar beat with inclination between the beat plane and the plane-of-flattening of the head were not predicted to glide along surfaces, as has been observed in mouse sperm. Instead, cells initially with the head approximately 1.5–3 μm from the surface were predicted to turn away and escape. The simulation model was also used to examine rolling motility due to elliptical helicoid flagellar beating. The head was found to rotate by approximately 240° over one beat cycle and due to the time-varying torques associated with the flagellar beat was found to exhibit ‘looping’ as has been observed in cells swimming against coverslips