Based on the Gale–Ryser theorem [2, 6], for the existence of suitable $(0,1)$-matrices for different partitions of a natural number, we revisit the classical result of Lorentz [4] regarding the characterization of a plane measurable set, in terms of its cross-sections, and extend it to general measure spaces.

Polygon meshes and particularly triangulated meshes can be used to describe the shape of different types of geometry such as bicycles, bridges, or runways. In engineering, such polygon meshes can be supplied as finite element meshes, resulting from topology optimization or from laser scanning. Especially from topology optimization, frame-like polygon meshes with slender parts are typical and often have to be converted into a CAD (Computer-Aided Design) format, e.g., for further geometrical detailing or performing additional shape optimization. Especially for such frame-like geometries, CAD designs are constructed as beams with cross-sections and beam-lines, whereby the cross-section is extruded along the beam-lines or beam skeleton. One major task in the recognition of beams is the classification of the cross-section type such as I, U, or T, which is addressed in this article. Therefore, a dataset consisting of different cross-sections represented as binary images is created. Noisy dilatation, the distance transformation, and main axis rotation are applied to these images to increase the robustness and reduce the necessary amount of samples. The resulting images are applied to a convolutional neuronal network.

The elliptical and hyperbolic outcrop patterns characteristic of periclinal folds can be used to classify structures according to different curvature attributes. Elliptical patterns indicate domal-basinal structures with synclastic curvature, that is, principal curvatures of the same sign. Hyperbolic patterns are diagnostic of anticlastic curvature (saddle-like structures). Such outcrop geometries are geological examples of Dupin's indicatrix, the geometrical figure obtained by sectioning a curved surface on a plane parallel and almost coincident with the tangent plane. The aspect ratio of Dupin's indicatrix is theoretically related to the ratio of the principal curvature values for the part of the structure being considered. This new method allows quantitative assessment of structures on maps and on remote sensing images. Illustrations are given from Wyoming, USA, and Yorkshire, England.

One approach being explored as a route to practical fusion energy uses heavy ion beams focused on an indirect drive target. Such beams will lose electrons while passing through background gas in the target chamber, and therefore it is necessary to assess the rate at which the charge state of the incident beam evolves on the way to the target. Accelerators designed primarily for nuclear physics or high energy physics experiments utilize ion sources that generate highly stripped ions in order to achieve high energies economically. As a result, accelerators capable of producing heavy ion beams of 10 to 40 MeV/amu with charge state 1 currently do not exist. Hence, the stripping cross sections used to model the performance of heavy ion fusion driver beams have, up to now, been based on theoretical calculations. We have investigated experimentally the stripping of 3.4 MeV/amu Kr+7 and Xe+11 in N2; 10.2 MeV/amu Ar+6 in He, N2, Ar, and Xe; 19 MeV/amu Ar+8 in He, N2, Ar, and Xe; 30 MeV He+1 in He, N2, Ar, and Xe; and 38 MeV/amu N+6 in He, N2, Ar, and Xe. The results of these measurements are compared with the theoretical calculations to assess their applicability over a wide range of parameters.

Detailed mapping, stratigraphic logging and structural analysis of the Hawasina Window culmination in the central Oman Mountains of Arabia reveals an extremely complex thrust geometry and structural history. The initial thrust sequence involved a southwestward propagating stacking during telescoping of the Arabian continental margin slope (Sumeini complex), and time-equivalent, more distal Tethyan basin (Hawasina and Haybi complexes) facies rocks. The Semail thrust, carrying the 12 km-thick ophiolite sequence, progressively overlaps Haybi and Hawasina duplexes towards the SW. Late stage “leap-frog” thrusts have punched Sumeini duplexes higher up into the earlier thrust stack locally reversing the normal stacking order. SW-directed thrusts and SW-facing folds in the SW and NE-directed backthrusts and NE-facing backfolds in the NE have created a fan structure cored by the Jebel Rais “pop-up” composed of Sumeini slope facies rocks.

The palaeogeographic presence of a large NE-facing promontory in the Cretaceous shelf edge is inferred beneath the Window. The frontal ramp of this promontory was sufficiently large to inhibit the overthrusting of large volumes of Hawasina and Haybi complex rocks. The whole central part of the Window shows NE-facing and verging backfolds and backthrusts affecting all duplexes from the lowest Sumeini up to the Semail ophiolite. The promontory is bounded by major lateral ramps to the NW (Wadi Shafan area) and SE (Jebel Milh area). A map and four balanced cross-sections are presented here to promote a complex model for the structural evolution of the Hawasina Window.