Impact of Different Components and Boundary Conditions on the Eigenfrequencies of a Magnet–Girder Assembly

Synchrotron radiation facilities are very important in different areas of fundamental and applied science to investigate structures or processes at small scales. Magnet–girder assemblies play a key role for the performance of such accelerator machines. High structural eigenfrequencies of the magnet–girder assemblies are required to assure a sufficient particle beam stability. The objective of the present parametric study was to numerically investigate and quantify the impact of different boundary conditions and components on the magnet–girder eigenfrequencies. As case studies, two 3 m long girder designs following the specifications of the PETRA IV project at DESY (German Electron Synchrotron, Hamburg, Germany) were selected. High magnet–girder assembly eigenfrequencies were achieved by, e.g., positioning the magnets close to the upper girder surface, increasing the connection stiffness between the magnets and the girder and between the girder and the bases, and positioning the girder support points as high as possible in the shape of a large triangle. Comparing the E/ρ ratio (E: Young’s modulus, ρ: material density) of different materials was used as a first approach to evaluate different materials for application to the girder. Based on the findings, general principles are recommended to be considered in the future girder design development processes.

Large triangle packings and Tuza’s conjecture in sparse random graphs

The triangle packing number v(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2v(G) edges intersecting every triangle in G. We show that Tuza’s conjecture holds in the random graph G = G(n, m), when m ⩽ 0.2403n3/2 or m ⩾ 2.1243n3/2. This is done by analysing a greedy algorithm for finding large triangle packings in random graphs.

Dense Graphs With a Large Triangle Cover Have a Large Triangle Packing

It is well known that a graph with m edges can be made triangle-free by removing (slightly less than) m/2 edges. On the other hand, there are many classes of graphs which are hard to make triangle-free, in the sense that it is necessary to remove roughly m/2 edges in order to eliminate all triangles.We prove that dense graphs that are hard to make triangle-free have a large packing of pairwise edge-disjoint triangles. In particular, they have more than m(1/4+cβ) pairwise edge-disjoint triangles where β is the density of the graph and c ≥ is an absolute constant. This improves upon a previous m(1/4−o(1)) bound which follows from the asymptotic validity of Tuza's conjecture for dense graphs. We conjecture that such graphs have an asymptotically optimal triangle packing of size m(1/3−o(1)).We extend our result from triangles to larger cliques and odd cycles.

Triangle Free Sets and Arithmetic Progressions – Two Pisier Type Problems

Let ${\cal P}_f({\bf N})$ be the set of finite nonempty subsets of ${\bf N}$ and for $F,G\in{\cal P}_f({\bf N})$ write $F < G$ when $\max F < \min G$. Let $X=\{(F,G):F,G\in{\cal P}_f({\bf N})$ and $F < G\}$. A triangle in $X$ is a set of the form $\{(F\cup H,G),(F,G),(F,H\cup G)\}$ where $F < H < G$. Motivated by a question of Erdős, Nešetríl, and Rödl regarding three term arithmetic progressions, we show that any finite subset $Y$ of $X$ contains a relatively large triangle free subset. Exact values are obtained for the largest triangle free sets which can be guaranteed to exist in any set $Y\subseteq X$ with $n$ elements for all $n\leq 14$.

Kinetic Occlusion by Apparent Movement

A small square and a large triangle below it were presented in the first frame. These were switched off and replaced by a triangle alone in the second frame, shifted horizontally and upwards. The triangle appeared to move obliquely, as expected, but most observers also saw the square moving horizontally and hiding behind the triangle, although there was no stimulus corresponding to it in the second frame. The visual system invokes the occlusion ‘hypothesis’ in order to explain the otherwise mysterious disappearance of the square. The experiment suggests that apparently intelligent solutions can be rapidly computed by the visual system.