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Every finite graph admits a simple (topological) drawing, that is, a drawing where every pair of edges intersects in at most one point. However, in combination with other restrictions simple drawings do not universally exist. For instance, k-planar graphs are those graphs that can be drawn so that every edge has at most k crossings (i.e., they admit a k-plane drawing). It is known that for k≤3 , every k-planar graph admits a k-plane simple drawing. But for k≥4 , there exist k-planar graphs that do not admit a k-plane simple drawing. Answering a question by Schaefer, we show that there exists a function Open image in new window such that every k-planar graph admits an f(k)-plane simple drawing, for all Open image in new window. Note that the function f depends on k only and is independent of the size of the graph. Furthermore, we develop an algorithm to show that every 4-planar graph admits an 8-plane simple drawing. 

We introduce and study the problem of constructing geometric graphs that have few vertices and edges and that are universal for planar graphs or for some sub-class of planar graphs; a geometric graph is universal for a class H of planar graphs if it contains an embedding, i.e., a crossing-free drawing, of every graph in H . Our main result is that there exists a geometric graph with n vertices and O(nlogn) edges that is universal for n-vertex forests; this extends to the geometric setting a well-known graph-theoretic result by Chung and Graham, which states that there exists an n-vertex graph with O(nlogn) edges that contains every n-vertex forest as a subgraph. Our O(nlogn) bound on the number of edges is asymptotically optimal. We also prove that, for every h>0 , every n-vertex convex geometric graph that is universal for the class of the n-vertex outerplanar graphs has Ωh(n2−1/h) edges; this almost matches the trivial O(n2) upper bound given by the n-vertex complete convex geometric graph. Finally, we prove that there is an n-vertex convex geometric graph with n vertices and O(nlogn) edges that is universal for n-vertex caterpillars. 

Abstract We describe a collection of T1-, diffusion- and functional T2*-weighted magnetic resonance imaging data from human individuals with albinism and achiasma. This repository can be used as a test-bed to develop and validate tractography methods like diffusion-signal modeling and fiber tracking as well as to investigate the properties of the human visual system in individuals with congenital abnormalities. The MRI data is provided together with tools and files allowing for its preprocessing and analysis, along with the data derivatives such as manually curated masks and regions of interest for performing tractography. 

The continued miniaturization of nanoelectronic devices approaches its fundamental physical limits due to power dissipation. Negative capacitance field-effect transistors using ferroelectric gate insulators are promising to overcome these limits, which would allow further device scaling. However, the microscopic details of negative capacitance are not well understood so far, since mainly Landau based mean-field theories are used to model these phenomena. Here we use an educational and simplified approach to better understand the basic microscopic origin of ferroelectric negative capacitance. Our “toy” model shows that negative capacitance originates from the thermodynamic instability of the ferroelectric polarization and is bounded by the saturation of microscopic dipole polarizability. This shows that negative capacitance is strongly connected to the origin of ferroelectricity itself. Furthermore, our microscopic model results in the same qualitative behavior as mean-field Landau based approaches. 

Ferroelectric hafnium and zirconium oxides have undergone rapid scientific development over the last decade, pushing them to the forefront of ultralow-power electronic systems. Maximizing the potential application in memory devices or supercapacitors of these materials requires a combined effort by the scientific community to address technical limitations, which still hinder their application. Besides their favorable intrinsic material properties, HfO2–ZrO2 materials face challenges regarding their endurance, retention, wake-up effect, and high switching voltages. In this Roadmap, we intend to combine the expertise of chemistry, physics, material, and device engineers from leading experts in the ferroelectrics research community to set the direction of travel for these binary ferroelectric oxides. Here, we present a comprehensive overview of the current state of the art and offer readers an informed perspective of where this field is heading, what challenges need to be addressed, and possible applications and prospects for further development.