We show that if a graph G with n ≥ 3 vertices can be drawn in the plane such that each of its edges is involved in at most four crossings, then G has at most 6n− 12 edges. This settles a conjecture of Pach, Radoičić, Tardos, and Tóth. As a corollary we also obtain a better bound for the Crossing Lemma which gives a lower bound for the minimum number of crossings in a drawing of a graph.

Background Given that school-age students, as active road users, are more vulnerable to injury compared with other pedestrians, a large number of them, following an injury, may either fail to go to school at least for a short time or even suffer from disabilities for the rest of their lives. The aim of this study was to determine safe behavior in road crossing using the theory of planned behavi...

Abstract Enhancement of magneto-optical effects in hybrid magneto-plasmonic systems has attracted considerable recent attention because their potential for building non-reciprocal nanophotonic devices. Quantitative understanding the fundamental origin and contributing mechanisms enhancement is crucial optimizing applications. Here, we unravel different physical origins giant Faraday rotation el...

In this paper, we discuss a direct measure for aperiodic energy and periodic energy in speech signals. Most measures for aperiodicity have been indirect, such as zero crossing rate, highfrequency energy and the ratio of high-frequency energy to lowfrequency energy. Such indirect measurements will usually fail in situations where there is both strong periodic and aperiodic energy in the speech s...

Meiotic recombination is an essential step in gametogenesis, and is one that also generates genetic diversity. Genome-wide association studies (GWAS) and molecular studies have identified genes that influence of human meiotic recombination. RNF212 is associated with total or average number of recombination events, and PRDM9 is associated with the locations of hotspots, or sequences where crossi...

The decrease of meiotic exchanges (crossing over and conversion) in two mutants of Sordaria macrospora correlated strongly with a reduction of chiasmata and of both types of "recombination nodules." Serial section reconstruction electron microscopy was used to compare the synapsis pattern of meiotic prophase I in wild type and mutants. First, synapsis occurred but the number of synaptonemal com...

The expected value for the weighted crossing number of a randomly weighted graph is studied. A variation of the Crossing Lemma for expectations is proved. We focus on the case where the edge-weights are independent random variables that are uniformly distributed on [0, 1].

HE models for genetic recombination which were designed by WHITEHOUSE T(1963) and HOLLIDAY (1 964) differ in some details, but both include a stage at which two chromatids of the tetrad carry corresponding segments of hybrid DNA (a region in which the two complementary strands come from different chromatids). The models were designed to account for both reciprocal and nonreciprocal recombinatio...

ciprocal exchanges (conversions) that are the result of Meiosis reduces the chromosome complement of a euDNA lost from one parent in the neighborhood of the karyote from diploidy to haploidy. It is typically marked DSB site and then compensated for by synthesis temby crossing over, the reciprocal exchange of segments plated on the uncut homologue, as called for by the of DNA between homologous ...