Bent functions embedded into the recursive framework of {\mathbb{Z}} -bent functions

Suppose that n is even. Let ${\mathbb{F}_2}$ denote the two-element field and ${\mathbb{Z}}$ the set of integers. Bent functions can be defined as ± 1-valued functions on ${\mathbb{F}_2^n}$ with ± 1-valued Fourier transform. More generally we call a mapping f on ${\mathbb{F}_2^n}$ a ${\mathbb{Z}}$ -bent function if both f and its Fourier transform ${\widehat{f}}$ are integer-valued. ${\mathbb{Z}}$ -bent functions f are separated into different levels, depending on the size of the maximal absolute value attained by f and ${\widehat{f}}$ . It is shown how ${\mathbb{Z}}$ -bent functions of lower level can be built up recursively by gluing together ${\mathbb{Z}}$ -bent functions of higher level. This recursion comes down at level zero, containing the usual bent functions. In the present paper we start to study bent functions in the framework of ${\mathbb{Z}}$ -bent functions and give some guidelines for further research.     

The “Hockey” Assist Makes the Difference—Validation of a Defensive Disruptiveness Model to Evaluate Passing Sequences in Elite Soccer

With the growing availability of position data in sports, spatiotemporal analysis in soccer is a topic of rising interest. The aim of this study is to validate a performance indicator, namely D-Def, measuring passing effectiveness. D-Def calculates the change of the teams’ centroid, centroids of formation lines (e.g., defensive line), teams’ surface area, and teams’ spread in the following three seconds after a pass and therefore results in a measure of disruption of the opponents’ defense following a pass. While this measure was introduced earlier, in this study we aim to prove the usefulness to evaluate attacking sequences. In this study, 258 games of Dutch Eredivisie season 2018/19 were included, resulting in 13,094 attacks. D-Def, pass length, pass velocity, and pass angle of the last four passes of each attack were calculated and compared between successful and unsuccessful attacks. D-Def showed higher values for passes of successful compared to unsuccessful attacks (0.001 < p ≤ 0.029, 0.06 ≤ d ≤ 0.23). This difference showed the highest effects sizes in the penultimate pass (d = 0.23) and the maximal D-Def value of an attack (d = 0.23). Passing length (0.001 < p ≤ 0.236, 0.08 ≤ d ≤ 0.17) and passing velocity (0.001 < p ≤ 0.690, −0.09 ≤ d ≤ 0.12) showed inconsistent results in discriminating between successful and unsuccessful attacks. The results indicate that D-Def is a useful indicator for the measurement of pass effectiveness in attacking sequences, highlighting that successful attacks are connected to disruptive passing. Within successful attacks, at least one high disruptive action (pass with D-Def > 28) needs to be present. In addition, the penultimate pass (“hockey assist”) of an attack seems crucial in characterizing successful attacks.     

Homogeneous non-degenerate 3-(α,δ)-Sasaki manifolds and submersions over quaternionic Kähler spaces

Annals of Global Analysis and Geometry - We show that every 3- $$(\alpha ,\delta )$$ -Sasaki manifold of dimension $$4n + 3$$ admits a locally defined Riemannian submersion over a quaternionic...     

Comparative Nucleosomal Reactivity of 5-Formyl-Uridine and 5-Formyl-Cytidine

5-Formyl-deoxyuridine (fdU) and 5-formyl-deoxycytidine (fdC) are formyl-containing nucleosides that are created by oxidative stress in differentiated cells. While fdU is almost exclusively an oxidative stress lesion formed from deoxythymidine (T), the situation for fdC is more complex. Next to formation as an oxidative lesion, it is particularly abundant in stem cells, where it is more frequently formed in an epigenetically important oxidation reaction performed by α-ketoglutarate dependent TET enzymes from 5-methyl-deoxycytidine (mdC). Recently, it was shown that genomic fdC and fdU can react with the ϵ-aminogroups of nucleosomal lysines to give Schiff base adducts that covalently link nucleosomes to genomic DNA. Here, we show that fdU features a significantly higher reactivity towards lysine side chains compared with fdC. This result shows that depending on the amounts of fdC and fdU, oxidative stress may have a bigger impact on nucleosome binding than epigenetics.     

L-Carnitine. Novel Synthesis and Determination of the Optical Purity

A new route to L-carnitine ( 1 ) based on the resolution of th trimethylammonium derivative 5 is described. The enantiomeric purity of 1 is determined by ¹H-NMR of its O-acetyl hydrochloride 11 using [Eu(hfc)₃] as chiral shift reagent. The optical rotation of 1 with an enantiomeric purity ≥99% is [α] =?31.3° (c = 10, H₂O).     

Superheavy elements and variations in the nuclear skin thickness

Although the interpretation of the data is uncertain, the recently claimed empirical evidence of element Z = 126 and some of its neighbours, provides the impetus for a reexamination of the nuclear one-body potentials, on the basis of which extrapolations have been made up to now. In particular the skin thickness parameters of the neutrons and protons are of interest and the authors presently consider the possibility that the nucleus may adjust its skin thickness to achieve a maximal shell energy.