Landau's function for one million billions

Let ${\mathfrak S}_n$ denote the symmetric group with $n$ letters, and $g(n)$ the maximal order of an element of ${\mathfrak S}_n$. If the standard factorization of $M$ into primes is M=q_1^{\al_1}q_2^{\al_2}... q_k^{\al_k}, we define $\ell(M)$ to be q_1^{\al_1}+q_2^{\al_2}+... +q_k^{\al_k}; one century ago, E. Landau proved that $g(n)=\max_{\ell(M)\le n} M$ and that, when $n$ goes to infinity, $\log g(n) \sim \sqrt{n\log(n)}$. There exists a basic algorithm to compute $g(n)$ for $1 \le n \le N$; its running time is \co(N^{3/2}/\sqrt{\log N}) and the needed memory is \co(N); it allows computing $g(n)$ up to, say, one million. We describe an algorithm to calculate $g(n)$ for $n$ up to $10^{15}$. The main idea is to use the so-called {\it $\ell$-superchampion numbers}. Similar numbers, the {\it superior highly composite numbers}, were introduced by S. Ramanujan to study large values of the divisor function \tau(n)=\sum_{d\dv n} 1

Enabling Communication Technologies for Automated Unmanned Vehicles in Industry 4.0

Within the context of Industry 4.0, mobile robot systems such as automated guided vehicles (AGVs) and unmanned aerial vehicles (UAVs) are one of the major areas challenging current communication and localization technologies. Due to stringent requirements on latency and reliability, several of the existing solutions are not capable of meeting the performance required by industrial automation applications. Additionally, the disparity in types and applications of unmanned vehicle (UV) calls for more flexible communication technologies in order to address their specific requirements. In this paper, we propose several use cases for UVs within the context of Industry 4.0 and consider their respective requirements. We also identify wireless technologies that support the deployment of UVs as envisioned in Industry 4.0 scenarios.Comment: 7 pages, 1 figure, 1 tabl

Density-independent mortality at early life stages increases the probability of overlooking an underlying stock-recruitment relationship

Abstract Beverton and Holt’s (1957) monograph contributed a widely used stock–recruitment relationship (BH-SRR) to fisheries science. However, because of variation around a presumed relationship between spawning biomass and recruits, the BH-SRR is often considered inadequate and approached merely as a curve-fitting exercise. The commonly used and simplified version of the BH-SRR has eclipsed the fact that in their classic monograph, the derivation accounted for mechanistic recruitment processes, including multi-stage recruitment with explicit cohort-dependent and -independent mortality terms that represent competition between recruits and extrinsic, cohort-independent factors such as the environment or predation as two independent sources of mortality. The original BH-SRR allows one to recreate recruitment patterns that correspond to observed ones. Doing so shows that variation in density-independent mortality increases the probability of overlooking an underlying stock–recruitment relationship. Intermediate coefficients of variation in mortality (75–100%) are sufficient to mask stock–recruitment relationships and recreate recruitment time series most similar to empirical data. This underlines the importance of variation in survival for recruitment and that Beverton and Holt’s work still provides a fundamental and useful tool to model the dynamics of populations.publishedVersio

How geometry affects sensitivity of a differential transformer for contactless characterization of liquids

The electrical and dielectric properties of liquids can be used for sensing. Specific appli-cations, e.g., the continuous in-line monitoring of blood conductivity as a measure of the sodium concentration during dialysis treatment, require contactless measuring methods to avoid any contam-ination of the medium. The differential transformer is one promising approach for such applications, since its principle is based on a contactless, magnetically induced conductivity measurement. The objective of this work is to investigate the impact of the geometric parameters of the sample or medium under test on the sensitivity and the noise of the differential transformer to derive design rules for an optimized setup. By fundamental investigations, an equation for the field penetration depth of a differential transformer is derived. Furthermore, it is found that increasing height and radius of the medium is accompanied by an enhancement in sensitivity and precision. © 2021 by the authors. Licensee MDPI, Basel, Switzerland

Differential inductive sensing system for truly contactless measuring of liquids’ electromagnetic properties in tubing

Certain applications require a contactless measurement to eliminate the risk of sensorinduced sample contamination. Examples can be found in chemical process control, biotechnology or medical technology. For instance, in critically ill patients requiring renal replacement therapy, continuous in‐line monitoring of blood conductivity as a measure for sodium should be considered. A differential inductive sensing system based on a differential transformer using a specific flow chamber has already proven suitable for this application. However, since the blood in renal replacement therapy is carried in plastic tubing, a direct measurement through the tubing offers a contactless method. Therefore, in this work we present a differential transformer for measuring directly through electrically non‐conductive tubing by winding the tube around the ferrite core of the transformer. Here, the dependence of the winding type and the number of turns of the tubing on the sensitivity has been analyzed by using a mathematical model, simulations and experimental validation. A maximum sensitivity of 364.9 mV/mol/L is measured for radial winding around the core. A longitudinal winding turns out to be less effective with 92.8 mV/mol/L. However, the findings prove the ability to use the differential transformer as a truly contactless sensing system. © 2021 by the authors. Licensee MDPI, Basel, Switzerland

Institutional and cultural barriers to transferring Lean production to China: evidence from a German automotive components manufacturer

Our qualitative study of a globally operating German automotive components manufacturer demonstrates how certain cognitive dispositions and behaviours of Chinese employees acted as barriers to the transfer of the firm’s Lean production system to its subsidiaries in China. We analyse how these barriers were rooted in the Chinese institutional and cultural context, thereby contributing a new micro-level perspective to business systems literature. Our findings further suggest that manufacturing in China will not be truly ‘Lean’ in the near future, which may place constraints on China’s technological development at a larger scale

Comparison of different lattice definitions of the topological charge

We present a comparison of different definitions of the topological charge on the lattice, using a small-volume ensemble with 2 flavours of dynamical twisted mass fermions. The investigated definitions are: index of the overlap Dirac operator, spectral projectors, spectral flow of the Hermitian Wilson-Dirac operator and field theoretic with different kinds of smoothing of gauge fields (HYP and APE smearings, gradient flow, cooling). We also show some results on the topological susceptibility.Comment: 7 pages, 2 figures, presented at the 32nd International Symposium on Lattice Field Theory (Lattice 2014), 23-28 June 2014, Columbia University, New York, NY, US