On the kth derivative of meromorphic functions with zeros of multiplicity at least k+1

AbstractIn this paper, we prove the following TheoremLet f(z) be a transcendental meromorphic function on C, all of whose zeros have multiplicity at least k+1 (k⩾2), except possibly finitely many, and all of whose poles are multiple, except possibly finitely many, and let the function a(z)=P(z)exp(Q(z))≢0, where P and Q are polynomials such that lim¯r→∞(T(r,a)T(r,f)+T(r,f)T(r,a))=∞. Then the function f(k)(z)−a(z) has infinitely many zeros

Ni-based bimetallic heterogeneous catalysts for energy and environmental applications

Bimetallic catalysts have attracted extensive attention for a wide range of applications in energy production and environmental remediation due to their tunable chemical/physical properties. These properties are mainly governed by a number of parameters such as compositions of the bimetallic systems, their preparation method, and their morphostructure. In this regard, numerous efforts have been made to develop “designer” bimetallic catalysts with specific nanostructures and surface properties as a result of recent advances in the area of materials chemistry. The present review highlights a detailed overview of the development of nickel-based bimetallic catalysts for energy and environmental applications. Starting from a materials science perspective in order to obtain controlled morphologies and surface properties, with a focus on the fundamental understanding of these bimetallic systems to make a correlation with their catalytic behaviors, a detailed account is provided on the utilization of these systems in the catalytic reactions related to energy production and environmental remediation. We include the entire library of nickel-based bimetallic catalysts for both chemical and electrochemical processes such as catalytic reforming, dehydrogenation, hydrogenation, electrocatalysis and many other reactions

Robust estimation of bacterial cell count from optical density

Optical density (OD) is widely used to estimate the density of cells in liquid culture, but cannot be compared between instruments without a standardized calibration protocol and is challenging to relate to actual cell count. We address this with an interlaboratory study comparing three simple, low-cost, and highly accessible OD calibration protocols across 244 laboratories, applied to eight strains of constitutive GFP-expressing E. coli. Based on our results, we recommend calibrating OD to estimated cell count using serial dilution of silica microspheres, which produces highly precise calibration (95.5% of residuals <1.2-fold), is easily assessed for quality control, also assesses instrument effective linear range, and can be combined with fluorescence calibration to obtain units of Molecules of Equivalent Fluorescein (MEFL) per cell, allowing direct comparison and data fusion with flow cytometry measurements: in our study, fluorescence per cell measurements showed only a 1.07-fold mean difference between plate reader and flow cytometry data