Difference operators for partitions under the Littlewood decomposition

Let n be a positive integer. Let \(\delta _3(n)\) denote the difference between the number of (positive) divisors of n congruent to 1 modulo 3 and the number of those congruent to 2 modulo 3. In 2004, Farkas proved that the arithmetic convolution sum

$$\begin{aligned} D_3(n):=\sum _{j=1}^{n-1}\delta _3(j)\delta _3(n-j) \end{aligned}$$

satisfies the relation

$$\begin{aligned} 3D_3(n)+\delta _3(n)={\sum _{\mathop {_{d \mid n}}\limits _{3 \not \mid d}}}d. \end{aligned}$$

In this paper, we use a result about binary quadratic forms to prove a general arithmetic convolution identity which contains Farkas’ formula and two other similar known formulas as special cases. From our identity, we deduce a number of analogous new convolution formulas.

     

Upgrading Solid Digestate from Anaerobic Digestion of Agricultural Waste as Performance Enhancer for Starch-Based Mulching Biofilm

Developing a green and sustainable method to upgrade biogas wastes into high value-added products is attracting more and more public attention. The application of solid residues as a performance enhancer in the manufacture of biofilms is a prospective way to replace conventional plastic based on fossil fuel. In this work, solid digestates from the anaerobic digestion of agricultural wastes, such as straw, cattle and chicken manures, were pretreated by an ultrasonic thermo-alkaline treatment to remove the nonfunctional compositions and then incorporated in plasticized starch paste to prepare mulching biofilms by the solution casting method. The results indicated that solid digestate particles dispersed homogenously in the starch matrix and gradually aggregated under the action of a hydrogen bond, leading to a transformation of the composites to a high crystalline structure. Consequently, the composite biofilm showed a higher tensile strength, elastic modulus, glass transition temperature and degradation temperature compared to the pure starch-based film. The light, water and GHG (greenhouse gas) barrier properties of the biofilm were also reinforced by the addition of solid digestates, performing well in sustaining the soil quality and minimizing N₂O or CH₄ emissions. As such, recycling solid digestates into a biodegradable plastic substitute not only creates a new business opportunity by producing high-performance biofilms but also reduces the environmental risk caused by biogas waste and plastics pollution.     

Phase‐Junction Electrocatalysts towards Enhanced Hydrogen Evolution Reaction in Alkaline Media

To ensure sustainable hydrogen production by water electrolysis, robust, earth‐abundant, and high‐efficient electrocatalysts are required. Constructing a hybrid system could lead to further improvement in electrocatalytic activity. Interface engineering in composite catalysts is thus critical to determine the performance, and the phase‐junction interface should improve the catalytic activity. Here, we show that nickel diphosphide phase junction (c‐NiP₂/m‐NiP₂) is an effective electrocatalyst for hydrogen production in alkaline media. The overpotential (at 10 mA cm^?2) for NiP₂‐650 (c/m) in alkaline media could be significantly reduced by 26 % and 96 % compared with c‐NiP₂ and m‐NiP₂, respectively. The enhancement of catalytic activity should be attributed to the strong water dissociation ability and the rearrangement of electrons around the phase junction, which markedly improved the Volmer step and benefited the reduction process of adsorbed protons.     

Is PF-00835231 a Pan-SARS-CoV-2 Mpro Inhibitor? A Comparative Study

The COVID-19 outbreak continues to spread worldwide at a rapid rate. Currently, the absence of any effective antiviral treatment is the major concern for the global population. The reports of the occurrence of various point mutations within the important therapeutic target protein of SARS-CoV-2 has elevated the problem. The SARS-CoV-2 main protease (Mpro) is a major therapeutic target for new antiviral designs. In this study, the efficacy of PF-00835231 was investigated (a Mpro inhibitor under clinical trials) against the Mpro and their reported mutants. Various in silico approaches were used to investigate and compare the efficacy of PF-00835231 and five drugs previously documented to inhibit the Mpro. Our study shows that PF-00835231 is not only effective against the wild type but demonstrates a high affinity against the studied mutants as well.     

A copper-based coordination polymer formed through synergistic bridging of 1,2,4-triazole and acetate anions: synthesis,crystal structure and magnetic properties

Transition Metal Chemistry - A Cu(II) coordination polymer, [Cu2(pztrz)2(μ-CH3COO)(CH3COO)]·3H2O (1), derived from mono-substituted 1,2,4-triazole derivative 3-(pyrazinyl)-1,2,4-triazole...     

AN EXPLORATION OF STRUCTURALLY CHARACTERIZED HETERO-TRINUCLEAR [Cu(II)2Ca(II)] FLEXIBLE-BIS(SALAMO)-TYPE COMPLEX

Journal of Structural Chemistry - A new bis(salamo)-type ligand has been designed and synthesized, and its hetero-trinuclear [Cu(II)2Ca(II)] complex has been successfully prepared. It is...     

A novel approach to synthesize dichlorobenzonitriles by selective ammoxidation

Research on Chemical Intermediates - Dichlorobenzonitriles are important organic intermediates for the production of many fine chemicals and are produced most economically and environment-friendly...     

Inclusion of thymol into cucurbiturils: density functional theory approach with dispersion correction and natural bond orbital analysis

Journal of Inclusion Phenomena and Macrocyclic Chemistry - Stability of inclusion complexes of thymol (a natural flavour) with cucurbit[n]urils was interpreted by using density functional theory...     

Simple and green synthesis of calcium alginate/hydroxyapatite hybrid material without high temperature treatment and its flame retardancy

Cellulose - Aiming to improve the thermal stability and flame retardant properties of calcium alginate, calcium alginate (CaAlg)/hydroxyapatite (HAP) hybrid material was synthesized in situ by the...     

Isochromatic fringe sharpening and interpolation along an isoclinic countour,with application to fracture mechanics

A method is presented for photoelastic analyses whereby fractional fringe orders are determined with enhanced accuracy. It utilizes data from the Tardy compensation method and combines the data by a robust algorithm to produce highly sharpened isochromatic fringe contours in the immediate vicinity of an isoclinic line. Their sharpness provides accurate assessment of fractional fringe orders at a point, relative to that achieved with the broad fringes of the standard Tardy method. Errors introduced by small isoclinic deviations are quantified and found to be inconsequential. The method is well suited to the application of linear elastic fracture mechanics in fracture studies of threedimensional photoelasticity models of complex structures. It is demonstrated for the determination of stress intensity factor at a crack in the solid fuel element of a rocket motor, when the solid fuel is subjected to internal pressure. Other applications are practical, too.