Chase’s lemma and its context

Chase’s lemma provides a powerful tool for translating properties of (co)products in abelian categories into chain conditions. This note discusses the context in which the lemma is used, making explicit what is often neglected in the literature because of its technical nature.     

On the complexity of extending the convergence ball of Wang’s method for finding a zero of a derivative

Ball convergence results are very important, since they demonstrate the complexity in choosing initial points for iterative methods. One of the most important problems in the study of iterative methods is to determine the convergence ball. This ball is small in general restricting the choice of initial points. We address this problem in the case of Wang’s method utilized to determine a zero of a derivative. Finding such a zero has many applications in computational fields, especially in function optimization. In particular, we find the convergence ball of Wang’s method using hypotheses up to the second derivative in contrast to earlier studies using hypotheses up to the fourth derivative. This way, we also extend the applicability of Wang’s method. Numerical experiments used to test the convergence criteria complete this study.     

Some unsolvable conjectures in finite p-groups

We survey some unsolvable conjectures in finite p-groups and their research progress.     

Saussurea costus for sustainable and eco-friendly synthesis of palladium nanoparticles and their biological activities

Palladium nanoparticles have been evaluated as a viable candidate in the realm of biological applications due to their unique features. Saussurea costus extract was used as a stabilizing and reducing agent for the synthesis of palladium nanoparticles with average grain size of 17.6 ± 1.2 nm. The synthesized PdNPs were evaluated for their antioxidant activity, anti Alzheimer's activity, antibacterial and anticancer activities. The nanocharacterization was carried out using different spectroscopic techniques, including UV–visible spectroscopy, Transmission Electron Microscopy, Fourier Transformed Infrared spectroscopy, X-ray Diffraction analysis, X-ray Photoelectron spectroscopy, Energy Dispersive X-ray Spectroscopy, Size distribution, and Zeta potential. The characterization data explained the PdNPs mediated by S.costus extract have spherical form and are disseminated without agglomeration. FTIR and XPS supported the hypotheisis that the biomolecules of S.costus are activing as a reducing and stabilizing agents. The antioxidant activity of PdNPs was assessed using a free radical scavenging assay (DPPH) which exhibited similar results to the ABTS assay i.e. 90 μg/mL IC₅₀ value. Moreover Alzheimer's disease can easily be inhibited by S.costus@PdNPs at 400 mg/mL, with 79.23 ± 1.11 % of inhibition rate against AChE and 76.13 ± 0.43 % towards BChE. S.costus@PdNPs showed comparatively greater antibacterial activity against all four Staphylococcus aureus, Bacillus subtilis Escherichia coli and Pseudomonas aeruginosa microorganisms. Supplementary research carried out on the anti-tumor effects of the generated PdNPs using the colon cancer (HCT-116), hepatocellular carcinoma (HepG2), and breast adenocarcinoma (MCF-7) cell lines. PdNPs showed potent anticancerous activity against all the cell lines. Thus we recommend S.costus@PdNPs as a thearapeutic agent after successful clinical trails in future.     

New lower bounds on the radius of spatial analyticity for the KdV equation

The radius of spatial analyticity for solutions of the KdV equation is studied. It is shown that the analyticity radius does not decay faster than $t^{?1/4}$ as time t goes to infinity. This improves the works of Selberg and da Silva (2017) [30] and Tesfahun (2017) [34]. Our strategy mainly relies on a higher order almost conservation law in Gevrey spaces, which is inspired by the I-method.     

Boundedness of semilinear Duffing equations at resonance in a critical situation

In this paper, we propose a sufficient and necessary condition for the boundedness of all the solutions for the equation $\ddot{x}+n^{2}x+g(x)=p(t)$ with the critical situation that $|{\int }_0^{2\pi }p(t)e^{?int}dt|=2|g(+\infty )?g(?\infty )|$ on g and p, where $n\in {\mathbb{N}}^{+}$, $p(t)$ is periodic and $g(x)$ is bounded.     

Convergence of solutions for the fractional Cahn–Hilliard system

This paper deals with the Cauchy–Dirichlet problem for the fractional Cahn–Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called ?ojasiewicz–Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but $C^1$) nonlinearities.     

Synthesis of Highly Functionalized N,N-Diarylamides by an Anodic C,N-Coupling Reaction

We report an innovative, sustainable and straightforward protocol for the synthesis of N,N-diarylamides equipped with nonprotected hydroxyl groups by using electrosynthesis. The concept allows the application of various substrates furnishing diarylamides with yields up to 57 % within a single and direct electrolytic protocol. The method is thereby easy to conduct in an undivided cell with constant current conditions offering a versatile and short-cut alternative to conventional pathways.