On a family of elliptic curves of rank at least 2

Let C_m:y² = x³ ? m²x + p²q² be a family of elliptic curves over ?, where m is a positive integer and p, q are distinct odd primes. We study the torsion part and the rank of C_m(?). More specifically, we prove that the torsion subgroup of C_m(?) is trivial and the ?-rank of this family is at least 2, whenever m ? 0 (mod 3), m ? 0 (mod 4) and m ≡ 2 (mod 64) with neither p nor q dividing m.

     

Correction to: Macrocycles in dual role: ancillary ligands in metal complexes and organocatalysts for the ring‑opening polymerization of lactide

Journal of Inclusion Phenomena and Macrocyclic Chemistry - A correction to this paper has been published: https://doi.org/10.1007/s10847-021-01060-y     

Co-Assembly between Fmoc Diphenylalanine and Diphenylalanine within a 3D Fibrous Viscous Network Confers Atypical Curvature and Branching

Supramolecular polymer co-assembly is a useful approach to modulate peptide nanostructures. However, the co-assembly scenario where one of the peptide building blocks simultaneously forms a hydrogel is yet to be studied. Herein, we investigate the co-assembly formation of diphenylalanine (FF), and Fmoc-diphenylalanine (FmocFF) within the 3D network of FmocFF hydrogel. The overlapping peptide sequence between the two building blocks leads to their co-assembly within the gel state modulating the nature of the FF crystals. We observe the formation of branched microcrystalline aggregates with an atypical curvature, in contrast to the FF assemblies obtained from aqueous solution. Optical microscopy reveal the sigmoidal kinetic growth profile of these aggregates. Microfluidics and ToF-SIMS experiments exhibit the presence of co-assembled structures of FF and FmocFF in the crystalline aggregates. Molecular dynamics simulation was used to decipher the mechanism of co-assembly formation.     

Unveiling the binding interaction of zinc (II) complexes of homologous Schiff-base ligands on the surface of BSA protein: A combined experimental and theoretical approach

Four new zinc (II) complexes [Zn (HL¹H)Br₂] (1), [Zn (HL¹H)Cl₂] (2), [Zn₂(HL²)Br₃] (3), and [Zn (HL²)Cl] (4) have been synthesized by adopting template synthetic strategy and utilizing two homologous Schiff base ligands (H₂L¹ = 4-bromo-2-{[2-(2-hydroxyethylamino)-ethylimino]-methyl}-6-methoxyphenol, H₂L² = 4-bromo-2-{[3-(2-hydroxyethylamino)propylimino]methyl}-6-methoxyphenol), differing in one -CH₂- unit in the ligating backbone, by adopting template synthetic strategy. All the complexes have been characterized by single crystal X-ray diffraction analysis as well as by other routine physicochemical techniques. Ligand mediated structural variations have been observed and rationalized by density functional theoretical (DFT) calculations. Interaction of the complexes 1–4 with Bovine Serum Albumin protein (BSA) has been studied by different spectroscopic techniques. A complete thermodynamic profile (ΔH^o, ΔS^o and ΔG^o) was evaluated initially from the change in absorption and fluorescence spectra upon addition of BSA to the complexes. Appreciable binding constant values in the range ~ 0.94–4.51 × 10⁴ M⁻¹ indicate efficient binding tendency of the complexes to BSA with the sequence 1 ≅ 2 > 3 ≅ 4. Circular dichroism (CD), isothermal calorimetric titration experiments, molecular docking and molecular dynamics have been performed to gain deep insight into the binding regions of complex 1 to BSA. Experimental evidences suggest an interaction of zinc complexes at the surface of BSA protein and this particular binding has been exploited to determine unknown concentration of BSA protein. For this purpose complex 1 was explored as a BSA protein quantification tool.     

C−C Bond Formation of Benzyl Alcohols and Alkynes Using a Catalytic Amount of KOtBu: Unusual Regioselectivity through a Radical Mechanism

We report a C?C bond‐forming reaction between benzyl alcohols and alkynes in the presence of a catalytic amount of KO^tBu to form α‐alkylated ketones in which the C=O group is located on the side derived from the alcohol. The reaction proceeds under thermal conditions (125 °C) and produces no waste, making the reaction highly atom efficient, environmentally benign, and sustainable. Based on our mechanistic investigations, we propose that the reaction proceeds through radical pathways.     

Hierarchically porous organic polymers: highly enhanced gas uptake and transport through templated synthesis

Porous organic polymers (POPs) possessing meso- and micropores can be obtained by carrying out the polymerization inside a mesoporous silica aerogel template and then removing the template after polymerization. The total pore volume (tpv) and specific surface area (ssa) can be greatly enhanced by modifying the template (up to 210% increase for tpv and 73% for ssa) as well as by supercritical processing of the POPs (up to an additional 142% increase for tpv and an additional 32% for ssa) to include larger mesopores. The broad range of pores allows for faster transport of molecules through the hierarchically porous POPs, resulting in increased diffusion rates and faster gas uptake compared to POPs with only micropores.     

Probing the Crystal Structure Landscape by Doping: 4‐Bromo, 4‐Chloro,and 4‐Methylcinnamic Acids

Accessing the data points in the crystal structure landscape of a molecule is a challenging task, either experimentally or computationally. We have charted the crystal structure landscape of 4‐bromocinnamic acid (4BCA) experimentally and computationally: experimental doping is achieved with 4‐methylcinnamic acid (4MCA) to obtain new crystal structures; computational doping is performed with 4‐chlorocinnamic acid (4CCA) as a model system, because of the difficulties associated in parameterizing the Br atom. The landscape of 4CCA is explored experimentally in turn, also by doping it with 4MCA, and is found to bear a close resemblance to the landscape of 4BCA, justifying the ready miscibility of these two halogenated cinnamic acids to form solid solutions without any change in crystal structure. In effect, 4MCA, 4CCA and 4BCA form a commutable group of crystal structures, which may be realized experimentally or computationally, and constitute the landscape. Unlike the results obtained by Kitaigorodskii, all but two of the multiple solid solutions obtained in the methyl‐doping experiments take structures that are different from the hitherto observed crystal forms of the parent compounds. Even granted that the latter might be inherently polymorphic, this unusual observation provokes the suggestion that solid solution formation may be used to probe the crystal structure landscape. The influence of π???π interactions, weak hydrogen bonds and halogen bonds in directing the formation of these new structures is also seen.     

Atomically Precise Nanocluster Assemblies Encapsulating Plasmonic Gold Nanorods

The self‐assembled structures of atomically precise, ligand‐protected noble metal nanoclusters leading to encapsulation of plasmonic gold nanorods (GNRs) is presented. Unlike highly sophisticated DNA nanotechnology, this strategically simple hydrogen bonding‐directed self‐assembly of nanoclusters leads to octahedral nanocrystals encapsulating GNRs. Specifically, the p‐mercaptobenzoic acid (pMBA)‐protected atomically precise silver nanocluster, Na₄[Ag₄₄(pMBA)₃₀], and pMBA‐functionalized GNRs were used. High‐resolution transmission and scanning transmission electron tomographic reconstructions suggest that the geometry of the GNR surface is responsible for directing the assembly of silver nanoclusters via H‐bonding, leading to octahedral symmetry. The use of water‐dispersible gold nanoclusters, Au_≈250(pMBA)_n and Au₁₀₂(pMBA)₄₄, also formed layered shells encapsulating GNRs. Such cluster assemblies on colloidal particles are a new category of precision hybrids with diverse possibilities.