The Möbius Transformation and Smirnov’s Inequality for Polynomials

Differential inequalities for polynomials generalizing the well-known Smirnov, Rahman, Schmeisser, and Bernstein inequalities are obtained.

     

Diastereoselective Cyclobutenol Synthesis: A Heterogeneous Palladium-Catalyzed Oxidative Carbocyclization-Borylation of Enallenols

A highly selective and efficient oxidative carbocyclization/borylation of enallenols catalyzed by palladium immobilized on amino-functionalized siliceous mesocellular foam (Pd-AmP-MCF) was developed for diastereoselective cyclobutenol synthesis. The heterogeneous palladium catalyst can be recovered and recycled without any observed loss of activity or selectivity. The high diastereoselectivity of the reaction is proposed to originate from a directing effect of the enallenol hydroxyl group. Optically pure cyclobutenol synthesis was achieved by the heterogeneous strategy by using chiral enallenol obtained from kinetic resolution.     

Rapid,Traceless, AgI‐Promoted Macrocyclization of Peptides Possessing an N‐Terminal Thioamide

Peptide macrocyclization is often a slow process, plagued by epimerization and cyclodimerization. Herein, we describe a new method for peptide macrocyclization employing the Ag^I‐promoted transformation of peptide thioamides. The Ag^I has a dual function: chemoselectively activating the thioamide and tethering the N‐terminal thioamide to the C‐terminal carboxylate. Extrusion of Ag₂S generates an isoimide intermediate, which undergoes acyl transfer to generate the native cyclic peptide, resulting in a rapid, traceless macrocylization process. Cyclic peptides are furnished in high yields within 1 hour, free of epimerization and cyclodimerization.     

Water-induced formation of an alkali-ion dimer in cryptomelane nanorods

Tunneled metal oxides such as α-Mn₈O₁₆ (hollandite) have proven to be compelling candidates for charge-storage materials in high-density batteries. In particular, the tunnels can support one-dimensional chains of K⁺ ions (which act as structure-stabilizing dopants) and H₂O molecules, as these chains are favored by strong H-bonds and electrostatic interactions. In this work, we examine the role of water molecules in enhancing the stability of K⁺-doped α-Mn₈O₁₆ (cryptomelane). The combined experimental and theoretical analyses show that for high enough concentrations of water and tunnel-ions, H₂O displaces K⁺ ions from their natural binding sites. This displacement becomes energetically favorable due to the formation of K²⁺ dimers, thereby modifying the stoichiometric charge of the system. These findings have potentially significant technological implications for the consideration of cryptomelane as a Li⁺/Na⁺ battery electrode. Our work establishes the functional role of water in altering the energetics and structural properties of cryptomelane, an observation that has frequently been overlooked in previous studies.

Water displaces potassium ions and initiates the formation of a homonuclear dimer ion (K²⁺) in the tunnels of hollandite.