A Talenti Comparison Result for Solutions to Elliptic Problems with Robin Boundary Conditions

Comparison results of Talenti type for elliptic problems with Dirichlet boundary conditions have been widely investigated in recent decades. In this paper, we deal with Robin boundary conditions. Surprisingly, contrary to the Dirichlet case, Robin boundary conditions make the comparison sensitive to the dimension, and while the planar case seems to be completely settled, in higher dimensions some open problems are yet unsolved. © 2023 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.     

An explicit computation of the Bergman kernel function

We give an explicit computation of the Bergman kernel function on the domain

Continuous modified Newton’s-type method for nonlinear operator equations

A nonlinear operator equation F(x)=0, F:H→H, in a Hilbert space is considered. Continuous Newton’s-type procedures based on a construction of a dynamical system with the trajectory starting at some initial point x ₀ and becoming asymptotically close to a solution of F(x)=0 as t→+∞ are discussed. Well-posed and ill-posed problems are investigated. Received: June 29, 2001; in final form: February 26, 2002?Published online: February 20, 2003 This paper was finished when AGR was visiting Institute for Theoretical Physics, University of Giessen. The author thanks DAAD for support     

On circular–linear one-factorizations of the complete graph

In Korchmáros et al. (2018)one-factorizations of the complete graph $K_n$ are constructed for $n=q+1$ with any odd prime power $q$ such that either $q\equiv 1(mod4)$ or $q=2^h?1$. The arithmetic restriction $n=q+1$ is due to the fact that the vertices of $K_n$ in the construction are the points of a conic $\Omega$ in the finite plane of order $q$. Here we work on the Euclidean plane and describe an analogous construction where the role of $\Omega$ is taken by a regular $n$-gon. This allows us to remove the above constraints and construct one-factorizations of $K_n$ for every even $n\ge 6$.     

On the structure of 3‐acetylamino‐5‐methyl‐1,2,4‐oxadiazole and on the fully degenerate rearrangements (FDR) of its anion: a stimulating comparison between the results of ‘in‐silicon chemistry’ and ‘laboratory chemistry’

An accurate crystal structure determination has provided evidence for a planar conformation for 3‐acetylamino‐5‐methyl‐1,2,4‐oxadiazole ( 5 ), in agreement with quantum‐mechanical calculations in the gas phase. In the crystal, a series of strong intermolecular N7H7^….O9 hydrogen bonds link the amido groups of different molecules, causing the formation of infinite parallel ordered chains. The effect of the DMSO solvent on the energy and charge distribution of compound 5 and on its relevant 5 ^? anion, involved in a fully degenerate rearrangement (FDR), has been deepened by quantum‐mechanical DFT calculations. The calculated energy barrier for the FDR increases in going from in vacuo to DMSO, in agreement with previsions based on the Hughes and Ingold rules concerning the nucleophilic substitution of an anionic reagent (the deprotonated amido group in the side chain) on a neutral substrate (the 1,2,4‐oxadiazole ring). Copyright © 2009 John Wiley & Sons, Ltd.     

7‐Chloro‐3,6‐di­methyl‐1H‐indene‐2‐carb­aldehyde

In the crystal structure of C₁₂H₁₁ClO, the (planar) mol­ecules give rise to a parallel packing. A model crystal obtained by semi‐empirical and packing‐energy calculations is consistent with the observed structure.     

Methylene‐Bridged Metallocenes with 2,2′‐Methylenebis[1H‐inden‐1‐yl] Ligands: Synthesis,Characterization, and Polymerization Catalysis of a Synthetically Simple Class of C2‐ and C2v‐Symmetric ansa‐Metallocenes

The acid‐catalyzed reaction between formaldehyde and 1H‐indene, 3‐alkyl‐ and 3‐aryl‐1H‐indenes, and six‐membered‐ring substituted 1H‐indenes, with the 1H‐indene/CH₂O ratio of 2 : 1, at temperatures above 60° in hydrocarbon solvents, yields 2,2′‐methylenebis[1H‐indenes] 1 – 8 in 50–100% yield. These 2,2′‐methylenebis[1H‐indenes] are easily deprotonated by 2 equiv. of BuLi or MeLi to yield the corresponding dilithium salts, which are efficiently converted into ansa‐metallocenes of Zr and Hf. The unsubstituted dichloro{(1,1′,2,2′,3,3′,3a,3′a,7a,7′a‐η)‐2,2′‐methylenebis[1H‐inden‐1‐yl]}zirconium ([ZrCl₂( 1′ )]) is the least soluble in organic solvents. Substitution of the 1H‐indenyl moieties by hydrocarbyl substituents increases the hydrocarbon solubility of the complexes, and the presence of a substituent larger than a Me group at the 1,1′ positions of the ligand imparts a high diastereoselectivity to the metallation step, since only the racemic isomers are obtained. Methylene‐bridged ‘ansa‐zirconocenes’ show a noticeable open arrangement of the bis[1H‐inden‐1‐yl] moiety, as measured by the angle between the planes defined by the two π‐ligands (the ‘bite angle’). In particular, of the ‘zirconocenes’ structurally characterized so far, the dichloro{(1,1′,2,2′,3,3′,3a,3′a,7a,7′a‐η)‐2,2′‐methylenebis[4,7‐dimethyl‐1H‐inden‐1‐yl]}zirconium ([ZrCl₂( 5′ )] is the most open. The mixture [ZrCl₂( 1′ )]/methylalumoxane (MAO) is inactive in the polymerization of both ethylene and propylene, while the metallocenes with substituted indenyl ligands polymerize propylene to atactic polypropylene of a molecular mass that depends on the size of the alkyl or aryl groups at the 1,1′ positions of the ligand. Ethene is polymerized by rac‐dichloro{(1,1′,2,2′,3,3′,3a,3′a,7a,7′a‐η)‐2,2′‐methylenebis[1‐methyl‐1H‐inden‐1‐yl]}zirconium ([ZrCl₂( 2′ )])/MAO to polyethylene waxes (average degree of polymerization ca. 100), which are terminated almost exclusively by ethenyl end groups. Polyethylene with a high molecular mass could be obtained by increasing the size of the 1‐alkyl substituent.     

Theoretical and spectroscopic study of the effect of ring substitution on the adsorption of anisole on platinum

The adsorption of anisole, 3,5-dimethylanisole, and 3,5-bis-(trifluoromethyl)-anisole on Pt(111) was studied theoretically and compared to the adsorption of benzene using relativistically corrected density functional theory. A cluster of 31 platinum atoms was used to simulate the surface. The three anisoles were found to be less strongly adsorbed than the parent molecule benzene, 3,5-bis-(trifluoromethyl)-anisole showing weakest adsorption, with an adsorption energy of only one-third that of benzene. The theoretical study was complemented by in situ ATR-IR spectroscopy of the adsorption of the anisole derivatives on a polycrystalline Pt film. The spectroscopic study indicated that the adsorption strength of the anisoles follows the same order as predicted by the calculations. In addition, catalytic hydrogenation tests showed that the propensity to aromatic ring hydrogenation can also be correlated to the mode and strength of adsorption of the anisoles. The degree of saturation followed the same order as the adsorption strength found by the calculations and indicated by spectroscopy. Although 3,5-dimethyl substitution on anisole resulted in only a partial loss of adsorption energy and reactivity toward ring hydrogenation as compared to anisole, the substitution by CF(3) groups led to a large loss of adsorption energy and complete loss of reactivity toward aromatic ring saturation. Along with the study of the substituent effect on the adsorption of aromatic molecules, the correlation between adsorption and propensity to saturation of aromatic substrates could be corroborated.