Asymmetric Catalytic Assembly of Triple-Columned and Multilayered Chiral Folding Polymers Showing Aggregation-Induced Emission (AIE)

Unprecedented chiral multilayer folding 3D polymers have been assembled and regulated by uniform and differentiated aromatic chromophore units between naphthyl piers. Screening catalysts, catalytic systems and monomers were proven to be crucial for asymmetric catalytic Suzuki–Miyaura polycouplings for this assembly. X-ray crystallography of the corresponding dimers and trimers revealed the absolute configuration and the intermolecular packing pattern. Up to 61 960 M_w/41 900 M_n and m/z 4317 for polymers and oligomers, as confirmed by gel permeation chromatography (GPC) and MALDI-TOF MS, indicated that these frameworks were composed of multiple stacked layers. The resulting multiple π-assemblies exhibited remarkable optical properties in aggregated states (photoluminescence in solids and aggregation-induced emission in solutions), as well as reversible redox properties in electrochemical performance.     

Functions generating (m,M, $$\varvec{\Psi }$$ )-Schur-convex sums

Aequationes mathematicae - The notion of $$(m,M,\Psi )$$ -Schur-convexity is introduced and functions generating $$(m,M,\Psi )$$ -Schur-convex sums are investigated. An extension of the...     

Smoothing arithmetic error terms: the case of the Euler φ function

In many cases known methods of detecting oscillations of arithmetic error terms involve certain smoothing pro‐cedures. Usually an application of the smoothing operator does not change significantly the order of magnitude of the error under consideration. This is so for instance in the case of the classical error terms known in the prime number theory. The main purpose of this paper is to show that the situation for primes is not general. Considering the error term in the asymptotic formula for the Euler totient function we show that just one application of an integral smoothing operator changes situation dramatically: the order of magnitude of drops from x to √x (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Around Jensen’s inequality for strongly convex functions

In this paper we use basic properties of strongly convex functions to obtain new inequalities including Jensen type and Jensen–Mercer type inequalities. Applications for special means are pointed out as well. We also give a Jensen’s operator inequality for strongly convex functions. As a corollary, we improve the Hölder-McCarthy inequality under suitable conditions. More precisely we show that if \(Sp\left( A \right) \subset \left( 1,\infty \right) \), then

$$\begin{aligned} {{\left\langle Ax,x \right\rangle }^{r}}\le \left\langle {{A}^{r}}x,x \right\rangle -\frac{{{r}^{2}}-r}{2}\left( \left\langle {{A}^{2}}x,x \right\rangle -{{\left\langle Ax,x \right\rangle }^{2}} \right) ,\quad r\ge 2 \end{aligned}$$

and if \(Sp\left( A \right) \subset \left( 0,1 \right) \), then

$$\begin{aligned} \left\langle {{A}^{r}}x,x \right\rangle \le {{\left\langle Ax,x \right\rangle }^{r}}+\frac{r-{{r}^{2}}}{2}\left( {{\left\langle Ax,x \right\rangle }^{2}}-\left\langle {{A}^{2}}x,x \right\rangle \right) ,\quad 0<r<1 \end{aligned}$$

for each positive operator A and \(x\in \mathcal {H}\) with \(\left\| x \right\| =1\).

     

Why and How Much Brouwer’s Fixed Point Theorem Fails in Noncompact Setting?

The celebrated Brouwer’s Fixed Point Theorem is dated in 1912. Its extension to compact set setting in Banach spaces due to Schauder appeared in 1930. Immediately it raised the question whether the Theorem can be extended to noncompact setting. The works of Kakutani, Klee, Benyamini and Sterfeld, Sternfeld and Lim solved the qualitative part of the problem. Lack of compactness makes the statement of the theorem false. However, there are some quantitative aspects of the question. The two basic are called minimal displacement problem, and optimal retraction problem. The aim of this article is to present the historical back ground and possibly, up to date state of investigations in this field. A list of open problems with comments will be discussed.     

Calix[4]arene-thiacrown-5 di(carboxylic acid) regioisomers as metal ion extractants

A regioisomer of a previously reported p-tert-butylcalix[4]arene-1,3-thiacrown-5 di(carboxylic acid) is prepared in which the thiacrown ring attachment sites are moved from distal phenolic oxygens of a cone p-tert-butylcalix[4]arene scaffold to proximal phenolic oxygens. The influence of this structural variation in the di-ionizable calixcrown ligand on competitive solvent extractions of alkali metal cations and of alkaline earth metal cations and single species solvent extractions of Hg²⁺ and Pb²⁺ from aqueous solutions into chloroform is evaluated.     

The effect of corona discharge on free convection heat transfer from a horizontal cylinder

Free convection heat transfer from an isothermal horizontal cylinder in the presence of DC positive corona discharge with a blade edge emitter electrode has been studied experimentally and numerically. A Mach–Zehnder interferometer was used to determine the local Nusselt numbers. The effect of corona discharge on heat transfer from the cylinder was investigated at Rayleigh numbers in the range between 1500 and 5000. To find the details of the flow patterns and to further verify the experimental results, numerical simulations were also performed. It was found that the numerical results are in good agreement with experimental data. By increasing the applied voltage up to 15.5 kV, the corona discharge generates a recirculation zone around the blade and below the lower stagnation point of the cylinder. The effect of the recirculation zone becomes stronger near the breakdown voltage (17 kV) and it is responsible for a local decrease in the cooling of the cylinder around the lower stagnation point. The results indicate that corona discharge has a significant effect on the average Nusselt number at lower Rayleigh numbers whereas it has smaller effect at higher Rayleigh numbers.     

Phase equilibrium in the system of water,alcohol or ketone,and sodium chloride

The occurrence of various regions of phase equilibrium in three-component systems of water–alcohol (ketone)–sodium chloride was studied. As for methanol and ethanol there are two regions: the liquid single-phase region and the two-phase region of liquid + solid. For propanol and acetone (of a complete miscibility with water) there also occurs, however, the two-phase region of liquid + liquid, and the three-phase region of liquid + liquid + solid. Both phenomena occur while salting-out the organic solvents from the water solution by sodium chloride. The systems containing butanol, pentanol, methylethyl- and diethylketone (of an incomplete miscibility with water) confirm the occurrence of the system regions, similar to those for propanol or acetone. The results of the experiments were explained by considering competive molecular interactions between: water and sodium chloride; water and organic solvent; organic solvent and sodium chloride.     

A Novel Method of Spectral Analysis of Oscillatory Belousov–Zhabotinsky Reaction

This paper is a preliminary study of time–frequency behavior of homogenous Belousov–Zhabotinsky reaction in a batch reactor. A series of measurements performed for a variable concentration of the reaction substrates led to the conclusion that there exists time-dependency of an instantaneous frequency of fundamental frequency and harmonics and of electrode potential fluctuations generated during the process discussed. The Short Time Fourier Transformation (STFT) was proposed for analyzing electrode potential registers. On the basis of STFT analysis a decrease in frequency of oscillations of exponential character has been postulated.