On some fixed-point theorems for generalized contractions and their perturbations

The aim of this paper is to prove a collection of fixed-point theorems for mappings which can be roughly called generalized contractions or their perturbations. In particular, we are going to consider operators (single-valued or multi-valued) in Banach spaces with a quasimodulus, in hyperconvex subsets of normed spaces, or finally in non-Archimedean spaces. A particular attention will be paid to Krasnoselskii-type fixed-point theorems as well as to a Schaefer-type fixed-point theorem. Some applications to nonlinear functional-integral equations will be given. Our results extend and complement some commonly known theorems.     

Spectroscopic and Theoretical Studies of Fluorescence Effects in 2-Methylamino-5-(2,4-dihydroxyphenyl)-1,3,4-thiadiazole Induced by Molecular Aggregation

The article presents the results of fluorescence analyses of 2-methylamino-5-(2,4-dihydroxyphenyl)-1,3,4-thiadiazole (MDFT) in an aqueous environment. MDFT dissolved in aqueous solutions with a pH value in the range from 1 to 4.5 yielded an interesting effect of two clearly separated fluorescence emissions. In turn, a single fluorescence was observed in MDFT dissolved in water solutions with a pH value from 4.5 to 12. As it was suggested in the previous investigations of other 1,3,4-thiadiazole compounds, these effects may be associated with conformational changes in the structure of the analysed molecule accompanied by aggregation effects. Crystallographic data showed that the effect of the two separated fluorescence emissions occurred in a conformation with the –OH group in the resorcyl ring bound on the side of the sulphur atom from the 1,3,4-thiadiazole ring. The hypothesis of aggregation as the mechanism involved in the change in the spectral properties at low pH is supported by the results of (Time-Dependent) Density Functional Theory calculations. The possibility of rapid analysis of conformational changes with the fluorescence spectroscopy technique may be rather important outcome obtained from the spectroscopic studies presented in this article. Additionally, the presented results seem to be highly important as they can be easily observed in solutions and biologically important samples.     

Importance of the Anchor Group Position (Para versus Meta) in Tetraphenylmethane Tripods: Synthesis and Self‐Assembly Features

The efficient synthesis of tripodal platforms based on tetraphenylmethane with three acetyl‐protected thiol groups in either meta or para positions relative to the central sp³ carbon for deposition on Au (111) surfaces is reported. These platforms are intended to provide a vertical arrangement of the substituent in position 4 of the perpendicular phenyl ring and an electronic coupling to the gold substrate. The self‐assembly features of both derivatives are analyzed on Au (111) surfaces by low‐temperature ultra‐high‐vacuum STM, high‐resolution X‐ray photoelectron spectroscopy, near‐edge X‐ray absorption fine structure spectroscopy, and reductive voltammetric desorption studies. These experiments indicated that the meta derivative forms a well‐ordered monolayer, with most of the anchoring groups bound to the surface, whereas the para derivative forms a multilayer film with physically adsorbed adlayers on the chemisorbed para monolayer. Single‐molecule conductance values for both tripodal platforms are obtained through an STM break junction experiment.     

‘Double chiral’ new members of calixsalen family

Synthesis and X-ray diffraction studies on the first examples of ‘double chiral’ calixsalens are presented. In these molecules, one can clearly distinguish two chiral zones. The first one is made by the macrocycle base, whereas the second chiral zone is set up of the additional chirality elements in the tail of the molecule. The ‘double chiral’ calixsalens are formed through cyclocondensation between chiral vicinal diamine of trans-1,2-diaminocyclohexane type and chiral C-5 substituted 2-hydroxyisophthalaldehyde derivatives. The absolute configuration of the dialdehyde did not affect the yield of the macrocyclization reaction. The presence of secondary amides in the tail part of the macrocycle leads to formation of hydrogen bonding network in the solid state, while sterical hindrance preserve interdigitation, thus, ‘double chiral’ calixsalens do not form aggregates typical for other calixsalens.     

The optimal general upper bound for the distinguishing index of infinite graphs

The distinguishing index $D\prime \left(G\right)$ of a graph $G$ is the least cardinal number $d$ such that $G$ has an edge-coloring with $d$ colors, which is preserved only by the trivial automorphism. We prove a general upper bound $D\prime \text{◂≤▸}\left(G\right)\le \mathrm{\Delta }-1$ for any connected infinite graph $G$ with finite maximum degree $\mathrm{\Delta }\ge 3$. This is in contrast with finite graphs since for every $\mathrm{\Delta }\ge 3$ there exist infinitely many connected, finite graphs $G$ with $\text{◂,▸}D\prime \left(G\right)=\mathrm{\Delta }$. We also give examples showing that this bound is sharp for any maximum degree $\mathrm{\Delta }$.