Words that almost commute

The Hamming distance $\mathrm{ham}(u,v)$ between two equal-length words u, v is the number of positions where u and v differ. The words u and v are said to be conjugates if there exist non-empty words $x,y$ such that $u=xy$ and $v=yx$. The smallest value $\mathrm{ham}(xy,yx)$ can take on is 0, when x and y commute. But, interestingly, the next smallest value $\mathrm{ham}(xy,yx)$ can take on is 2 and not 1. In this paper, we consider conjugates $u=xy$ and $v=yx$ where $\mathrm{ham}(xy,yx)=2$. More specifically, we provide an efficient formula to count the number $h(n)$ of length-n words $u=xy$ over a k-letter alphabet that have a conjugate $v=yx$ such that $\mathrm{ham}(xy,yx)=2$. We also provide efficient formulae for other quantities closely related to $h(n)$. Finally, we show that $h(n)$ grows erratically: cubically for n prime, but exponentially for n even.     

The Bergman kernel for the Vekua equation

The paper extends some well-known results for analytic functions onto solutions of the Vekua equation $\partial \overline{{}_z}W=aW+b\overline{W}$ regarding the existence and construction of the Bergman kernel and of the corresponding Bergman projection operator.     

Asymptotic expansions of Kummer hypergeometric functions with three asymptotic parameters a,b and z

In a recent paper (Temme, 2021) new asymptotic expansions are given for the Kummer functions $M(a,b,z)$ and $U(a,b+1,z)$ for large positive values of $a$ and $b$, with $z$ fixed and special attention for the case $a～b$. In this paper we extend the approach and also accept large values of $z$. The new expansions are valid when at least one of the parameters $a$, $b$, or $z$ is large. We provide numerical tables to show the performance of the expansions.     

Spin-state dynamics of a photochromic iron(II) complex and its immobilization on oxide surfaces via phenol anchors

This work presents a detailed study of the photo-induced spin-state dynamics of the photochromic iron(II) complex 1, where the metal ion is in the field of a tripodal hexa-imine ligand with protolysable phenol groups. The nature of the complex’s ground state has been identified as a spin singlet by ¹H NMR and steady-state UV/vis spectroscopies, and its distorted octahedral structure was analyzed via crystal structure determination. Sub-picosecond and nanosecond time-resolved laser flash photolysis experiments identify the long-lived quintet state of 1 as the selective product of photoexcitation in the UV/vis spectral region. Thermal barriers of spin-state interconversion as a function of solvent and added base are derived from temperature-dependent rates of transient decay. Ground-state recovery is found to be significantly affected by the solvent and is strongly enhanced, in particular, by base-driven solvolysis of the ligand’s phenol groups. Partial spontaneous deprotonation of the phenolic hydroxyl groups of 1 seems to prevail on metal oxide surfaces, i.e. on alumina. Composite materials, like 1 at Al₂O₃, that retain the characteristic spectral features of the parent iron(II) complex can be readily obtained by wet impregnation of hydrous alumina with solutions of 1.     

Linear representations of formal loops

A representation of an object in a category is an abelian group in the corresponding comma category. In this paper, we derive the formulas describing linear representations of objects in the category of formal loops and formal loop homomorphisms and apply them to obtain a new approach to the representation theory of formal Moufang loops and Malcev algebras based on Moufang elements. Certain ‘non-associative Moufang symmetry’ of groups is revealed.     

Design,synthesis and theoretical analysis of functionalized dicarboxylate moieties based on organotin(IV) dinuclear complexes: crystal structure elucidation and biological studies

Dinuclear tin(IV) dicarboxylate complexes of the types [(n-Bu)₂Sn(oda)(4-pic)]₂·2H₂O (1) and [(n-Bu)₂Sn(pda)(H₂O)]₂ (2) [H₂oda = oxydiacetic acid; H₂pda = pyridine 2,6-dicarboxylic acid) were synthesized and characterized via physicochemical and spectroscopic studies. The spectroscopic results indicated that Sn is seven-coordinate having pentagonal bipyramidal (pbp) geometry in both complexes. The X-ray study of complex 2 further specified pentagonal bipyramidal geometry with dinuclear structural arrangement due to the involvement of carboxylate bridges formed by pda^2? moiety. The crystal structure is further stabilized by different weak interactions viz., C–C?H, C–C?O, C–H?O, C–C?O, and C–H?H. These interactions are further supported by Hirshfeld surface analysis along with 2-D fingerprint plots of complex 2. In vitro DNA-binding studies of both complexes were evaluated using spectroscopic techniques (absorbance and fluorescence) which ascertained optimum binding affinity of both complexes. However, cleavage activity of the complexes was assessed using supercoiled DNA (pBR322) via gel electrophoresis technique which demonstrated significant cleavage pattern of both complexes at different concentrations. Furthermore, chemotherapeutic potential of complexes 1 and 2 against MCF-7 and Hep carcinoma cell lines also suggested significant antiproliferative effect of complexes. These results revealed momentous exploration of drug–DNA interactions which may engender new insinuation for the advancement of metallo-pharmaceuticals.     

DABCO-mediated aza-Michael addition of 4-aryl-1H-1,2,3-triazoles to cycloalkenones. Regioselective synthesis of disubstituted 1,2,3-triazoles

Aza-Michael addition of 4-aryl-1H-1,2,3-triazoles to 2-cycloalken-1-ones has been studied in the presence of DABCO as organic base. The reactions were carried out in acetonitrile at room temperature to provide 2,4-disubstituted 2H-1,2,3-triazoles as major adducts and 1,4-disubstituted 1H-1,2,3-triazoles as minor adducts. Though the reaction times are longer (4–8 days), the two regioisomers were separated by using column chromatography and the adducts were obtained in very good to excellent combined chemical yields. The electron-rich and electron-poor substituents on aryl moiety of 4-aryl-triazoles could tolerate the reaction conditions to afford the title adducts.     

Subgroups of totally imprimitive permutation groups

In this work the study of subgroups of totally imprimitive permutation groups is continued. Some sufficient conditions for the existence of minimal non-FC-subgroups are given. These results depend mainly on the properties of self-normalizing subgroups, which are closed in the topology of point-wise convergence. Finally, new properties of a totally imprimitive permutation p-group satisfying the cyclic-block property are given which might be a suitable example to look for minimal non-FC-subgroups.