Symmetry-itemized enumeration of quadruplets of RS-stereoisomers. II. The partial-cycle-index method of the USCI approach combined with the stereoisogram approach

The symmetry-itemized enumeration of quadruplets of stereoisograms is discussed by starting from a tetrahedral skeleton, where the partial-cycle-index (PCI) method of the unit-subduced-cycle-index approach (Fujita in Symmetry and combinatorial enumeration of chemistry. Springer, Berlin, 1991) is combined with the stereoisogram approach (Fujita in J Org Chem 69:3158–3165, 2004, Tetrahedron 60:11629–11638, 2004). Such a tetrahedral skeleton as contained in the quadruplet of a stereoisogram belongs to an RS-stereoisomeric group denoted by $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ , where the four positions of the tetrahedral skeleton accommodate achiral and chiral proligands to give quadruplets belonging to subgroups of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ according to the stereoisogram approach. The numbers of quadruplets are calculated as generating functions by applying the PCI method. They are itemized in terms of subgroups of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ , which are further categorized into five types. Quadruples for stereoisograms of types I–V are ascribed to subgroups of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ , where their features are discussed in comparison between RS-stereoisomeric groups and point groups.     

Hierarchical enumeration of Kekulé’s benzenes,Ladenburg’s benzenes,Dewar’s benzenes,and benzvalenes by using combined-permutation representations

The group hierarchy for each skeleton of ligancy 6 is formulated to be: point group (PG \({\varvec{G}}_{\sigma }\)) \(\subseteq \) RS-stereoisomeric group (RS-SIG \({\varvec{G}}_{\sigma \widetilde{\sigma }\widehat{I}}\)) \(\subseteq \) stereoisomeric group (SIG \(\widetilde{{\varvec{G}}}_{\sigma \widetilde{\sigma }\widehat{I}}\)) \(\subseteq \) isoskeletomeric group (ISG \(\widetilde{\widetilde{{\varvec{G}}}}_{\sigma \widetilde{\sigma }\widehat{I}}\) = \({\varvec{S}}^{[6]}_{\sigma \widehat{I}}\)), where we start from the PG \({\varvec{G}}_{\sigma }\) = \({\varvec{D}}_{6h}\) for the Kekulé benzene skeleton, from the PG \({\varvec{G}}_{\sigma }\) = \({\varvec{D}}_{3h}\) for the Ladenburg benzene skeleton, from the PG \({\varvec{G}}_{\sigma }\) = \({\varvec{C}}_{2v}\) for the Dewar benzene skeleton, or from the PG \({\varvec{G}}_{\sigma }\) = \({\varvec{C}}_{2v}\) for the benzvalene skeleton. After these groups are constructed as combined-permutation representations, the calculation of the respective cycle indices with chirality fittingness (CI-CFs) and the introduction of ligand-inventory functions are conducted to give generation functions for 3D-based enumerations (for PGs and RS-SIGs) and 2D-based enumerations (for SIGs and ISGs). The enumeration results are discussed by means of isomer-classification diagrams, in which equivalence classes under enantiomerism (for PGs), RS-stereoisomerism (for RS-SIGs), stereoisomerism (for SIGs), and isoskeletomerism (for ISGs) are illustrated schematically. The implicit connotations of the conventional terms “skeletal isomerism”, “positional isomerism”, and “constitutional isomerism” are discussed, where the effects of the concept of isoskeletomerism are emphasized.     

Importance of the proligand-promolecule model in stereochemistry. I. The unit-subduced-cycle-index (USCI) approach to geometric features of prismane derivatives

Among the four methods of the unit-subduced-cycle-index (USCI) approach (Fujita in Symmetry and Combinatorial Enumeration in Chemistry. Springer, Berlin, Heidelberg, 1991), the fixed-point-matrix (FPM) method and the partial-cycle-index (PCI) method have been applied to the combinatorial enumeration of prismane derivatives. These enumeration processes are based on the proligand-promolecule model, which enables us to take account of achiral and chiral proligands. Prochirality in a geometric meaning has been discussed in general by emphasizing the presence of enantiospheric orbits in enumerated prismane derivatives. An enantiospheric orbit accommodating chiral proligands (along with achiral ones) has been shown to exhibit prochirality by using various prismane derivatives as examples. On the other hand, the scope of pseudoasymmetry has been extended to cover such a rigid skeleton as prismane in addition to a usual pseudoasymmetric center as a single atom, where the proligand-promolecule model plays an essential role.     

Symmetry-itemized enumeration of quadruplets of RS-stereoisomers: I—the fixed-point matrix method of the USCI approach combined with the stereoisogram approach

The RS-stereoisomeric group $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ is examined to characterize quadruplets of RS-stereoisomers based on a tetrahedral skeleton and found to be isomorphic to the point group $\mathbf{O}_{h}$ of order 48. The non-redundant set of subgroups (SSG) of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ is obtained by referring to the non-redundant SSG of $\mathbf{O}_{h}$ . The coset representation for characterizing the orbit of the four positions of the tetrahedral skeleton is clarified to be $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{3v\widetilde{\sigma }\widehat{I}})$ , which is closely related to the $\mathbf{O}_{h}(/\mathbf{D}_{3d})$ . According to the unit-subduced-cycle-index (USCI) approach (Fujita in Symmetry and combinatorial enumeration in chemistry. Springer, Berlin, 1991), the subdution of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{3v\widetilde{\sigma }\widehat{I}})$ is examined so as to generate unit subduced cycle indices with chirality fittingness (USCI-CFs). The fixed-point matrix method of the USCI approach is applied to the USCI-CFs. Thereby, the numbers of quadruplets are calculated in an itemized fashion with respect to the subgroups of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ . After the subgroups of $\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$ are categorized into types I–V, type-itemized enumeration of quadruplets is conducted to illustrate the versatility of the stereoisogram approach.     

Graphs to chemical structures 3. General theorems with the use of different sets of sphericity indices for combinatorial enumeration of nonrigid stereoisomers

The extended sphericity indices of k-cycles, which were defined in Part 2 of this series (S. Fujita, Theor Chem Acc, Online: http://www.springerlink.com/index/10.1007/s00214-004-0606-z) according to the enantiospheric, homospheric, or hemispheric nature of each k-cycle, are further extended to prove more general theorems for enumerating nonrigid stereoisomers with rotatable ligands. One of the extended points is the use of different sets of sphericity indices to treat one or more orbits contained in skeletons and ligands. Another is to take account of chirality in proligands and sub-proligands, the latter of which are introduced to consider further inner structures of ligands. Two theorems for enumerating nonrigid stereoisomers are proved by adopting two schemes of their derivation, i.e., the scheme ``positions of a skeleton ⇐ proligands ⇐ ligands (positions of a ligand ⇐ sub-proligands)' and the scheme ``(positions of a skeleton ⇐ proligands ⇐ ligands (positions of a ligand)) ⇐ sub-proligands'. The theorems are applied to the stereoisomerism of trihydroxyglutaric acids. Thereby, it is demonstrated where Pólya's theorem and other previous methods are deficient, when applied to the enumeration of stereoisomers.     

Integrated Discussion on Stereogenicity and Chirality for Restructuring Stereochemistry

To discuss the difference between stereogenicity and chirality, we propose the concept of RS-stereoisomeric groups. Beginning with this concept, we have further proposed the concepts of holantimers, stereoisograms, and RS-stereogenicity. Thereby, we have clarified that the concept of RS-stereogenicity, but not conventional stereogenicity, is closely related to chirality. Thus, five RS-stereogenicity types are defined and examined to discuss the difference between stereogenicity and chirality. Combinatorial enumerations have also been studied by considering the RS-stereogenicity.     

Symmetry-itemized enumeration of RS-stereoisomers of allenes. I. The fixed-point matrix method of the USCI approach combined with the stereoisogram approach

After the RS-stereoisomeric group \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) of order 16 has been defined by starting point group \(\mathbf{D}_{2d}\) of order 8, the isomorphism between \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) and the point group \(\mathbf{D}_{4h}\) of order 16 is thoroughly discussed. The non-redundant set of subgroups (SSG) of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) is obtained by referring to the non-redundant set of subgroups of \(\mathbf{D}_{4h}\) . The coset representation for characterizing the orbit of the four positions of an allene skeleton is clarified to be \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{s\widetilde{\sigma }\widehat{I}})\) , which is closely related to the \(\mathbf{D}_{4h}(/\mathbf{C}_{2v}^{\prime \prime \prime })\) . According to the unit-subduced-cycle-index (USCI) approach (Fujita, Symmetry and combinatorial enumeration of chemistry. Springer, Berlin 1991), the subduction of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{s\widetilde{\sigma }\widehat{I}})\) is examined so as to generate unit subduced cycle indices with chirality fittingness (USCI-CFs). Then, the fixed-point matrix method of the USCI approach is applied to the USCI-CFs. Thereby, the numbers of quadruplets are calculated in an itemized fashion with respect to the subgroups of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) . After the subgroups of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) are categorized into types I–V, type-itemized enumeration of quadruplets is conducted to illustrate the versatility of the stereoisogram approach.     

Local convergence of a relaxed two-step Newton like method with applications

We present a local convergence analysis for a relaxed two-step Newton-like method. We use this method to approximate a solution of a nonlinear equation in a Banach space setting. Hypotheses on the first Fréchet derivative and on the center divided-difference of order one are used. In earlier studies such as Amat et al. (Numer Linear Algebra Appl 17:639–653, 2010, Appl Math Lett 25(12):2209–2217, 2012, Appl Math Comput 219(24):11341–11347, 2013, Appl Math Comput 219(15):7954–7963, 2013, Reducing Chaos and bifurcations in Newton-type methods. Abstract and applied analysis. Hindawi Publishing Corporation, Cairo, 2013) these methods are analyzed under hypotheses up to the second Fréchet derivative and divided differences of order one. Numerical examples are also provided in this work.     

Synthesis,characterization, thermal,antimicrobial and antioxidant studies of some transition metal dithiocarbamates

Metal dithiocarbamate complexes of Co(II) [1], Cu(II) [2], Mn(II) [3], Cr(III) [4], and Pd(II) [5] have been synthesized using sodium N-ethyl-N-phenyldithiocarbamate (NaL). The complexes were characterized by elemental analyses, FTIR and UV–vis spectroscopic techniques, magnetic moment, molar conductance and thermal analyses (TGA and DSC). The infrared spectra indicated the coordination of dithiocarbamate through the two sulphur atoms in a symmetrical bidentate fashion. The thermal behavior of these complexes showed that the hydrated complexes lost water molecules in the first step, followed by decomposition of the ligand molecules in the final steps. Mass loss considerations at these final decomposition steps indicate conversion of the complexes to sulphides. The antimicrobial potentials of the complexes were evaluated against some selected bacteria strains (Escherichia coli, Pseudomonas aureginosa, Salmonella typhi and Staphylococcus aureus) and fungi organisms (Aspergillus flavus and Fasiparium oxysporium). The compounds showed a broad spectrum of fungicidal and bactericidal activities which exceeds that of the control drugs at a 100 μg/mL concentration. The antioxidant properties of the ligand and its Cu(II) complex were evaluated in vitro using DPPH assay, and the complex was found to exhibit better radical scavenging ability than the free ligand.     

Partition coefficients for the SAMPL5 challenge using transfer free energies

SAMPL challenges (Mobley et al. in J Comput Aided Mol Des 28:135–150, 2014; Skillman in J Comput Aided Mol Des 26:473–474, 2012; Geballe in J Comput Aided Mol Des 24:259–279, 2010; Guthrie in J Phys Chem B 113:4501–4507, 2009) provide excellent opportunities to assess theoretical approaches on new data sets with a goal of gaining greater insight towards protein and ligand modeling. In the SAMPL5 experiment, cyclohexane–water partition coefficients were determined using a vertical solvation scheme in conjunction with the SMD continuum solvent model. Several DFT functionals partnered with correlation consistent basis sets were evaluated for the prediction of the partition coefficients. The approach chosen for the competition, a B3PW91 vertical solvation scheme, yields a mean absolute deviation of 1.9 logP units and performs well at estimating the correct hydrophilicity and hydrophobicity for the full SAMPL5 molecule set.     

Expected distance based on random walks

By considering a graph as a network of resistances, Klein and Randi? (J Math Chem 12(1):81–95, 1993) proposed the definition of a distance measure. Indeed, if each edge of the graph represents a resistance of \(1 \varOmega \), the equivalent resistance of the graph between each pair of vertices may be used as a distance. Based upon random walks in graphs, Stephenson and Zelen (Soc Netw 11(1):1–37, 1989) built a computational model to find the probability that each edge is used. From a mathematical point of view, both articles are based upon exactly the same model and the link between random walks and the electrical representation was established by Newman (Soc Netw 27(1):39–54, 2005) when defining an alternative to Freeman’s (Sociometry 40:35–41, 1977, Soc Netw 1(3):215–239, 1979) betweenness centrality based upon random walks. In the present paper, the similitude between these two processes is exploited to propose a new random walks based distance measure that may be defined as the expected length of a walk between any pair of vertices. We call it the expected distance and we prove that it is actually a distance. From this new definition, the RW Index is proposed that sums the expected walks lengths between pairs of vertices exactly in the same way as the Wiener index sums the shortest paths distances or the Kirchhoff index sums the equivalent resistances. We compare the three indices and establish the vertex and the edge decompositions for both. We compute some value of the RW index for some families of graphs and conjecture the upper and lower bounds of the RW index.     

On the mechanism of the unwanted acetylation of polysaccharides by 1,3-dialkylimidazolium acetate ionic liquids: part 2—the impact of lignin on the kinetics of cellulose acetylation

Cellulose acetylation has been reported as a side reaction of cellulose treatment with the ionic liquid 1-ethyl-3-methylimidazolium acetate ([EMIm][OAc]) (Karatzos et al. in Cellulose 19:307–312, 2012) and other 1,3-dialkylimidazolium acetate ionic liquids. 1-Acetylimidazole (AcIm), an [EMIm][OAc] impurity, has been found to be the actual acetylating agent (Zweckmair et al. in Cellulose 22:3583–3596, 2015), and the degree of acetylation was relatively low, below a DS of approx. 0.1%. Higher degrees of cellulose acetylation (DS > 10%) have been observed when the entire wood was mixed with [EMIm][OAc] instead of cellulosic pulp only (Abushammala et al. in Carbohydr Polym 134:609–616, 2015). In this paper, we explore the impact of wood constituents, mainly lignin, on cellulose acetylation using AcIm. The results demonstrate that lignin itself can be readily acetylated upon mixing with AcIm, and—noteworthy—that lignin presence significantly accelerates cellulose acetylation. The initial rate of cellulose acetylation by AcIm increased from 1.8 to 4.7%/h when only 1% of lignin, based on cellulose mass, was added. A mechanistic study employing cellulose and lignin model compounds showed lignin to be more susceptible to acetylation than cellulose and to act as an intermediate acetyl group source for further cellulose acetylation in a catalytic scenario.     

Second derivative free sixth order continuation method for solving nonlinear equations with applications

In this paper, we deal with the study of convergence analysis of modified parameter based family of second derivative free continuation method for solving nonlinear equations. We obtain the order of convergence is at least five and especially, for parameter \(\alpha =2\) sixth order convergence. Some application such as Max Planck’s conservative law, multi-factor effect are discussed and demonstrate the efficiency and performance of the new method (for \(\alpha =2\)). We compare the absolutely value of function at each iteration \(|f(x_n)|\) and \(|x_n-\xi |\) with our method and Potra and Pták method [1], Kou et al. method [2]. We observed that our method is more efficient than existing methods. Also, the Dynamics of the method are studied for a special case of the parameter in convergence.