Multiplying after the turn

In education multiplying is usually viewed as repeated joining together and dividing as repeated taking away or, which comes to the same thing, as an equal distribution. This presentation springs from Antiquity, when thought was mostly concrete. In modern mathematics we have relation-numbers instead of, image-numbers and likewise multiplying is a facet of relational thinking. The view that children merely can learn through the concrete is often biassedly understood in the sense that the concrete has to be abstracted, which characterizes substantial thinking. However, in the case of relational thinking, learning through the concrete means that to achieve insight the mathematical activities have to be applied to reality, a crucial point, for most people have difficulties with applying multiplication, much more than with, the inherent algorithms. It appears that they do not really know what multiplication is, particularly not its space structure. The more general the structure the more and wider the applications. This thesis infers that multiplying as multiple of classes is much less useful than multiplying as space form. Questing for the essence of multiplication is the major topic of this paper. Which changes has its structure undergone and how can education deal with them? At the end it is illustratively explained why probability, based on the established multiplication, is usually such a tough domain.     

3′-(1,2,3-Triazol-1-yl)-2′,3′-dideoxythymidine and 3′-(1,2,3-triazol-1-yl)-2′,3′-dideoxyuridine

Cycloaddition of different acetylenic compounds on the azido function of 3′-azido-2′,3′-dideoxythymidine and 3′-azido-2′,3′-dideoxyuridine afforded products with a 1,2,3-triazol-1-yl substituent in the 3′-position. In contrast with the parent compounds, these triazolyl derivatives had no appreciable activity against human immunodeficiency virus (HIV-1).     

Macromolecular metal complexes as catalysts with improved stability

Polymer-bound tertiary amine–copper complexes and polymer-bound phosphite–rhodium complexes were studied as catalysts for oxidative coupling of phenols and hydroformylation of alkenes, respectively. The activity and stability of these catalysts could be tuned or optimized by adapting the structure of the ligands and by changing the distance between adjacent ligands along the polymer chains. The latter effect has been described in terms of strain in the intermediate chain segments in the copper complexes or enhancement of the effective local ligand concentration around the rhodium complexes. So-called immobilized homogeneous catalysts were obtained by end-grafting of both types of macromolecular catalysts on to inert and insoluble silica particles. These immobilized polymeric catalysts could easily be separated and recovered. Under proper conditions a grafted polymerbound imidazole–copper complex and a new type of polymer-bound triphenyl–phosphite–rhodium complex showed excellent stability in continuous processes.     

Computing the Trivariate Chain Length/Degree of Branching/Number of Combination Points Distribution for Radical Polymerization with Transfer to Polymer and Recombination Termination

Summary: Polymer molecules made by radical polymerization with transfer to polymer and recombination termination contain branch points (connecting branch arms to backbones) and combination points (connecting various molecular structures). The trivariate chain length/degree of branching/number of combination points distribution (CLD/DBD/CPD) was calculated using a two‐dimensional version of a previously used pseudo‐distribution approach. This yielded the CPD moments for given chain length n and number of branches i. Both DBD and CPD at given chain length resemble a binomial distribution. For the construction of the full DBD and CPD a set of orthogonal polynomials, the Krawtchouk polynomials, in combination with the binomial distribution was employed. A first‐order Krawtchouk approximation enabled to compute the full CLD/DBD/CPD from the three CPD moments as a function of n and i. Results agree well with those from a Monte Carlo (MC) simulation method. However, the large scatter due to the small numbers of molecules collected in the MC method at longer chain lengths prevents comparison in this range.

Solutions of 3D CLD/DBD/CPD: CPD at constant chain length (20 000) and number of branch points (20).     

The crystal structure of the CeNA:RNA hybrid ce(GCGTAGCG):r(CGCUACGC)

Cyclohexenyl nucleic acids (CeNA) are characterised by the carbon–carbon double bond replacing the O4′‐oxygen atom of the natural D ‐2′‐deoxyribose sugar ring in DNA. CeNAs exhibit a high conformational flexibility, are stable against nuclease activity and their hybridisation is RNA selective. Additionally, CeNA has been shown to induce an enhanced biological activity when incorporated in siRNA. This makes CeNA a good candidate for siRNA and synthetic aptamer applications. The crystal structure of the synthetic CeNA:RNA hybrid ce(GCGTAGCG):r(CGCUACGC) has been solved with a resolution of 2.50 Å. The CeNA:RNA duplex adopts an anti‐parallel, right‐handed double helix with standard Watson–Crick base pairing. Analyses of the helical parameters revealed the octamer to form an A‐like double helix. The cyclohexenyl rings mainly adopt the ³H₂ conformation, which resembles the C3′‐endo conformation of RNA ribose ring. This C3′‐endo ring puckering was found in most of the RNA residues and is typical for A‐family helices. The crystal structure is stabilised by the presence of hexahydrated magnesium ions. The fact that the CeNA:RNA hybrid adopts an A‐type double helical conformation confirms the high potential of CeNAs for the construction of efficient siRNAs which can be used for therapeutical applications.     

Molecular‐Weight‐Distribution Modelling of Radical Polymerization in Batch and Continuous Reactors with Transfer to Polymer Leading to Gel Formation

Full chain‐length distribution (CLD) modelling applying the Galerkin finite‐element method^[1] (FEM) to polymerization reactors featuring a certain degree of gel formation is confronted with extremely long computation times. The paper describes a new method to predict CLDs for systems where gel formation may occur. The new concept is to model a part of the CLD up to a cut‐off length L, while satisfying the full set of population balances. With transfer to polymer as the mechanism responsible for gelation, this gives rise to a closure problem, which has been solved by assuming the dead CLD beyond L to be represented by a part of a Flory distribution. The method could be proved to work by performing simulations and comparing cut‐off CLDs to full CLDs for non‐gelling systems and comparing results for different L for systems with gelation. The model is demonstrated for polymerization reactors, the batch reactor and the continuous stirred‐tank reactor (CSTR), with either disproportionation or recombination termination. Reliable results are obtained for systems with moderate gel formation. Comparing these results to those from moment models including balance equations up to the fourth moment, a number of interesting differences have been found.     

Remarkable solvent-dependent excited-state chirality: a molecular modulator of circularly polarized luminescence

The photochemical control of ground- and excited-state chirality of (M)-cis-(1) and (P)-trans-(2)-2-nitro-7-(dimethylamino)-9-(2',3'-dihydro-1'H-naphtho[2,1-b]-thiopyran-1'-ylidene)-9H-thioxanthene is described. It is shown that while ground state chirality can be controlled photochemically by irradiation with light of different wavelengths, the excited state chirality can be tuned either photochemically in a similar way or by appropriate choice of solvent. In benzene solution, circularly polarized luminescence of the two isomers with opposite ground-state helicity, (M)-cis-1 and (P)-trans-2, revealed corresponding excited states of opposite helicity. On the contrary, in n-hexane solution, circularly polarized luminescence was identical for the two forms indicating identical excited state chirality. Circularly polarized luminescence (CPL), steady-state and time-dependent fluorescence, and time-resolved microwave conductivity (TRMC) measurements in both n-hexane and benzene are reported, which provide an explanation for the remarkable solvent dependence of excited-state chirality.     

Structure–Activity Relationship Study of a Potent α-Thrombin Binding Aptamer Incorporating Hexitol Nucleotides

The replacement of one or more nucleotide residues in the potent α-thrombin-binding aptamer NU172 with hexitol-based nucleotides has been devised to study the effect of these substitutions on the physicochemical and functional properties of the anticoagulant agent. The incorporation of single hexitol nucleotides at the T9 and G18 positions of NU172 substantially retained the physicochemical features of the parent oligonucleotide, as a result of the biomimetic properties of the hexitol backbone. Importantly, the NU172- T ^H9 mutant exhibited a higher binding affinity toward human α-thrombin than the native aptamer and an improved stability even after 24 h in 90 % human serum, with a significant increase in the estimated half-life. The anticoagulant activity of the modified oligonucleotide was also found to be slightly preferable to NU172. Overall, these results confirm the potential of hexitol nucleotides as biomimetic agents, while laying the foundations for the development of NU172-inspired α-thrombin-binding aptamers.     

Comprehensive two‐dimensional GC for the analysis of low‐molecular‐weight oxygenates in three different matrices from a petrochemical pilot plant using a single calibration

A method using comprehensive ²DGC with flame ionisation detection was developed to quantify 17 low‐molecular‐weight oxygenates in three different matrices, namely water, oil and gas, using a single calibration. The method was required for the pilot‐plant experiments of a chemical process unit. From an analytical perspective, the first task was to find a suitable analytical method with sufficient selectivity and sensitivity to analyse the selected oxygenates at low levels in the presence of high levels of hydrocarbons. The second was the accurate quantitation of oxygenates in the water, oil and gas fractions, using the same instrument and calibration. Both these requirements were met by using comprehensive ²DGC in the inverse configuration and calibrating the detector with the number of moles injected versus response. The method was successfully applied for the characterisation of the reactor product stream of the chemical process unit and made it possible to determine the fate of the selected oxygenates after passing through the reactor. The development of the method and some of the results are described in this paper.