Reporting gamma-ray spectrometric measurement results near the natural limit: primary measurement results,best estimates calculated with the Bayesian posterior and best estimates calculated with the probability density distribution resembling shifting

When reporting best estimates as measurement results, systematic influences near the natural limit are introduced that originate not only in the possible inconsistency of the analytical procedure but also in the conversion of the primary measurement results to best estimates. It has been shown that the probability density distribution resembling shifting of values observed in the unfeasible region to the limit of the feasible range (the natural limit) introduces a smaller systematic influence than the Bayesian posterior, resembling censoring with repetition of the measurements that result in observed values in the unfeasible region. The drawback that the uncertainty intervals calculated with the Bayesian posterior do not encompass the natural limit and may not encompass the observed value, although it lies within the feasible region, is avoided by using the probability density distribution resembling shifting. Therefore, such best estimates introduce a smaller systematic influence into the measurement results and are better suited as inputs for subsequent analyses.     

α-Amino acid derived enaminones and their application in the synthesis of N-protected methyl 5-substituted-4-hydroxypyrrole-3-carboxylates and other heterocycles

A new and simple synthesis of novel N-protected methyl 5-substituted-4-hydroxypyrrole-3-carboxylates, which exist in equilibrium with their 4-oxo tautomers, has been developed in two steps starting from N-protected α-amino acids. The key intermediates are enaminones, which can also be isolated, characterized, and used for the construction of other functionalized heterocycles, before they spontaneously decompose to pyrrole products. 4-Hydroxypyrroles are prone to partial aerial oxidation but can be efficiently alkylated or reduced to stable polysubstituted pyrrolidine derivatives.     

Effect of vinylene and 1,4-phenylene spacers on efficiency of the ground-state intramolecular charge-transfer in enlarged 4-dimethylamino-1-methylpyridinium cations

¹⁵N NMR chemical shifts of the exo- and endocyclic nitrogen atoms show how efficient is the ground-state intramolecular charge transfer between these sites in 4-dimethylamino-1-methylpyridinium cation (increased contribution of the quinoid resonance form results in a shielding and deshielding effect of their NMR signals, respectively). As it was anticipated, insertion of vinylene and/or 1,4-phenylene spacers to the cation considerably hinders the ground-state charge transfer. This hypothesis is further supported by an analysis of the C–NMe₂ bond lengths (X-ray data show that spacers elongate this bond). The selected valence angles in the compounds studied are also linearly dependent on δ(¹⁵N_endo) and δ(¹⁵N_exo) values. Although the correlation coefficient for the δ(¹⁵N_endo) versus δ(¹⁵N_exo) dependence is equal to 0.983, decrease of the net charge on one nitrogen atom is not compensated entirely by its increase on another nitrogen atom. This shows that exocyclic nitrogen atom is not the only acceptor of the positive charge in the molecule. The natural population analysis shows that the positive charge is transferred not only to the exocyclic N but also to, e.g., 1- and N-methyl C as well as to C3 and C5 atoms in pyridine ring. Ground-state charge transfer through the p-phenylene moiety was found to be less effective than through the trans-vinylene bridge. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Automated statistical control of activities measured by gamma-ray spectrometry

Statistical tests were performed for reviewing the results of routinely performed measurements. Within the operating procedure for routine spectral analysis the current results were compared with the average values obtained from measurements of samples of the same type collected at the same location. The agreement of the current result with the population of the results was assessed by the u-statistic, where the expected spread of the population of the results is given by their standard deviation and their uncertainties. After checking for reliability, the current result was used for updating the average and the spread of the population. Some examples are presented to illustrate how the average and the spread follow changes in the population of the measurement results.     

Regular polyhedra—old and new

Although it is customary to define polygons as certain families of edges, when considering polyhedra it is usual to view polygons as 2-dimensional pieces of the plane. If this rather illogical point of view is replaced by consistently understanding polygons as 1-dimensional complexes, the theory of polyhedra becomes richer and more satisfactory. Even with the strictest definition of regularity this approach leads to 17 individual regular polyhedra in the Euclidean 3-space and 12 infinite families of such polyhedra, besides the traditional ones (which consist of 5 Platonic polyhedra, 4 Kepler—Poinsot polyhedra, 3 planar tessellations and 3 Petrie—Coxeter polyhedra). Among the many still open problems that naturally arise from the new point of view, the most obvious one is the question whether the regular polyhedra found in the paper are the only ones possible in the Euclidean 3-space.This work was supported in part by National Science Foundation Grant MPS74-07547 A01.     

Musings on an example of Danzer’s

Some twenty years ago Ludwig Danzer provided the example of a (35₄) geometric configuration for which no visually attractive presentation was found despite its great combinatorial symmetry. For a long time no other (35₄) configurations was known. This changed recently, and the newly found symmetric configurations, and missteps in finding them, are presented in some detail.