In this article we introduce a subjective notion of the predictive power of nuclear Hamiltonians (an objective one does not exist) and examine it in the particular context of the single-nucleon energy spectra. We consider various types of uncertainties originating both from the experiment and theory stressing the dominating character of the theoretical errors. The latter originate from the complexity of the nuclear many body systems that is not matched adequately by the formalism behind the present day nuclear Hamiltonians. The related inverse problem is formulated and the presence of errors (ignorance, lack of knowledge) is parametrized in terms of the associated probability distributions. Various hypotheses concerning the input uncertainties ('numerical noise') are formulated and the impact of the input-uncertainties in the adjustment procedures down to the final parameter values and theoretical spectra is illustrated and discussed. A number of Open Problems are formulated and listed at the end of the article.

In the framework of our study, three homologous series of ionic liquids containing the 1-alkyl-4-hydroxy-1-methylpiperidinium cation with the alkyl chain length ranging from ethyl to hexadecyl and anions of natural origin derived from three forms (S, R, and RS) of mandelic acid were obtained in high yields that exceeded 90%. The obtained compounds were subjected to a thorough analysis of their physicochemical properties, including the specific rotation, density, refractive index, surface activity, and solubility in solvents with diverse ranges of polarity. The minimum concentration at which the ionic liquids exhibited antimicrobial activity was also examined. Among the tested ionic liquids, compounds comprising an alkyl chain longer than octyl possessed noticeable biological activity toward model pathogenic microorganisms, including Gram-positive bacteria, Gram-negative bacteria, and fungi. In addition, a correlation between the surface properties and biological activity was observed for all homologous series of synthesized mandelates; this observation can significantly facilitate the design of chemical structures of new and more potent antiseptics.

In this article we discuss the problem of a reliable determination of parameters of the nuclear mean-field Hamiltonians in view of obtaining high quality comparisons with the experimental single particle spectra: a. In the case of extrapolations to extreme (exotic) isospin nuclei as well as b. In terms of predictions of the energy levels that have not been so far measured also in 'non-exotic' nuclei. We formulate our discussion in terms of the Inverse Problem, a quickly growing sub-field of Applied Mathematics, and introduce a number of mathematical properties which are illustrated in Ref.(1) We focus on those mechanisms that may significantly limit the predictive power of the resulting Hamiltonians