PubMed · 13963664
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J Z HEARON. 1963-05-10. Theorems on linear systems.. https://doi.org/10.1111/j.1749-6632.1963.tb13364.x
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Many contemporary problems in science involve making predictions based on partial observation of extremely complicated spatially extended systems with many degrees of freedom and with physical instabilities on both large and small scale. Various new ensemble filtering strategies have been developed recently for these applications, and new mathematical issues arise. Because ensembles are extremely expensive to generate, one such issue is whether it is possible under appropriate circumstances to take long time steps in an explicit difference scheme and violate the classical Courant-Friedrichs-Lewy (CFL)-stability condition yet obtain stable accurate filtering by using the observations. These issues are explored here both through elementary mathematical theory, which provides simple guidelines, and the detailed study of a prototype model. The prototype model involves an unstable finite difference scheme for a convection-diffusion equation, and it is demonstrated below that appropriate observations can result in stable accurate filtering of this strongly unstable spatially extended system.
Traditional wisdom and empirical observation unite in recommending a 7 year unit for measuring human life - including individual and institutional science. But, because of astronomy and the decimal system, things tend to be measured either in years, five years or in decades. A year is too short while a decade is too long to measure the trends and transitions of individual or institutional life. And the half decade, such as the 'five year plan' beloved by politicians and bureaucrats seems too short. Therefore, seven years should become the standard unit for tracking trends and measuring attainment. Precedents for using a seven year unit include the notorious Jesuit saying: 'Give me the child until he is seven, and I will show you the man'; and the 'ninth commandment' of Leo Szilard: 'Do your work for six years; but in the seventh, go into solitude or among strangers, so that the memory of your friends does not prevent you from being what you have become'. In a scientific career, seven years is approximately the time spent at high school, the time taken for a traditional basic scientific training of first degree and doctorate, and the period after the doctorate building the knowledge to become an expert specialist. There seems to be enough anecdotal evidence to support the idea that we should reconsider the universal but un-reflective use of decimal units in planning and evaluation. For instance, seven year fellowships and program grants might replace the current five year versions. A new - and previously unconsidered - field of research beckons.
Assessment of the perturbed matrix method (PMM) ability in reproducing valence UV absorption spectra is carried out on two model systems: 1,2,3-triazine in methanol solution and uracil in water solution. Results show that even using the simplest definition of the quantum center, i.e. the portion of the system explicitly treated quantum mechanically, PMM provides rather good results. This paper further confirms the possibility of using PMM as a theoretical-computational tool, complementary to other methodologies, for addressing the electronic properties in molecular systems of high complexity.