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Register a new userOn the use of group theory to generalize elements of pairwise comparisons matrix: a cautionary note
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This paper examines the constricted use of group theory in the studies of pairwise comparisons. The presented approach is based on the application of the famous Levi Theorems of 1942 and 1943 for orderable groups. The theoretical foundation for multiplicative (ratio) pairwise comparisons has been provided. Counterexamples have been provided to support the theory. In our opinion, the scientific community must be made aware of the limitations of using the group theory in pairwise comparisons. Groups, which are not torsion free, cannot be used for ratios by Levis theorems.
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This note is concerned with potentially misleading concepts in the treatment of cosmological magnetic fields by magnetohydrodynamical (MHD) modelling. It is not a criticism of MHD itself but rather a cautionary comment on the validity of its use in cosmology. Now that cosmological data are greatly improved compared with a few decades ago, and even better data are imminent, it makes sense to revisit original modelling assumptions and examine critically their shortcomings in respect of modern science. Specifically this article argues that ideal MHD is a poor approximation around recombination, since it inherently restricts evolutionary timescales, and is often misapplied in the existing literature.
Spherically symmetric solutions of generic gravitational models are optimally, and legitimately, obtained by expressing the action in terms of the two surviving metric components. This shortcut is not to be overdone, however: a one-function ansatz invalidates it, as illustrated by the incorrect solutions of [1].
This mathematical recreation extends the analysis of a recent paper, asking when a traveller at a bus stop and not knowing the time of the next bus is best advised to wait or to start walking toward the destination. A detailed analysis and solution is provided for a very general class of probability distributions of bus arrival time, and the solution characterised in terms of a function analogous to the hazard rate in reliability theory. The note also considers the question of intermediate stops. It is found that the optimal strategy is not always the laziest, even when headways are not excessively long. For the common special case where one knows the (uniform) headway but not the exact timetable, it is shown that one should wait if the headway is less than the walking time (less bus travel time), and walk if the headway is more than twice this much. In between it may be better to wait or to walk, depending on ones confidence in being able to catch up to a passing bus.
No abstract given; compares pairs of languages from World Atlas of Language Structures.
About 160 years ago, the Italian mathematician Fa`a di Bruno published two notes dealing about the now eponymous formula giving the derivative of any order of a composition of two functions. We reproduce here the two original notes, Fa`a di Bruno (1855, 1857), written respectively in Italian and in French, and propose a translation in English.
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