Natural operators on frame bundles

summary:Let $F_1$ be a natural bundle of order $r_1$; a basis of the $s$-th order differential operators of $F_1$ with values in $r_2$-th order bundles is an operator $D$ of that type such that any other one is obtained by composing $D$ with a suitable zero-order operator. In this article a basis is found in the following two cases: for $F_1=\text{semi} F^{r_1}$ (semi-holonomic $r_1$-th order frame bundle), $s=0$, $r_2<r_1$ and $F_1=F^1$ ($1$-st order frame bundle), $r_2\le s$. The author uses here the so-called method of orbit reduction which provides one with a criterion for checking a basis in terms of the $K^{r_1+s,r_2}_n$-action on the type fiber of the concerned bundle, where $K^{r_1+s,r_2}_n$ denotes the kernel of the projection of the $(n,r_1+s)$ jet group onto the $(n,r_2)$ jet group [see I. Kolár, P. Michor and J. Slovák, `Natural operations in differential geometry' (Springer-Verlag, Berlin) (1993; Zbl 0782.53013) or D. Krupka, Loc

Grouping fuzzy sets by similarity

a b s t r a c t The paper presents results on factorization of systems of fuzzy sets. The factorization consists in grouping those fuzzy sets which are pairwise similar at least to a prescribed degree a. An obstacle to such factorization, well known in fuzzy set theory, is the fact that &apos;&apos;being similar at least to degree a&quot; is not an equivalence relation because, in general, it is not transitive. As a result, ordinary factorization using equivalence classes cannot be used. This obstacle can be overcome by considering maximal blocks of fuzzy sets which are pairwise similar at least to degree a. We show that one can introduce a natural complete lattice structure on the set of all such maximal blocks and study this lattice. This lattice plays the role of a factor structure for the original system of fuzzy sets. Particular examples of our approach include factorization of fuzzy concept lattices and factorization of residuated lattices

Investigation of the tribological performance of ionic liquids in non-conformal EHL contacts under electric field activation

Real-time external alteration of the internal properties of lubricants is highly desirable in all mechanical systems. However, fabricating a suitable and effective smart lubricant is a long-lasting experimental process. In this study, the film thickness and frictional response of ionic liquid-lubricated non-conformal contacts to an electric field excitation under elastohydrodynamic conditions were examined. Film thickness was evaluated using a "ball-on-disc" optical tribometer with an electric circuit. Friction tests were carried on a mini traction machine (MTM) tribometer with a "ball-on-disc" rotation module and an electric circuit for contact area excitation. The results demonstrate that there is a difference in the behaviour of the ionic liquid during electric field excitation at the evaluated film thicknesses. The results of evaluated film thicknesses demonstrate that there is a difference in the behaviour of the ionic liquid during electric field excitation. Therefore, the ionic liquids could be a new basis for the smart lubrication of mechanical components. Moreover, the proposed experimental approach can be used to identify electrosensitive fluids

Carbonyl content determination in bio-oils with increased nitrogen content

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Anti-holonomic jets and the Lie bracket

summary:Second order anti-holonomic jets as anti-symmetric parts of second order semi-holonomic jets are introduced. The anti-holonomic nature of the Lie bracket is shown. A general result on universality of the Lie bracket is proved