We show that a Pettis integrable function from a closed interval to a Banach space is Henstock-Kurzweil integrable. This result can be considered as a continuous version of the celebrated Orlicz-Pettis theorem concerning series in Banach spaces.
For lower-semicontinuous and convex stochastic processes Zn and nonnegative random variables ϵn we investigate the pertaining random sets A(Zn,ϵn) of all ϵn-approximating minimizers of Zn. It is shown that, if the finite dimensional distributions of the Zn converge to some Z and if the ϵn converge in probability to some constant c, then the A(Zn,ϵn) converge in distribution to A(Z,c) in the hyperspace of Vietoris. As a simple corollary we obtain an extension of several argmin-theorems in the literature. In particular, in contrast to these argmin-theorems we do not require that the limit process has a unique minimizing point. In the non-unique case the limit-distribution is replaced by a Choquet-capacity.
Any given increasing [0,1]2→[0,1] function is completely determined by its contour lines. In this paper we show how each individual uninorm property can be translated into a property of contour lines. In particular, we describe commutativity in terms of orthosymmetry and we link associativity to the portation law and the exchange principle. Contrapositivity and rotation invariance are used to characterize uninorms that have a continuous contour line.
The chromosome complements of thirteen species of the planthopper family Dictyopharidae are described and illustrated. For each species, the structure of testes and, on occasion, ovaries is additionally outlined in terms of the number of seminal follicles and ovarioles. The data presented cover the tribes Nersiini, Scoloptini and Dictyopharini of the subfamily Dictyopharinae and the tribes Ranissini, Almanini, and Orgeriini of the Orgeriinae. The data on the tribes Nersiini and Orgeriini are provided for the first time. Males of Hyalodictyon taurinum and Trimedia cf. viridata (Nersiini) have 2n = 26 + X; Scolops viridis, S. sulcipes, and S. abnormis (Scoloptini) 2n = 36 + X; Callodictya krueperi (Dictyopharini) 2n = 26 + X; Ranissus edirneus and Schizorgerius scytha (Ranissini) 2n = 26 + X. Males of Almana longipes and Bursinia cf. genei (Almanini) have 2n = 26 + X and 2n = 24 + XY, respectively. The latter chromosome complement was not recorded previously for the tribe Almanini. Males of Orgerius ventosus and Deserta cf. bipunctata (Orgeriini) have 2n = 26 + X. The testes of males of A. longipes and B. cf. genei each have 4 seminal follicles, which is characteristic of the tribe Almanini. Males of all other species have 6 follicles per testis. When the ovaries of a species were also studied, the number of ovarioles was coincident with that of seminal follicles. For comparison, Capocles podlipaevi (2n = 24 + X and 6 follicles per testis in males) from the Fulgoridae, the sister family to Dictyopharidae, was also studied. We supplemented all the data obtained with our earlier observations on Dictyopharidae. The chromosomal complement of 2n = 28 + X or that of 2n = 26 + X and 6 follicles per testis are suggested to be the ancestral condition among Dictyopharidae, from which taxa with various chromosome numbers and testes each with 4 follicles have differentiated.