If (M,∇) is a manifold with a symmetric linear connection, then T*M can be endowed with the natural Riemann extension g¯ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to g¯g¯ initiated by C. L.Bejan and O.Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure PP on (T*M, g¯) and prove that P is harmonic (in the sense of E.Garciá-Río, L.Vanhecke and M. E.Vázquez-Abal (1997)) if and only if g¯ reduces to the classical Riemann extension introduced by E.M. Patterson and A.G. Walker (1952)., Cornelia-Livia Bejan, Şemsi Eken., and Obsahuje bibliografii
Cairaella henrii gen. n., sp. n. (Proteocephalidea: Proteocephalinae) is described from the intestine of Norops trachyderma (Cope) (Polychrotidae) from San Pablo de Kantesiya in Ecuador. The new genus differs from the 12 other known genera of the Proteocephalinae (and all other proteocephalidean genera) by the presence of a dense network of osmoregulatory canals situated in the cortex and by the morphology of the scolex which is flattened dorsoventrally, with elongated deeply embedded suckers possessing a well-developed circular musculature situated in the anterolateral region, and by eggs with a three-layered embryophore possessing small outgrowths on its external surface. Ophiotaenia nicoleae sp. n. is described from the intestine of Thecadactylus rapicauda (Houttuyn) (Gekkonidae) from San Pablo de Kantesiya in Ecuador. This new species is characterised by the testes arranged in two fields, numbering 142-204, the cirrus-sac length representing 21-33% of proglottis width, the genital pore situated in the middle of the proglottis or slightly anteriorly, and the ovary width representing 68-88% of proglottis width. It differs from 20 of 27 Ophiotaenia species parasitic in New World reptiles by the presence of an apical organ and from the remaining species by one to several other morphological characters, such as the number of testes, diameter and shape of the scolex, position of ventral and dorsal osmoregulatory canals, or the presence of a vaginal sphincter. Both taxa represent the first record of proteocephalidean tapeworms in polychrotid and gekkonid lizards, respectively.
In this paper, necessary and sufficient conditions for equality in the inequalities of Oppenheim and Schur for positive semidefinite matrices are investigated.