In this paper we introduce a new type of orthogonality for real normed planes which coincides with usual orthogonality in the Euclidean situation. With the help of this type of orthogonality we derive several characterizations of the Euclidean plane among all normed planes, all of them yielding also characteristic properties of inner product spaces among real normed linear spaces of dimensions $d\geq 3$.
The most recent representative of the semi-aquatic insect family Chresmodidae is described from the Lebanese Cenomanian marine lithographic limestone. Its highly specialized legs, with a high number of tarsomeres, never observed in other orders of insects, were probably adapted for water surface skating. We hypothesize the occurrence of a unique, extraordinary "antenna" mutation affecting the distal part of the legs of the Chresmodidae, maybe homeotic or affecting some genes that participate in the leg development and segmentation. The Chresmodidae had a serrate ovipositor adapted to endophytic egg laying in floating or aquatic plants. They were probably predaceous on nektonic small animals. As the Chresmodidae and the aquatic water skaters of the bug families Veliidae and Gerridae were contemporaneous during at least the Lower Cretaceous, these insects probably did not cause the extinction of this curious group.
Pseudo $\star $-autonomous lattices are non-commutative generalizations of $\star $-autonomous lattices. It is proved that the class of pseudo $\star $-autonomous lattices is a variety of algebras which is term equivalent to the class of dualizing residuated lattices. It is shown that the kernels of congruences of pseudo $\star $-autonomous lattices can be described as their normal ideals.
Angiosarcoma is a soft tissue tumour with a dismal prognosis. We present a 74 year old male presenting with a non healing ulcer on the scalp. On histopathology a diagnosis of angiosarcoma was made. An early diagnosis and tumour size play a pivotal role in the survival of the patient., Deepal J Deshpande, Chitra S Nayak, Sunil N Mishra, and Literatura 6
We prove a non-archimedean Dugundji extension theorem for the spaces $C^{\ast }(X,\mathbb {K})$ of continuous bounded functions on an ultranormal space $X$ with values in a non-archimedean non-trivially valued complete field $\mathbb {K}$. Assuming that $\mathbb {K}$ is discretely valued and $Y$ is a closed subspace of $X$ we show that there exists an isometric linear extender $T\colon C^{\ast }(Y,\mathbb {K})\rightarrow C^{\ast }(X,\mathbb {K})$ if $X$ is collectionwise normal or $Y$ is Lindelöf or $\mathbb {K}$ is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace $Y$ of an ultraregular space $X$ is a retract of $X$.