This paper gives a complete classification of conics in \(PE_2(\mathbb{R})\). The classification has been made earlier (Reveruk 5), but it showed to be incomplete and not possible to cite and use in ...further studies of properties of conics, pencil of conics, and of quadratic forms in pseudo-Euclidean spaces. This paper provides that. A pseudo-orthogonal matrix, pseudo-Euclidean values of a matrix, diagonalization of a matrix in a pseudo-Euclidean way are introduced. Conics are divided in families and by types, giving both of them geometrical meaning. The invariants of a conic with respect to the group of motions in \(PE_2(\mathbb{R})\) are determined, making it possible to determine a conic without reducing its equation to canonical form. An overview table is given.
U radu dajemo pregled nekih svojstava potpunog četverovrha ABCD u euklidskoj ravnini. Proučavamo kružnice s promjerima AB, AC, AD, BC, BD, CD, kao i nožišne trokute i nožišne kružnice točaka A, B, C, ...D s obzirom na trokute BCD, ACD, ABD, ABC redom navedene. Svi prikazani rezultati su poznati iz literature, ali ih ovdje dokazujemo koristeći istu metodu.
The objective of this study was to develop a successful freezing protocol for cryopreservation of Danube barbel sperm, and to identify the presence of different spermatozoa subpopulations. By testing ...different concentrations of different cryoprotectants, we determined that the use of 5% dimethyl sulfoxide (DMSO) yielded the highest total motility of ~25%. Cooling rates influenced by frame height and cooling time in liquid nitrogen vapor showed that a frame height of 3 cm and cooling time of 2 min yielded the highest post-thaw motility. Supplementation of cryomedia with 0.1 M of sugars led to an increase in the total post-thaw motility by ~50%, while protein supplementation lowered post-thaw motility. Motile spermatozoa hierarchically clustered according to their motility parameters, displaying a four-subpopulation (SP1-SP4) structure. SP1 was defined by low values of velocity but high overall linearity; SP2 was comprised of fast non-linear spermatozoa, that had high velocity values but low linearity; SP3 was characterized by fast linear spermatozoa, and SP4 by slow non-linear spermatozoa. Protocols developed in this study will lead to the creation of new and enhanced conservation strategies for this species.
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