In this paper, affine and projective graphs are obtained from affine and projective planes of order p r by accepting a line as a path. Some properties of these affine and projective graphs are investigated. Moreover, a definition of distance is given in the affine and projective planes of order p r and, with the help of this distance definition, the point or points having the most advantageous (central) position in the corresponding graphs are determined, with some examples being given. In addition, the concepts of a circle, ellipse, hyperbola, and parabola, which are well known for the Euclidean plane, are carried over to these finite planes. Finally, the roles of finite affine and projective Klingenberg planes in all the results obtained are considered and their equivalences in graph applications are discussed.

Despite there have been many experiments conducted about antioxidants, the best sole or combination use of antioxidants to include as a standard ingredient to freezing extenders yet to be found. This study conducted to investigate the different doses of methionine (2.5 and 5mM), cysteine (1 and 2mM) and butylated hydroxytoluene (BHT) (1 and 2mM) for ram semen cryopreservation on post-thaw and postincubation (6 h) time points over spermatological parameters. Semen samples collected from Kivircik rams via electro-ejaculator in breeding season. After essential spermatological evaluations, appropriate samples were pooled then split into 7 equal aliquots to create study groups (antioxidant free control, 2.5mM methionine, 5mM methionine, 1mM cysteine, 2mM cysteine, 1mM BHT and 2mM BHT). Semen samples put into French straws (0.25mL) and freezing procedure (two-step) conducted via a programmable gamete freezer. At both time points; motility, HOST, PSA-FITC and TUNEL assays were conducted to discover the impacts of cryopreservation and incubation process over sperm cells.Antioxidant supplemented groups yielded better results comparing to control groups in terms of various spermatological parameters not only at post-thaw time point but after incubation for 6 h of time. The study demonstrated that supplementing sperm freezing extenders with abovesaid antioxidants may create new approaches to cryopreservation procedures and through increasing success rate of freezing, fertility results may increase to better results in near future.

In this paper, we deal with a new class of Moufang–Klingenberg planes and study with the concepts of 4-transitivity and 6-figures on the class of planes. So, we reach some algebraic and geometric results related on the concepts. For instance, we show that the group of collineations of these planes acts transitively on 4-gons. Finally, we obtain some properties of 6-figures.

This paper deals with a certain class of projective Klingenberg planes over the local ring F[η]/<η^{m}> with F an arbitrary field, known as the plural algebra of order m. In particular addition and multiplication of points on a line is defined geometrically and interpreted algebraically, by using the coordinate ring.

In this article, we deal with a class of projective Klingenberg planes constructed over a plural algebra of order m. Thanks to this, the incidence matrices for some special cases of the class are obtained. Next, the number of collineations of the certain classes are found. Besides, an example of a collineation for these classes are given. Finally, we achieve to carry the obtained results to more general case.