PHENOTYPIC CONSOLIDATION DAUGHTERS OF BULL-SIRES ESTIMATED BY THE METHOD OF LINEAR CLASSIFICA-TION

The aim of our research was to determine the degree of phenotypic consolidation coefficients of bull-sires daughters in the Sumy intra-breed type of Ukrainian Black-and-White dairy breed, estimated by linear traits that characterize the conformation type of cows. The experimental base of research was information on the linear classification of first-calf cows of Sumy intra-breed type of Ukrainian Black-and-White dairy breed in the PE "Burynske" breeding farm in Pidlisnivskyi department of Sumy district. The estimation of the first-calf cows by type was carried out using the method of linear classification by two systems: 9-point, with a linear description of 18 conformation traits and 100-point including four complexes of breeding traits that characterize: dairy type, body, limbs and udder. The coefficients of phenotypic consolidation (K1 and K2) cows of selection groups for linear traits of the conformation were determined according to formulas proposed by Yu. P. Polupan (2005). The level of phenotypic consolidation daughters of bull-sires estimated by group conformation traits indicated that regardless of the origin, phenotypic consolidation coefficients by overall type assessment had a positive value with a wide variability of coefficient K1 from 0.056 to 0.512 and coefficient K2 – from 0.065 to 0.505. Among all estimated by phenotypic consolidation bulls, less consolidated by group linear traits were found sires of Ukrainian Black-and-White dairy breed – Motuzok and Front. Among sires of Holstein breed the best by group traits and overall assessment of the type were: V. Delight (K1 = 0.242-0.334; K2 = 0.247-0.349); D. Capris (K1 = 0.085-0.341; K2 = 0.089-0.348) and Hayes (K1 = 0.071-0.375; K2 = 0.082-0.370). From the sires of Ukrainian Black-and-White dairy breed the Iceberg was the best (К1 = 0.059-0.478; К2 = 0.047-0.472). Magnitude of the degree phenotypic consolidation of descriptive traits of linear estimation daughters of bull-sires, regardless of origin, differed significantly by higher variability in comparison with group traits. Coefficients of phenotypic consolidation calculated by the formula K1, varied according to the bull-sires assessment from -0.434 (teats length of Iceberg daughters) to 0.504 (rump angle of Milliam daughters). The higher variability of phenotypic consolidation coefficients was revealed according to the formula K2, whose level varied from -0.571 (udder depth of Lyubimy's daughters) to 0.546 (angularity of Milliam daughters). By the coefficients of phenotypic consolidation of descriptive traits K1 and K2, there wasn't found no one bull-sire only with positive values. The most consolidated for overwhelming majority of descriptive traits were the bulls-sires of Holstein breed – Lauer, Toprate and Milliam. The highest consolidation of descriptive traits had all daughters of bulls by angularity with the highest number of positive values of coefficients. The presence of a genetic influence of bull-sires on the degree of phenotypic consolidation of the majority of linear traits of daughters with a significant advantage of Holstein producers was established.

THE SECOND FUNDAMENTAL THEOREM OF INVARIANT THEORY FOR THE ORTHOSYMPLECTIC SUPERGROUP

The first fundamental theorem of invariant theory for the orthosymplectic supergroup scheme $\text{OSp}(m|2n)$ states that there is a full functor from the Brauer category with parameter $m-2n$ to the category of tensor representations of $\text{OSp}(m|2n)$ . This has recently been proved using algebraic supergeometry to relate the problem to the invariant theory of the general linear supergroup. In this work, we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. Specifically, we give a linear description of the kernel of the surjective homomorphism from the Brauer algebra to endomorphisms of tensor space, which commute with the orthosymplectic supergroup. The main result has a clear and succinct formulation in terms of Brauer diagrams. Our proof includes, as special cases, new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues, which are independent of the Capelli identities. The results of this paper have led to the result that the map from the Brauer algebra ${\mathcal{B}}_{r}(m-2n)$ to endomorphisms of $V^{\otimes r}$ is an isomorphism if and only if $r<(m+1)(n+1)$ .

The st-bond polytope on series-parallel graphs

The st-bond polytope of a graph is the convex hull of the incidence vectors of its st-bonds, where an st-bond is a minimal st-cut. In this paper, we provide a linear description of the st-bond polytope on series-parallel graphs. We also show that the st-bond polytope is the intersection of the st-cut dominant and the bond polytope.

Methodology of udder description and the effect on somatic cell count in Czech White Shorthaired goat breed

This study evaluated the effect of the linearly described shape traits of goat udders on somatic cell count. In a herd of 487 white shorthaired goats, seven traits (udder symmetry, udder depth, udder width, teat length, teat placement, rear udder attachment and udder cleft) were assessed in relation to somatic cell count in milk. The average somatic cell count was 1.3 mill cells/ml when considering the environmental effects (month and year of performance testing, lactation number. The somatic cell count is influenced by the depth (p = 0.0015) and width (p = 0.0268) of the udder. The results demonstrate that some traits of the udder shape influence the somatic cell count and can be considered as functional traits indicating animal health and herd profitability. After further studies, the methodology for linear description of the udder could be used for other dairy goat breeds, not only in the Czech Republic..

Evaluation of Stallions Based on Linear Description of Their Daughters

The purpose of our work was an objective evaluation of mares of the Czech warmblood horse based on the linear description, as well as the evaluation of the benefits of stallion breeding based on the linear description of their daughters and – for specific stallions – the evaluation of certain individual exterior traits which are passed on by stallions to their offspring. Stallion horses with at least 7 descendants were used for the evaluation and determination of the values, mares which underwent a linear description of traits at the age of 3 years. For this evaluation we used available data from the year 1996 to 2012, a total including 251 stallions and 4709 mares and more than 500 000 records related to the linear description. The data were gathered from the database of the Central Register of Horse Breeding at Slatińany in the Czech Republic. These data were manually compiled using Excel 2007 and then processed and evaluated according to the objectives of the present study using the linear model GLM as well as the statistical programme Scheffe. The results of the study showed a convincing statistical influence of the stallions on all the monitored exterior traits analyzed on the mares for the father‑factor, and after evaluating all the general exterior traits the statistical difference among the stallions was seen as convincing. We found out that in most cases the breed had no convincing statistical influence on the analyzed traits of the linear description. On the contrary, in terms of the other effects (father and year of measurement) we found a convincing statistical influence on all traits of the linear description. For some stallions we evaluated particular traits of linear description, which they pass on to their female offspring using charts and graphics. Afterwards we compared reciprocally certain stallions according to the traits of the linear description.