Molecular Classification of Vicia faba L Genotypes by Using RAPD-PCR Markers

The research included the molecular classification study of seven genotypes of the bean Vicia faba L. (FBSPN2, TLD1266, TLD1814, TLB1266, Luzdeotono, favad and Histal. Using the RAPD technique for DNA, as 13 random primers were used, the products of inflation were transferred within the agarose gel, and the results of the study showed the possibility of separating the genotypes from each other and determining the degree of genetic variation between them, as the primers used produced (1002) packages of them (417 normal bundles and (585) mixed bundles. The genetic differences of the studied genotypes were determined to be distinguished by the number of bundles, as they reached (28) bundles, including (13) unique bundles and (15) absent bundles. The ILB1266 genotype showed the highest number of unique bundles, which It reached 4 bundles, while the cultivar Favad showed the absence of unique bundles in it, either bundles are absent. The genotypes (ILD1266, IILB1266, Luzdeotono) were distinguished for having the highest number, which amounted to (3) bundles. As for the FBSPN2 genotype, it did not have any absent bundle, and the primers varied. Of the resulting bundle sizes, their sizes ranged between bp (1925-130), and the highest value for the genetic dimension ranged between (0.110 - 0.269), as the lowest genetic dimension was between the two structures (FBSPN2 and ILD1266), which amounted to 0.110, and the highest value for the genetic dimension was (0.2 69) between the genotypes (ILD1266, HISTAL) (ILD1266, Luzdeotono) The Dendrogram shows the separation of the studied genotypes into two main groups, and each of them into two subgroups.

The Development of Topological 4-manifold Theory

The development of topological 4-manifold theory is described in the form of a flowchart showing the interdependence among many key statements in the theory. In particular, the flowchart demonstrates how the theory crucially relies on the constructions in this book, what goes into the work of Quinn on smoothing, normal bundles, and transversality, and what is needed to deduce the famous consequences, such as the classification of closed, simply connected, topological 4-manifolds, the category preserving Poincaré conjecture, and the existence of exotic smooth structures on 4-dimensional Euclidean space. Precise statements of the results, brief indications of some proofs, and extensive references are provided.

Quotients of double vector bundles and multigraded bundles

<p style='text-indent:20px;'>We study quotients of multi-graded bundles, including double vector bundles. Among other things, we show that any such quotient fits into a tower of affine bundles. Applications of the theory include a construction of normal bundles for weighted submanifolds, as well as for pairs of submanifolds with clean intersection.</p>

Fields of geometric objects associated with compiled hyperplane-distribution in affine space

A compiled hyperplane distribution is considered in an n-dimensional projective space . We will briefly call it a -distribution. Note that the plane L(A) is the distribution characteristic obtained by displacement in the center belonging to the L-subbundle. The following results were obtained: a) The existence theorem is proved: -distribution exists with arbitrary (3n – 5) functions of n arguments. b) A focal manifold is constructed in the normal plane of the 1st kind of L-subbundle. It was obtained by shifting the cen­ter A along the curves belonging to the L-distribution. A focal manifold is also given, which is an analog of the Koenigs plane for the distribution pair (L, L). c) It is shown that a framed -distribution in the 1st kind normal field of H-distribution induces tangent and normal bundles. d) Six connection theorems induced by a framed -distri­bu­tion in these bundles are proved. In each of the bundles , the framed -distribution induces an intrin­sic torsion-free affine connection in the tangent bundle and a centro-affine connection in the corresponding normal bundle. e) In each of the bundles (d) in the differential neighborhood of the 2nd order, the covers of 2-forms of curvature and curvature tensors of the corresponding connections are constructed.

Some Characterizations of Semi-Invariant Submanifolds of Golden Riemannian Manifolds

In this paper, we study some characterizations for any submanifold of a golden Riemannian manifold to be semi-invariant in terms of canonical structures on the submanifold, induced by the golden structure of the ambient manifold. Besides, we determine forms of the distributions involved in the characterizations of a semi-invariant submanifold on both its tangent and normal bundles.

Plurisubharmonic functions on a neighborhood of a torus leaf of a certain class of foliations

Let C be a smooth elliptic curve embedded in a smooth complex surface X such that C is a leaf of a suitable holomorphic foliation of X. We investigate the complex analytic properties of a neighborhood of C under some assumptions on the complex dynamical properties of the holonomy function. As an application, we give an example of {(C,X)} in which the line bundle {[C]} is formally flat along C, however it does not admit a {C^{\infty}} Hermitian metric with semi-positive curvature. We also exhibit a family of embeddings of a fixed elliptic curve for which the positivity of normal bundles does not behave in a simple way.