A square matrix is said to be circular bidiagonal whenever (i) each nonzero entry on the diagonal, or subdiagonal, in top-right corner; (ii) subdiagonal nonzero, and corner nonzero. Let F denote a field, let V finite-dimensional vector space over F. We consider an ordered pair of F-linear maps A:V→V A⁎:V→V that satisfy following two conditions: there exists basis for with respect which represen...

Abstract We present the concept of cotorsion pairs cut along subcategories an abelian category. This provides a generalization complete pairs, and represents general framework to find approximations restricted certain subcategories. also exhibit some connections between Auslander–Buchweitz approximation theory, by considering relative analogs for Frobenius contexts. Several applications are giv...

abstract to identify the species, understanding the characteristics of the species, explore the possibility of speciation, mutations in the study population, and many other purposes. in this method for studying colchesin bath, karyotype use for immature worms. after reviewing the slide karyotype of polychaeta (nereis sp.) and observation and analysis of  50  plaque metaphase plate chromosome nu...