We investigate the local integrability and linearizability of three dimensional Lotka-Volterra equations at the origin. Necessary and sufficient conditions for both integrability and linearizability are obtained for (1,−1, 1), (2,−1, 1) and (1,−2, 1)-resonance. To prove sufficiency, we mainly use the method of Darboux with extensions for inverse Jacobi multipliers, and the linearizability of a ...

 Letbe a ring with an endomorphism and an -derivationAntoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced nil-Armendariz rings. In this paper we introduce and investigate the notion of nil--compatible rings. The class of nil--compatible rings are extended through various ring extensions and many classes of nil--compatible rings are constructed. We al...

Abstract: In this paper, we consider the skew circulant and skew left circulant matrices with the Fibonacci and Lucas numbers. Firstly, we discuss the invertibility of the skew circulant matrix and present the determinant and the inverse matrix by constructing the transformation matrices. Furthermore, the invertibility of the skew left circulant matrices are also discussed. We obtain the determ...

Let D be a Noetherian domain containing a field, a ∈ D a nonzero nonunit and z an indeterminate over D. We prove that the generic fiber of the extension D[[z]] →֒ D[1/a][[z]] has dimension ≥ dim(D/aD). 2000 Mathematics Subject Classification: 13F25, 13J10.

let $k$ be a field of characteristic$p>0$, $k[[x]]$, the ring of formal power series over $ k$,$k((x))$, the quotient field of $ k[[x]]$, and $ k(x)$ the fieldof rational functions over $k$. we shall give somecharacterizations of an algebraic function $fin k((x))$ over $k$.let $l$ be a field of characteristic zero. the power series $finl[[x]]$ is called differentially algebraic, if it satisfies...

In this paper, we prove that the Nakayama automorphism of a graded skew PBW extension over finitely presented Koszul Auslander-regular algebra has trivial homological determinant. For A = ?(R)<x1, x2> connected R, compute its P-determinant and inverse ?. particular case quasi-commutative extensions Artin-Schelter regular algebras, show explicitly connection between ring coefficients exten...