Narrow Results

This paper deals with the variety of commutative non associative algebras satisfying the identity , γ ∈ K. In [3] it is proved that if γ = 0, 1 then any finitely generated algebra is nilpotent. Here we generalize this result by proving that if γ ≠ -1, then any such algebra is locally nilpotent. Our results require characteristic ≠ 2, 3.

Using purely syntactical arguments, it is shown that every nontrivial pseudovariety of monoids contained in DO whose corresponding variety of languages is closed under unambiguous product, for instance DA, is local in the sense of Tilson.

In this paper, we continue to study the differential inverse power series ring $R[[x^{-1};\delta ]]$, where $R$ is a ring equipped with a derivation $\delta$. We characterize when $R[[x^{-1};\delta ]]$ is a local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, right stable range one, abelian, projective-free, $I$-ring, respectively. Furthermore, we prove that $R[[x^{-1};\delta ]]$ is a domain satisfying the $ACC$ on principal left ideals if and only if so does $R$. Also, for a piecewise prime ring (PWP) $R$ we determine a large class of the differential inverse power series ring $R[[x^{-1};\delta ]]$ which have a generalized triangular matrix representation for which the diagonal rings are prime. In particular, it is proved that, under suitable conditions, if $R$ has a (flat) projective socle, then so does $R[[x^{-1};\delta ]]$. Our results extend and unify many existing results.

In the present note, we continue the study of skew inverse Laurent series ring $R((x^{-1};\alpha ,\delta ))$ and skew inverse power series ring $R[[x^{-1};\alpha ,\delta ]]$, where $R$ is a ring equipped with an automorphism $\alpha$ and an $\alpha$-derivation $\delta$. Necessary and sufficient conditions are obtained for $R[[x^{-1};\alpha ,\delta ]]$ to satisfy a certain ring property which is among being local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, projective-free and $I$-ring, respectively. It is shown here that $R((x^{-1};\alpha ,\delta ))$ (respectively $R[[x^{-1};\alpha ,\delta ]]$) is a domain satisfying the ascending chain condition (Acc) on principal left (respectively right) ideals if and only if so does $R$. Also, we investigate the problem when a skew inverse Laurent series ring $R((x^{-1};\alpha ,\delta ))$ has the same Goldie rank as the ring $R$ and is proved that, if $R$ is a semiprime right Goldie ring, then $R((x^{-1};\alpha ,\delta ))$ is semiprimitive. Furthermore, we study on the relationship between the simplicity, semiprimeness, quasi-Baerness and Baerness property of a ring $R$ and these of the skew inverse Laurent series ring. Finally, we consider the problem of determining when $f(x)\in R((x^{-1};\alpha ,\delta ))$ is nilpotent.

2-Edge connectivity is an important fault tolerance property of a network because it maintains network communication despite the deletion of a single arbitrary edge. Planar spanning subgraphs have been shown to play a significant role for achieving local decentralized routing in wireless networks. Existing algorithmic constructions of spanning planar subgraphs of unit disk graphs (UDGs) such as Minimum Spanning Tree, Gabriel Graph, Nearest Neighborhood Graph, etc. do not always ensure connectivity of the resulting graph under single edge deletion. Furthermore, adding edges to the network so as to improve its edge connectivity not only may create edge crossings (at points which are not vertices) but it may also require edges of unbounded length. Thus we are faced with the problem of constructing 2-edge connected geometric planar spanning graphs by adding edges of bounded length without creating edge crossings (at points which are not vertices). To overcome this difficulty, in this paper we address the problem of augmenting the edge set (i.e., adding new edges) of planar geometric graphs with straight line edges of bounded length so that the resulting graph is planar and 2-edge connected. We provide bounds on the number of newly added straight-line edges, prove that such edges can be of length at most 3 times the max length of an edge of the original graph, and also show that the factor 3 is optimal. It is shown to be NP-Complete to augment a geometric planar graph to a 2-edge connected geometric planar graph with the minimum number of new edges of a given bounded length. We also provide a constant time algorithm that works in location-aware settings to augment a planar graph into a 2-edge connected planar graph with straight-line edges of length bounded by 3 times the longest edge of the original graph. It turns out that knowledge of vertex coordinates is crucial to our construction and in fact we prove that this problem cannot be solved locally if the vertices do not know their coordinates. Moreover, we provide a family of k-connected UDGs which does not have 2-edge connected spanning planar subgraphs, for any .