let $r$ be an infinite ring. here we prove that if $0_r$ belongs to ${x_1x_2cdots x_n ;|; x_1,x_2,dots,x_nin x}$ for every infinite subset $x$ of $r$, then $r$ satisfies the polynomial identity $x^n=0$. also we prove that if $0_r$ belongs to ${x_1x_2cdots x_n-x_{n+1} ;|; x_1,x_2,dots,x_n,x_{n+1}in x}$ for every infinite subset $x$ of $r$, then $x^n=x$ for all $xin r$.

In generalized semi-infinite programming the feasible set is known to be not closed in general. In this paper, under natural and generic assumptions, the closure of the feasible set is described in explicit terms.

Let R be a ring with unity and U(R) its group of units. Let ∆U = {a ∈ U(R) | [U(R) : CU(R)(a)] < ∞} be the FC-radical of U(R) and let ∇(R) = {a ∈ R | [U(R) : CU(R)(a)] < ∞} be the FC-subring of R. An infinite subgroup H of U(R) is said to be an ω-subgroup if the left annihilator of each nonzero Lie commmutator [x, y] in R contains only finite number of elements of the form 1− h, where x, y ∈ R ...

At this point, there are more additions than errors to report... 1.1. Pythagoras’ Constant. A geometric irrationality proof of √ 2 appears in [1] and a curious recursion is studied in [2]. More references on radical denestings include [3, 4, 5, 6]. 1.2. The Golden Mean. The cubic irrational χ = 1.8392867552... is mentioned elsewhere in the literature with regard to iterative functions [7, 8, 9]...

One of our main results is a classification all the possible quivers of selfinjective radical cube zero finite dimensional algebras over an algebraically closed field having finite complexity. In the paper [5] we classified all weakly symmetric algebras with support varieties via Hochschild cohomology satisfying Dade’s Lemma. For a finite dimensional algebra to have such a theory of support var...

in this paper we consider the group algebra $r(c_2times‎ ‎d_infty)$‎. ‎it is shown that $r(c_2times d_infty)$ can be‎ ‎represented by a $4times 4$ block circulant matrix‎. ‎it is also‎ ‎shown that $mathcal{u}(mathbb{z}_2(c_2times d_infty))$ is‎ ‎infinitely generated‎.

‎linear semi-infinite programming problem is an important class of optimization problems which deals with infinite constraints‎. ‎in this paper‎, ‎to solve this problem‎, ‎we combine a discretization method and a neural network method‎. ‎by a simple discretization of the infinite constraints,we convert the linear semi-infinite programming problem into linear programming problem‎. ‎then‎, ‎we use...

While the classical account of the linear continuum takes it to be a totality of points, which are its ultimate parts, Aristotle conceives of it as continuous and infinitely divisible, without ultimate parts. A formal account of this conception can be given employing a theory of quantification for nonatomic domains and a theory of region-based topology.

In this comment, I extend Cavalcanti and Nosal’s (2010) framework to include the case of perfectly divisible money and unrestricted money holdings. I show that when trade takes place in Walrasian markets, counterfeits circulate and the Friedman rule is still optimal.

This study discusses some variants of Ordered WeightedAveraging (OWA) operators and related information aggregation methods. Indetail, we define the Extended Ordered Weighted Sum (EOWS) operator and theExtended Ordered Weighted Averaging (EOWA) operator, which are applied inscientometrics evaluation where the preference is over finitely manyrepresentative works. As...