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Vol.21 No.2
Abstract
Keskin and Harmanci defined the family <TEX>$\mathcal{B}(M,X)=\{A{\leq}M{\mid}{\exists}Y{\leq}x,{\exists}f{\in}Hom_R(M,X/Y),Ker\;f/A{\ll}M/A\}$</TEX>. And Orhan and Keskin generalized projective modules via the class <TEX>$\mathcal{B}$</TEX>(M,X). In this note we introduce X-local summands and X-hollow modules via the class <TEX>$\mathcal{B}$</TEX>(M,X). Let R be a right perfect ring and let M be an X-lifting module. We prove that if every co-closed submodule of any projective module contains its radical, then M has an indecomposable decomposition. This result is a generalization of Kuratomi and Chang's result [9, Theorem 3.4]. Let X be an R-module. We also prove that for an X-hollow module H such that every non-zero direct summand K of H with <TEX>$K{\in}\mathcal{B}(H,X)$</TEX>, if <TEX>$H{\oplus}H$</TEX> has the internal exchange property, then H has a local endomorphism ring.
Abstract
The notion of intersection-soft ideal of CI-algebras is introduced, and related properties are investigated. A characterization of an intersection-soft ideal is provided, and a new intersection-soft ideal from the old one is established.
Abstract
In this paper, we first show that for any space X, there is a <TEX>${\sigma}$</TEX>-complete Boolean subalgebra of <TEX>$\mathcal{R}$</TEX>(X) and that the subspace {<TEX>${\alpha}{\mid}{\alpha}$</TEX> is a fixed <TEX>${\sigma}Z(X)^{\sharp}$</TEX>-ultrafilter} of the Stone-space <TEX>$S(Z({\Lambda}_X)^{\sharp})$</TEX> is the minimal basically disconnected cover of X. Using this, we will show that for any countably locally weakly Lindel<TEX>$\ddot{o}$</TEX>f space X, the set {<TEX>$M{\mid}M$</TEX> is a <TEX>${\sigma}$</TEX>-complete Boolean subalgebra of <TEX>$\mathcal{R}$</TEX>(X) containing <TEX>$Z(X)^{\sharp}$</TEX> and <TEX>$s_M^{-1}(X)$</TEX> is basically disconnected}, when partially ordered by inclusion, becomes a complete lattice.
Abstract
The Boolean rank of a nonzero m <TEX>$m{\times}n$</TEX> Boolean matrix A is the least integer k such that there are an <TEX>$m{\times}k$</TEX> Boolean matrix B and a <TEX>$k{\times}n$</TEX> Boolean matrix C with A = BC. In 1984, Beasley and Pullman showed that a linear operator preserves the Boolean rank of any Boolean matrix if and only if it preserves Boolean ranks 1 and 2. In this paper, we extend this characterization of linear operators that preserve the Boolean ranks of Boolean matrices. We show that a linear operator preserves all Boolean ranks if and only if it preserves two consecutive Boolean ranks if and only if it strongly preserves a Boolean rank k with <TEX>$1{\leq}k{\leq}min\{m,n\}$</TEX>.
Abstract
In this paper, we present a detailed comparison of the performance of the numerical solvers such as the biconjugate gradient stabilized, operator splitting, and multigrid methods for solving the two-dimensional Black-Scholes equation. The equation is discretized by the finite difference method. The computational results demonstrate that the operator splitting method is fastest among these solvers with the same level of accuracy.
Abstract
In the paper, by directly verifying an inequality which gives a lower bound for the first order modified Bessel function of the first kind, the authors supply a new proof for the complete monotonicity of a difference between the exponential function <TEX>$e^{1/t}$</TEX> and the trigamma function <TEX>${\psi}^{\prime}(t)$</TEX> on (0, <TEX>${\infty}$</TEX>).