let $mathcal {a}$ be an abelian category with enough projective objects and $mathcal {x}$ be a full subcategory of $mathcal {a}$. we define gorenstein projective objects with respect to $mathcal {x}$ and $mathcal{y}_{mathcal{x}}$, respectively, where $mathcal{y}_{mathcal{x}}$=${ yin ch(mathcal {a})| y$ is acyclic and $z_{n}yinmathcal{x}}$. we point out that under certain hypotheses, these two g...

In this paper we deepen, in the setting of coherence, some results obtained recent papers on notion p-entailment Adams and its relationship with conjoined iterated conditionals. We recall that conditionals are suitably defined framework conditional random quantities. Given a family $\mathcal{F}$ $n$ events $\{E_1|H_1,\ldots, E_n|H_n\}$ denote by $\mathscr{C}(\mathcal{F})=(E_1|H_1)\wedge \cdots ...

In this paper we define the reticulation of a residuated lattice, prove that it has “good properties“, present two constructions for it, prove its uniqueness up to an isomorphism, define the reticulation functor and give several examples of finite residuated lattices and their reticulations.

‎let $mathcal a$ and $mathcal b$ be unital rings‎, ‎and $mathcal m$‎ ‎be an $(mathcal a‎, ‎mathcal b)$-bimodule‎, ‎which is faithful as a‎ ‎left $mathcal a$-module and also as a right $mathcal b$-module‎. ‎let ${mathcal u}=mbox{rm tri}(mathcal a‎, ‎mathcal m‎, ‎mathcal‎ ‎b)$ be the triangular ring and ${mathcal z}({mathcal u})$ its‎ ‎center‎. ‎assume that $f:{mathcal u}rightarrow{mathcal u}$ is...

A compact metric space $(X, \rho)$ is given. Let $\mu$ be a Borel measure on $X$. By $r$-cluster we mean a measurable subset of $X$ with diameter at most $r$. A family of $k$ $2r$-clusters is called a $r$-cluster structure of order $k$ if any two clusters from the family are separated by a distance at least $r$. By measure of a cluster structure we mean a sum of clusters measures from the clust...

Previous computer-assisted problem-solving systems have incorporated all the problem-solving steps within a single stage, making it difficult to diagnose stages at which errors occurred when a student encounters difficulties, and imposing a too-high cognitive load on students in their problem solving. This study proposes a computer-assisted system named MathCAL, whose design is based on four pr...

We offer a new definition of $varepsilon$-orthogonality in normed spaces, and we try to explain some properties of which. Also we introduce some types of $varepsilon$-orthogonality in an arbitrary  $C^ast$-algebra $mathcal{A}$, as a Hilbert $C^ast$-module over itself, and investigate some of its properties in such spaces. We state some results relating range-kernel orthogonality in $C^*$-algebras.

A number of Arabidopsis (Arabidopsis thaliana) mutants exhibit leaf reticulation, having green veins that stand out against paler interveinal tissues, fewer cells in the interveinal mesophyll, and normal perivascular bundle sheath cells. Here, to examine the basis of leaf reticulation, we analyzed the Arabidopsis RETICULATA-RELATED (RER) gene family, several members of which cause leaf reticula...

Abstract Palu defined the index with respect to a cluster tilting object in suitable triangulated category, order better understand Caldero–Chapoton map that exhibits connection between algebras and representation theory. We push this further by proposing an contravariantly finite, rigid subcategory, we show behaves similarly classical index. Let ${\mathcal{C}}$ be skeletally small category spl...

Abstract For a braided fusion category $\mathcal{V}$, $\mathcal{V}$-fusion is $\mathcal{C}$ equipped with monoidal functor $\mathcal{F}:\mathcal{V} \to Z(\mathcal{C})$. Given fixed $(\mathcal{C}, \mathcal{F})$ and $G$-graded extension $\mathcal{C}\subseteq \mathcal{D}$ as an ordinary category, we characterize the enrichments $\widetilde{\mathcal{F}}:\mathcal{V} Z(\mathcal{D})$ of $\mathcal{D}$ ...