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...

‎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...

The LHCb pentaquarks -- the $P_c(4312)$, $P_c(4440)$ and $P_c(4457)$ have been theorized to be $\Sigma_c \bar{D}$ \bar{D}^*$ S-wave molecules. Here we explore possibility that two of these contain in addition a $\Lambda_c(2595) component P-wave. We will analyze effects this extra channel within effective field theories: first one standard contact-range theory second include non-diagonal pion dy...

Abstract In this study, we investigate the , $\bar{D}\Xi^\prime_c$?> $\bar{D}\Sigma_c^*$?> $\bar{D}\Xi_c^*$?> $\bar{D}^{*}\Sigma_c$?> $\bar{D}^{*}\Xi^\prime_c$?> $\bar{D}^{*}\Sigma_c^*$?> and $\bar{D}^{*}\Xi_c^*$?> pentaquark molecular states with without strangeness via QCD sum rules in detail, focusing on light flavor, $SU(3)$?> breaking effects, make predictions f...

Abstract In this work, we employ the light-cone QCD sum rule to calculate magnetic dipole moments of $$P_c(4440)$$ P c ( 4440 ) , $$P_c(4457)$$ 4457 and $$P_{cs}(4459)$$ c...

The effects of the $\Sigma_c\bar{D}^*$-$\Lambda_{c}(2595)\bar{D}$ coupled-channel dynamics and various one-boson-exchange (OBE) forces for LHCb pentaquark states, $P_c(4440)$ $P_c(4457)$, are reinvestigated. Both pion $\rho$-meson exchanges considered dynamics. It is found that role $\Lambda_{c}(2595)\bar{D}$ channel in descriptions $P_c(4457)$ states not significant with OBE parameters constra...

The effects of the $\Sigma_c\bar{D}^*$-$\Lambda_{c}(2595)\bar{D}$ coupled-channel dynamics and various one-boson-exchange (OBE) forces for LHCb pentaquark states, $P_c(4440)$ $P_c(4457)$, are reinvestigated. Both pion $\rho$-meson exchanges considered dynamics. It is found that role $\Lambda_{c}(2595)\bar{D}$ channel in descriptions $P_c(4457)$ states not significant with OBE parameters constra...