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...

The mass spectrum of hidden charm pentaquark states composed two diquarks and an antiquark are calculated by use effective Hamiltonian which includes explicitly the spin, color, flavor dependent interactions. results show that $$P_c(4312)^+$$ $$P_c(4440)^+$$ could be explained as with isospin spin-parity $$IJ^P=1/2\left( 3/2^-\right) $$ , $$P_c(4457)^+$$ state a 5/2^-\right) $$P_{cs}(4459)^+$$ ...