On the extended T-system of type C3C_3

We continue the study of extended T-systems of quantum affine algebras. We find a sub-system of the extended T-system of the quantum affine algebra $U_q \hat{\mathfrak{g}}$ of type $C_3$. The sub-system consisting of four systems which are denoted by I, II, III, and IV. Each of the systems I, II, III, IV is closed. The systems I-IV can be used to compute minimal affinizations with weights of the form $\lambda_1 \omega_1 + \lambda_2 \omega_2 + \lambda_3 \omega_3$, where at least one of $\lambda_1$, $\lambda_2$, $\lambda_3$ are zero. Using the systems I-IV, we compute the characters of the restrictions of the minimal affinizations in the systems to $U_q \mathfrak{g}$ and obtain some conjectural decomposition formulas for the restrictions of some minimal affinizations.Comment: arXiv admin note: substantial text overlap with arXiv:1208.482

Exploring a ΣcDˉ\Sigma_{c}\bar{D} state: with focus on Pc(4312)+P_{c}(4312)^{+}

Stimulated by the new discovery of $P_{c}(4312)^{+}$ by LHCb Collaboration, we endeavor to perform the study of $P_{c}(4312)^{+}$ as a $\Sigma_{c}\bar{D}$ state in the framework of QCD sum rules. Taking into account the results from two sum rules, a conservative mass range 4.07\sim4.97~\mbox{GeV} is presented for the $\Sigma_{c}\bar{D}$ hadronic system, which agrees with the experimental data of $P_{c}(4312)^{+}$ and could support its interpretation as a $\Sigma_{c}\bar{D}$ state.Comment: 9 pages, 6 figures. arXiv admin note: text overlap with arXiv:1801.0872

Efficient schemes on solving fractional integro-differential equations

Fractional integro-differential equation (FIDE) emerges in various modelling of physical phenomena. In most cases, finding the exact analytical solution for FIDE is difficult or not possible. Hence, the methods producing highly accurate numerical solution in efficient ways are often sought after. This research has designed some methods to find the approximate solution of FIDE. The analytical expression of Genocchi polynomial operational matrix for left-sided and right-sided Caputo’s derivative and kernel matrix has been derived. Linear independence of Genocchi polynomials has been proved by deriving the expression for Genocchi polynomial Gram determinant. Genocchi polynomial method with collocation has been introduced and applied in solving both linear and system of linear FIDE. The numerical results of solving linear FIDE by Genocchi polynomial are compared with certain existing methods. The analytical expression of Bernoulli polynomial operational matrix of right-sided Caputo’s fractional derivative and the Bernoulli expansion coefficient for a two-variable function is derived. Linear FIDE with mixed left and right-sided Caputo’s derivative is first considered and solved by applying the Bernoulli polynomial with spectral-tau method. Numerical results obtained show that the method proposed achieves very high accuracy. The upper bounds for th

Extended TT-System of Type G2G_2

We prove a family of 3-term relations in the Grothendieck ring of the category of finite-dimensional modules over the affine quantum algebra of type $G_2$ extending the celebrated $T$-system relations of type $G_2$. We show that these relations can be used to compute classes of certain irreducible modules, including classes of all minimal affinizations of type $G_2$. We use this result to obtain explicit formulas for dimensions of all participating modules