The solid torus core recognition problem is the problem that, given a knot in the solid tours, decides whether the knot is the core of the solid torus. This problem is in $\mathbf{\text{NP}}$ since the thickened torus recognition problem is in $\mathbf{\text{NP}}$. We give an alternate proof of the fact and prove that the problem is in $\mathbf{\text{co-NP}}$. It is also proved that the Hopf link recognition problem is in $\mathbf{\text{NP}}$ and $\mathbf{\text{co-NP}}$ as a corollary to our result.

All decompositions of $M_3\left(\mathrm{ℂ}\right)$ into a direct vector-space sum of two subalgebras such that none of the subalgebras contains the identity matrix are classified. Thus, the classification of all decompositions of $M_3\left(\mathrm{ℂ}\right)$ into a direct vector-space sum of two subalgebras as well as description of Rota–Baxter operators of nonzero weight on $M_3\left(\mathrm{ℂ}\right)$ is finished.

Extensions of the closed-form expressions for the series represented as a product of sine and logarithmic functions mentioned in reference [7] have been reformulated utilizing the Hurwitz zeta function and its first derivative. Furthermore, analogous closed-form expressions have been derived for these series that involve the cosine function instead.

We study one of multidimensional inverse scattering problems for quantum systems in a constant electric field, by utilization of the Enss-Weder time-dependent method. The main purpose of this paper is to propose some methods of sharpening key estimates in the analysis, which are much simpler than those in the previous works. Our methods give an appropriate class of short-range potentials which can be determined uniquely by scattering operators, that seems natural in terms of direct scattering problems.

For any positive integer $n$, let $C_n$ be the cyclic group of order $n$. We determine all possible values of the integer group determinant of ${\mathrm{C}}_4\times {\mathrm{C}}_2^2$, which is the only unsolved abelian group of order 16.

We compute the Artin $L$-function of a diagonal hypersurface $D_{\lambda }$ over a finite field associated to a character of a finite group acting on $D_{\lambda }$, and under some condition, express it in terms of hypergeometric functions and Jacobi sums over the finite field. As an application, we derive certain relations among hypergeometric functions over different finite fields from the Grothendieck-Lefschetz trace formula.