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## Vol. 52,2023

### No. 3

- NAKAMURA, Fumihiko;
*Accessibility and stabilization by infinite horizon optimal control with negative discounting.*- Hokkaido Mathematical Journal, 52 (2023) pp.353-379

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The present paper investigates systems exhibiting two attractors, and we discuss the problem of steering the state from one attractor to the other attractor by our idea of associating with the stabilization problem an optimal control problem. We first formulate the steering problem and give partial answers for the problem in a two-dimensional case by using the ordinary differential equation based on the infinite horizon optimal control model with negative discounts. Furthermore, under some conditions, we verify that the phase space can be separated into some openly connected components depending on the asymptotic behavior of the orbit starting from initial points in their components. This classification of initial points suggests that the system enables robust stabilizable control. Moreover, we illustrate some numerical results for the control obtained by applying our focused system for the Bonhoeffer-Van der Pol model.

Keywords | Accessibility; stabilization; infinite horizon; negative discounting; |
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msc2020(primary) | 49J15; |

msc2020(secondary) | 34D05; |

- HAN, Huhe;
*Maximum and minimum of support functions.*- Hokkaido Mathematical Journal, 52 (2023) pp.381-399

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For given continuous functions $\gamma_{{}_{i}}: S^{n}\to \mathbb{R}_{+}$ ($i=1, 2$), the functions $\gamma_{{}_{\max}}$ and $\gamma_{{}_{\min}}$ can be defined naturally. In this paper, by applying the spherical method, we first show that the Wulff shape associated to $\gamma_{{}_{\max}}$ is the convex hull of the union of Wulff shapes associated to $\gamma_{{}_1}$ and $\gamma_{{}_2}$, if $\gamma_{{}_1}$ and $\gamma_{{}_2}$ are convex integrands. Next, we show that the Wulff shape associated to $\gamma_{{}_{\min}}$ is the intersection of Wulff shapes associated to $\gamma_{{}_1}$ and $\gamma_{{}_2}$. Moreover, relationships between their dual Wulff shapes are given.

Keywords | support function; convex integrand; maximum; minimum; Wulff shape; |
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msc2020(primary) | 52A20; 52A55; 82D25; |

- URAKAWA, Hajime;
*Harmonic maps and biharmonic maps for double fibrations of compact Lie groups.*- Hokkaido Mathematical Journal, 52 (2023) pp.401-425

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This work is motivated by the works of W.Y. Hsiang and H.B. Lawson [7], Pages $12$ and $13$. In this paper, we deal with the following double fibration: \[ \xymatrix@R-0.5cm @C-0.5cm{ & (G,g) \ar[ld]_{\pi_1} \ar[rd]^{\pi_2} & \\ (G/H,h_1) && (K\backslash G,h_2) } \] We will show that every $K$-invariant minimal or biharmonic hypersurface $M$ in $(G/H,h_1)$ induces an $H$-invariant minimal or biharmonic hypersurface $\widetilde{M}$ in $(K\backslash G,h_2)$ by means of $\widetilde{M}:=\pi_2(\pi_1{}^{-1}(M))$ (cf. Theorems 3.2 and 4.1). We give a one to one correspondence between the class of all the $K$-invariant minimal or biharmonic hypersurfaces in $G/H$ and the one of all the $H$-invariant minimal or biharmonic hypersurfaces in $K\backslash G$ (cf. Theorem 4.2).

Keywords | harmonic map; biharmonic map; homogeneous space; triplet; Lie group; |
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msc2020(primary) | 58E10; |

msc2020(secondary) | 53C42; |

- WANG, Gefei;
*The artin braid group actions on the set of spin structures on a surface.*- Hokkaido Mathematical Journal, 52 (2023) pp.427-462

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We study the action of the Artin braid group $B_{2g+2}$ on the set of spin structures on a hyperelliptic curve of genus $g$, which reduces to that of the symmetric group $S_{2g+2}$. It has been already described in terms of the classical theory of Riemann surfaces. In this paper, we compute the $S_{2g+2}$-orbits of the spin structures of genus $g$ and the isotropy group $\mathfrak{G}_i$ of each orbit in a purely combinatorial way.

Keywords | spin structure; Dehn twist; $\mathfrak{S}_{2g+2}$ action; Artin braid group; |
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msc2020(primary) | 20F36; 57K20; 14H30; |

- CHIBA, Jun; MATSUMOTO, Keiji;
*An analogy of Jacobi's formula and its applications.*- Hokkaido Mathematical Journal, 52 (2023) pp.463-494

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We give an analogy of Jacobi's formula, which relates the hypergeometric function with parameters $(1/4,1/4,1)$ and theta constants. By using this analogy and twice formulas of theta constants, we obtain a transformation formula for this hypergeometric function. As its application, we express the limit of a pair of sequences defined by a mean iteration by this hypergeometric function.

Keywords | Hypergeometric function; Theta constants; Mean iteration; |
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msc2020(primary) | 33C05; |

msc2020(secondary) | 14K25; 33C90; |

- ASHIDA, Sohei;
*Upper bounds of local electronic densities in molecules.*- Hokkaido Mathematical Journal, 52 (2023) pp.495-516

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The eigenfunctions of electronic Hamiltonians determine the stable structures and dynamics of molecules through the local distributions of their densities. In this paper an a priori upper bound for such local distributions of the densities is given. The bound means that concentration of electrons is prohibited due to the repulsion between the electrons. A relation between one-electron and two-electron densities resulting from the antisymmetry of the eigenfunctions plays a crucial role in the proof.

Keywords | electronic density; eigenfunction; molecule; |
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msc2020(primary) | 81V55; |

msc2020(secondary) | 35B45; |

- NAYAK, Raj Kumar; BHUNIA, Pintu; PAUL, Kallol;
*Improvements of $A$-numerical radius bounds.*- Hokkaido Mathematical Journal, 52 (2023) pp.517-544

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We obtain upper and lower bounds for the $A$-numerical radius inequalities of operators and operator matrices which generalize and improve on the existing ones. We present new upper bounds for the $A$-numerical radius of the product of two operators. We also develop various inequalities for the $A$-numerical radius of $2 \times 2 $ operator matrices.

Keywords | $A$-numerical radius; $A$-operator seminorm; $A$-adjoint operator; Positive operator; Inequality; |
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msc2020(primary) | 47A12; |

msc2020(secondary) | 47A30; |