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## Vol. 51,2022

### No. 2

- HUANG, Ti-Ren; CHEN, Lu; CHU, Yu-Ming;
*Asymptotically sharp bounds for the complete $p$-elliptic integral of the first kind.*- Hokkaido Mathematical Journal, 51 (2022) pp.189-210

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In the article, we present several monotonicity and convexity properties involving the complete $p$-elliptic integral $\mathcal{K}_{p}(r)$ of the first kind, and provide several asymptotically sharp bounds for $\mathcal{K}_{p}(r)$.

Keywords | complete elliptic integral; complete $p$-elliptic integral; monotonicity; convexity; |
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msc2020(primary) | 33E05; 33C05; 26A48; 26D07.; |

*Invasion Speed of Predator in a Lattice Dynamical System.*

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This article studies the invasion dynamics of a lattice dynamical system with predator-prey type nonlinearity, which models that the predator invades the habitat of the prey. The system does not generate monotone semiflows and can not be studied by the well known conclusions. By constructing proper auxiliary equations, we obtain the rough invasion speed of the predator that initially occupies a habitat of finite size.

Keywords | upper-lower solutions; asymptotic spreading; nonmonotone system; |
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msc2020(primary) | 35C07; 35K57; 37C65; 92D25; |

*Morse theoretic aspects of Plücker embeddings.*

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Plücker embeddings of real and complex Grassmannians ${Gr_p({\boldsymbol F}^n)}$ (${\boldsymbol F} = {\boldsymbol R}$, ${\boldsymbol C}$) and manifolds associated to Plücker embeddings are analyzed from the viewpoint of critical point theory. It is proved that the Plücker embedding of ${Gr_p({\boldsymbol F}^n)}$ with $1\le p\le n-p$ is taut if and only if $1\le p\le 2$, in contrast to standard embeddings which are known to be taut for all $1\le p\le n$. As an application, we show a convexity property of Plücker embeddings, which is an analogue to Schur-Horn convexity on Hermitian matrices.

Keywords | Grassmannian; Plücker embedding; Taut embedding; Convexity; perfect Morse function; |
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msc2020(primary) | 51M35; |

msc2020(secondary) | 58E05; 14M15; 15A18; |

*Geometric realization and its variants.*

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In this paper, we compare different realization functors for simplicial objects in a topologically enriched model category, generalizing tom Dieck's theorem on homotopy type of classifying spaces.

Keywords | Geometric realization; classifying space; topologically enriched category; |
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msc2020(primary) | 18N50; 55R35; |

msc2020(secondary) | 55R37; 18N40; |

*The ring of modular forms of $\mathrm{O}(2,4;\mathbb{Z})$ with characters.*

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We show that the ring of modular forms with characters for $\mathrm{O}(2,4;\mathbb{Z})$ is generated by forms of weights 4, 4, 6, 8, 10, 10, 12, and 30 with three relations of weights 8, 20, and 60. The proof is based on the study of a moduli space of K3 surfaces.

Keywords | modular form; K3 surface; |
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msc2020(primary) | 14J28; 14J15; 11F55; |

*On the problem of convexity for the restricted three-body problem around the heavy primary.*

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In this note, we prove that below the first critical energy level, a proper combination of the Ligon-Schaaf and Levi-Civita regularization mappings provides a convex symplectic embedding of the double cover of the energy surfaces of the planar\linebreak rotating Kepler problem into ${\mathbb R}^{4}$ endowed with its standard symplectic structure. This convex embedding extends to the bounded component of the planar circular restricted three-body problem around the heavy body outside a small neighborhood of the collisions. This opens up new approaches to attack the Birkhoff conjecture about the existence of a global surface of section in the restricted planar circular three-body problem using holomorphic curve techniques.

Keywords | Global surfaces of section; convexity; restricted three-body problem; contact and symplectic topology; |
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msc2020(primary) | 53D05; 53D10; 70F05; 70F07; |

*Metric Growth of the Discrete $q$-Laplacian.*

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The metric growth of Laplacian $\Delta u$ in the theory of potentials on Riemannian manifolds by [13] means the growth of Green potential of $(\Delta u)^2$. This notion was introduced to obtain a Riesz representation for a biharmonic functions. On a hyperbolic network, we investigated a similar problem in [7], by taking the discrete Laplacian $\Delta$ and the (discrete) Green function. Let $q$ be a non-negative function such that $q \not\equiv 0$. Then $- \Delta + q$ is a Schrödinger operator. Let us put $\Delta _q := \Delta - q$ and call it $q$-Laplacian. The $q$-Green function $g_a$ of the network with pole at $a$ always exists. In this paper, we estimate the metric growth of $\Delta _q u$ by $q$-Green potential of $|\Delta _q u|$ and $|\Delta _q u|^2$. As an application, we prove Riesz representation theorem for $q$-biharmonic functions.

Keywords | discrete potential theory; Schrödinger operator; discrete $q$-Laplacian; $q$-Green function; metric growth of $q$-Laplacian; Riesz's decomposition; $q$-biharmonic function; |
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msc2020(primary) | 31C20; |

msc2020(secondary) | 31C25; |