We give a different proof of a theorem of O. Loos [5] which characterizes maximal tori of extrinsically symmetric spaces. On the way we show some facts on certain symmetric subspaces, so called meridians, which previously have been known only using classification.

In the previous paper, we classified $n$-noids of genus one into two classes, and considered one of the classes. As a sequel, we give a necessary and sufficient condition for the existence of an $n$-noid of genus one with prescribed flux in the other class. By using the condition, we give obstructions for a certain type of flux and arrangement of the ends. We also give new examples by deforming a quadruple covering of Berglund--Rossman's $n$-end catenoid.

Sections 7 and 8 of ``The Dorfmeister--Neher theorem on isoparametric hypersurfaces'', (Osaka J. Math. 46, 695--715) are the heart of the paper, but a lack of clear argument causes some questions, although the statement is true. The purpose of the present paper is to make it clear.

In this paper, we give a new approach to find a dually flat Finsler metric. As its application, we produce many new spherically symmetric dually flat Finsler metrics by using known projective spherically symmetric Finsler metrics.

Consider the problems \begin{equation*} \left\{ \begin{array}{@{}ll@{}} -\Delta_{p}u=f\ \text{in}\ \Omega{,} & u=0\ \text{on}\ \partial \Omega,\\ -\Delta_{p}v=g\ \text{in}\ \Omega{,} & v=0\ \text{on}\ \partial \Omega, \end{array} \right. \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial \Omega$, $\Delta_{p}z=\mathrm{div}(\lvert\nabla z\rvert^{p-2}\nabla z)$, $p>1$. We prove a strong comparison principle that allows $f-g$ to change sign. An application to singular asymptotically $p$-linear boundary problems is given.

Pseudodiagrams are knot or link diagrams where some of the crossing information is missing. Pseudoknots are equivalence classes of pseudodiagrams, where equivalence is generated by a natural set of Reidemeister moves. In this paper, we introduce a Gauss-diagrammatic theory for pseudoknots which gives rise to the notion of a virtual pseudoknot. We provide new, easily computable isotopy and homotopy invariants for classical and virtual pseudodiagrams. We also give tables of unknotting numbers for homotopically trivial pseudoknots and homotopy classes of homotopically nontrivial pseudoknots. Since pseudoknots are closely related to singular knots, this work also has implications for the classification of classical and virtual singular knots.

The gonality sequence $(d_{r})_{r \ge 1}$ of a curve of genus $g$ encodes, for $r<g$, important information about the divisor theory of the curve. Mostly it is very difficult to compute this sequence. In general it grows rather modestly (made precise below) but for curves with special moduli some ``unexpected jumps'' may occur in it. We first determine all integers $g>0$ such that there is no such jump, for all curves of genus $g$. Secondly, we compute the leading numbers (up to $r=19$) in the gonality sequence of an extremal space curve, i.e. of a space curve of maximal geometric genus w.r.t. its degree.

In this paper, we study the uniqueness of the direct decomposition of a toric manifold. We first observe that the direct decomposition of a toric manifold as algebraic varieties is unique up to order of the factors. An algebraically indecomposable toric manifold happens to decompose as smooth manifold and no criterion is known for two toric manifolds to be diffeomorphic, so the unique decomposition problem for toric manifolds as smooth manifolds is highly nontrivial and nothing seems known for the problem so far. We prove that this problem is affirmative if the complex dimension of each factor in the decomposition is less than or equal to two. A similar argument shows that the direct decomposition of a smooth manifold into copies of $\mathbb{C}P^{1}$ and simply connected closed smooth 4-manifolds with smooth actions of $(S^{1})^{2}$ is unique up to order of the factors.

R.E. Caflish and J.H. Maddocks analyzed the dynamics of a planar slender elastic rod. We consider a thin elastic rod $\gamma$ in an $N$-dimensional riemannian manifold. The former model represents an elastic rod with positive thickness, and the equation becomes a semilinear wave equation. Our model represents an infinitely thin elastic rod, and the equation becomes a 1-dimensional semilinear plate equation. We prove the short time existence of solutions. We also discuss the behaviour of the solution when the resistance goes to infinity, and find that the solution converges to a solution of a gradient flow equation.

The paper will characterise the separability of the forgetful functor from the category of comodules over a group coring to the category of modules over a suitable algebra. Then the applications of our results to group entwined modules, Doi--Hopf group modules and relative Hopf group modules are considered.

We obtain a simple presentation of the hyperelliptic mapping class group $\mathcal{M}^{h}(N)$ of a nonorientable surface $N$. As an application we compute the first homology group of $\mathcal{M}^{h}(N)$ with coefficients in $H_{1}(N; \mathbb{Z})$.

Let $a$, $b$, $c$ be distinct positive integers. Set $M = a+b+c$ and $N = abc$. We give an explicit description of the Mordell--Weil group of the elliptic curve $E_{(M, N)}\colon y^{2}-Mxy-Ny = x^{3}$ over $\mathbb{Q}$. In particular we determine the torsion subgroup of $E_{(M, N)}(\mathbb{Q})$ and show that its rank is positive. Furthermore there are infinitely many positive integers $M$ that can be written in $n$ different ways, $n\in\{2, 3\}$, as the sum of three distinct positive integers with the same product $N$ and $E_{(M, N)}(\mathbb{Q})$ has rank at least $n$.

For any group $G$ the involutions $\mathcal{I}$ in $G$ form a $G$-set under conjugation. The corresponding $kG$-permutation module $k\mathcal{I}$ is known as the involution module of $G$, with $k$ an algebraically closed field of characteristic two. In this paper we discuss the involution module of the projective special unitary group $\mathrm{PSU}_{3}(4^{f})$.

We consider spherically symmetric motions of a polytropic gas under the self-gravitation governed by the Euler--Poisson equations. The adiabatic exponent ($=$ the ratio of the specific heats) $\gamma$ is assumed to satisfy $6/5 < \gamma \leq 2$. Then there are equilibria touching the vacuum with finite radii, and the linearized equation around one of the equilibria has time-periodic solutions. To justify the linearization, we should construct true solutions for which this time-periodic solution plus the equilibrium is the first approximation. We solve this problem by the Nash--Moser theorem. The result will realize the so-called physical vacuum boundary. But the present study restricts $\gamma$ to the case in which $\gamma/(\gamma-1)$ is an integer. Other cases are reserved to the future as an open problem. The time-local existence of smooth solutions to the Cauchy problems is also discussed.