Let $M$ be a simple $3$-manifold with $F$ a boundary component of genus at least two. Let $\alpha$ and $\beta$ be separating slopes on $F$. It is shown that if both $2$-handle attachings $M[\alpha ]$ and $M[\beta ]$ are toroidal and one of them contains an essential torus whose intersection with $M$ is a thrice punctured essential torus, then $\Delta (\alpha ,\beta )\le 8$.

We show that each of the pseudo-Riemannian symmetric $R$-space associated with a non-degenerate Jordan triple system and a real form of an effective simple irreducible pseudo-Hermitian symmetric space can be realized as a para-real form of an effective semisimple para-Hermitian symmetric space of hyperbolic orbit type.

We consider perturbed half-linear ordinary differential equations near $\mathrm{\infty }$. We clarify asymptotic forms of the nontrivial solutions as $t\to \infty$ under $p$-th power integrability conditions imposed on the perturbations. Generalized Riccati equations associated with the half-linear equations under consideration are employed to prove the main results.

A bridge position of a knot is said to be perturbed if there exists a cancelling pair of bridge disks. Motivated by the examples of knots admitting unperturbed, strongly irreducible, non-minimal bridge positions due to Jang-Kobayashi-Ozawa-Takao, we derive examples of unperturbed, weakly reducible, non-minimal bridge positions. Also, a bridge version of Gordon’s Conjecture is proposed: the connected sum of unperturbed bridge positions is unperturbed.

We study the uniqueness problem of an entire function $f$ when it shares two small functions with its derivative $f^{(k)}(k\ge 1)$. This confirms the conjecture posed by Li and Yang [5].

We consider the problem of minimal volume vector fields on a given Riemann surface, specialising on the case of $M^{*}$, that is, the arbitrary radius $2$-sphere with two antipodal points removed. We discuss the homology theory of the unit tangent bundle $(T^1{M}^{*},\partial T^1{M}^{*})$ in relation with calibrations and a certain minimal volume equation. A particular family $X_{\text{m},k}$, $k\in \mathbb{N}$, of minimal vector fields on $M^{*}$ is found in an original fashion. The family has unbounded volume, ${\text{lim}}_k\text{vol}(X_{\text{m},k_{|\Omega }})=+\infty$, on any given open subset $\Omega$ of $M^{*}$ and indeed satisfies the necessary differential equation for minimality. Another vector field $X_{\ell }$ is discovered on a region ${\Omega }_1\subset {\mathbb{S}}^2$, with volume smaller than any other known optimal vector field restricted to ${\Omega }_1$.

Any three hypergeometric series whose respective parameters, $a$, $b$ and $c$, differ by integers satisfy a linear relation with coefficients that are rational functions of $a$, $b$, $c$ and the variable $x$. These relations are called three-term relations. This paper gives explicit formulas describing symmetries of the coefficients of three-term relations.