We consider the Wiener sausage up to time $t$ associated with a closed ball. A formula for the expected volume of the Wiener sausage is obtained in odd dimensions. In these cases, we also find that the formula leads to the asymptotic expansion for large $t$ and each coefficient is represented by zeros of a modified Bessel function of the second kind. Moreover we obtain a formula for the expected surface area of the Wiener sausage.

We give simple proofs of Laradji's theorem on blocks with central defect groups, Watanabe's theorem on the Glauberman--Watanabe correspondences of blocks and Robinson's theorem on defect groups of $p$-blocks of $p$-solvable groups attaining Brauer's upper bound for the number of irreducible characters.

We show that for the closure of a classical braid which satisfies certain conditions, the spun $T^{2}$-knot of the classical knot has the unknotting number one. This gives an alternative proof of the fact that the spun $T^{2}$-knot of a classical torus knot has the unknotting number one.

In the paper [12], Yang conjectured that a nonelementary subgroup $G$ of $\mathrm{SL}(2, \mathbb{C})$ containing elliptic elements is discrete if for each elliptic element $g \in G$ the group $\langle f, g \rangle$ is discrete, where $f \in \mathrm{SL}(2,\mathbb{C})$ is a test map being loxodromic or elliptic. By embedding $\mathrm{SL}(2,\mathbb{C})$ into $\mathrm{U}(1,1; \mathbb{H})$, we give an affirmative answer to this question. As an application, we show that a nonelementary and nondiscrete subgroup of $\mathrm{Isom}(H^{3})$ must contain an elliptic element of order at least 3.

Let $\Gamma$ be a chart with at most two crossings. In this paper, we show that if $\Gamma$ is a 2-minimal generalized $n$-chart with $n \ge 5$, then $\Gamma$ contains at least $4n-10$ black vertices. And we show that if the closure of the surface braid represented by $\Gamma$ is a disjoint union of spheres, then $\Gamma$ is a ribbon chart. Hence the closure is a ribbon surface.

We give a necessary and sufficient condition for the existence of an $n$-end catenoid of genus one with prescribed flux. By using the condition, we construct new examples of families whose flux data go near to that of ``the catenoid of genus one''.

In this paper it is proved that for any irrational $\alpha$ and some $0 < \theta \le 1.5/100$, there are infinitely many primes $p$ such that $p+2$ has at most two prime factors and $\lVert\alpha p+\beta\rVert < p^{-\theta}$ which improves K. Matomäki's result $\theta < 1/1000$.

In this article, we show that some semi-rigid $\mu$-stable sheaves on a projective K3 surface $X$ with Picard number $1$ are stable under Bridgeland's stability condition. As a consequence of our work, we show that the special set $U(X) \subset \mathrm{Stab}(X)$ introduced by Bridgeland reconstructs $X$ itself. This gives a sharp contrast to the case of an abelian surface.

In [4], J. Eells and L. Lemaire introduced $k$-energy and $k$-harmonic maps. In 1989, S.B. Wang [17] showed the first variation formula of the $k$-energy. In this paper, we give the second variation formula of $k$-energy and a notion of weakly stable and unstable. We also study $k$-harmonic maps into product Riemannian manifolds and $k$-harmonic curves into Riemannian manifolds with constant sectional curvature. Moreover, we give some non-trivial solutions of $3$-harmonic curves.

In this note we study lower bounds of energy growth for solutions to wave equations which are compact in space perturbations of the wave equation $\partial_{t}^{2}u - \Delta u = 0$. Assuming that there exists a null bicharacteristic $(x(t),\xi(t))$, parametrized by the time $t$, such that $x(t)$ remains inside a ball and $\xi(t)$ outside a ball for $t \geq 0$ we prove that the solution operator $R(t)$ is bounded from below by constant times $\sqrt{\lvert\xi(t)\rvert/\lvert\xi(0)\rvert}$ in the operator norm. We apply this result to examples constructed by the same idea as in Colombini and Rauch [1] and show that there exist compact in space perturbations which cause $\exp(ct^{\alpha})$ growth of the energy for any given $0\leq \alpha \leq 1$.

Let $(M, g, J)$ be a compact Hermitian manifold and $\Omega$ the fundamental 2-form of $(g, J)$. A Hermitian manifold $(M, g, J)$ is said to be locally conformal Kähler if there exists a closed 1-form $\omega$ such that $d\Omega=\omega \wedge \Omega$. The purpose of this paper is to investigate a relation between a locally conformal Kähler structure and the adapted differential operator on compact solvmanifolds.

We study finite time quenching for the radial solutions of a system of heat equations coupled at the boundary condition. This system exhibits simultaneous and non-simultaneous quenching. In particular, three kinds of simultaneous quenching profiles are obtained for different nonlinear exponent regions and appropriate initial data.