Staude's thread construction of ellipsoid is revisited from a new view-point concerning the length of geodesic segments. Thanks to the general nature of this view-point, one obtains similar thread construction on other stages, i.e., on “Liouville manifolds”.

In this paper, we deal with the inner boundary of random walk range, that is, the set of those points in a random walk range which have at least one neighbor site outside the range. If $L_n$ be the number of the inner boundary points of random walk range in the $n$ steps, we prove $\lim_{n\to \infty} ({L_n}/{n})$ exists with probability one. Also, we obtain some large deviation result for transient walk. We find that the expectation of the number of the inner boundary points of simple random walk on the two dimensional square lattice is of the same order as ${n}/{(\log n)^2}$.

For the first Painlevé equation we establish an orbifold polynomial Hamiltonian structure on the fibration of Okamoto's spaces and show that this geometric structure uniquely recovers the original Painlevé equation, thereby solving a problem posed by K. Takano.

Let $X$ be the complex Fermat variety of dimension $n=2d$ and degree $m > 2$. We investigate the submodule of the middle homology group of $X$ with integer coefficients generated by the classes of standard $d$-dimensional subspaces contained in $X$, and give an algebraic (or rather combinatorial) criterion for the primitivity of this submodule.

Constant mean curvature (CMC) surfaces in spheres are investigated under the extra condition of biharmonicity. From the work of Miyata, especially in the flat case, we give a complete description of such immersions and show that for any $h \in (0,1)$ there exist CMC proper-biharmonic planes and cylinders in ${\mathbb S}^5$ with $|H|=h$, while a necessary and sufficient condition on $h$ is found for the existence of CMC proper-biharmonic tori in ${\mathbb S}^5$.

Let $\Bbb K$ be an algebraically closed field of characteristic zero. We say that a polynomial automorphism $f: \Bbb K^n\to \Bbb K^n$ is special if the Jacobian of $f$ is equal to 1. We show that every $(n-1)$-dimensional component $H$ of the set ${\rm Fix}(f)$ of fixed points of a non-trivial special polynomial automorphism $f: \Bbb K^n\to\Bbb K^n$ is uniruled. Moreover, we show that if $f$ is non-special and $H$ is an $(n-1)$-dimensional component of the set ${\rm Fix}(f)$, then $H$ is smooth, irreducible and $H={\rm Fix}(f)$. Moreover, for $\Bbb K = \mathbb{C}$ if $f$ is non-special and ${\rm Jac}(f)$ has an infinite order in $\Bbb C^*$, then the Euler characteristic of $H$ is equal to 1.

Let $K$ be the result of a 1-fusion (band sum) of a knot $k$ and a distant trivial knot in $S^3$. From results of D. Gabai and of M. G. Scharlemann, we know that the genus of $K$ is at least that of $k$ and that equality holds if and only if the band sum is, in fact, a connected sum (in which case $K$ is ambient isotopic to $k$). In this paper, we consider a generalization of this result to an $m$-fusion of a link and a distant trivial link with $m$-components.

Given a link in $S^3$ we will use invariants derived from the Alexander module and the Blanchfield pairing to obtain lower bounds on the Gordian distance between links, the unlinking number and various splitting numbers. These lower bounds generalise results recently obtained by Kawauchi.

We give an application restricting the knot types which can arise from a sequence of splitting operations on a link. This allows us to answer a question asked by Colin Adams in 1996.

Starting from a model with a weakly compact cardinal, we construct a model in which the weak stationary reflection principle for $\omega_2$ holds but the Fodor-type reflection principle for $\omega_2$ fails. So the stationary reflection principle for $\omega_2$ fails in this model. We also construct a model in which the semi-stationary reflection principle holds but the Fodor-type reflection principle for $\omega_2$ fails.

Stokes phenomena with respect to parameters are investigated for the Gauss hypergeometric differential equation with a large parameter. For this purpose, the notion of the Voros coefficient is introduced for the equation. The explicit forms of the Voros coefficients are given as well as their Borel sums. By using them, formulas which describe the Stokes phenomena are obtained.

In this paper we give the Weierstrass equations and the generators of Mordell–Weil groups for Jacobian fibrations on the singular $K3$ surface of discriminant 3.

In this paper, we consider the normality or the integer decomposition property (IDP, for short) for Minkowski sums of integral convex polytopes. We discuss some properties on the toric rings associated with Minkowski sums of integral convex polytopes. We also study Minkowski sums of edge polytopes and give a sufficient condition for Minkowski sums of edge polytopes to have IDP.

Let $T$ be an $m$-linear Calderón–Zygmund operator with kernel $K$ and $T^*$ be the maximal operator of $T$. Let $S$ be a finite subset of $Z^+\times \{1,\dots,m\}$ and denote $d\vec{y}=dy_1\cdots dy_m$. Define the commutator $T_{\vec{b},S}$ of $T$, and $T^*_{\vec{b},S}$ of $T^*$ by $T_{\vec{b},S}(\vec{f})(x)= \int_{\mathbb{R}^{nm}}\prod_{(i,j)\in S}(b_i(x)-b_i(y_j)) K(x,y_{1},\dots,y_{m}) \prod_{j=1}^mf_j(y_j)d\vec{y}$ and $T^*_{\vec{b},S}(\vec{f})(x)= \sup_{\delta>0}\big|\int_{{\sum_{j=1}^m|x-y_j|^2>\delta^2}} \prod_{(i,j)\in S}(b_i(x)-b_i(y_j))K(x,y_{1},\dots,y_{m}) \prod_{j=1}^mf_{j}(y_j)d\vec{y}\big|$. These commutators are reflexible enough to generalize several kinds of commutators which already existed. We obtain the weighted strong and endpoint estimates for $T_{\vec{b},S}$ and $T^*_{\vec{b},S}$ with multiple weights. These results are based on an estimate of the Fefferman–Stein sharp maximal function of the commutators, which is proved in a pretty much more organized way than some known proofs. Similar results for the commutators of vector-valued multilinear Calderón–Zygmund operators are also given.

In this paper, we generalize the Hurwitz space which is defined by Fried and Völklein by replacing constant Teichmüller level structures with non-constant Teichmüller level structures defined by finite étale group schemes. As an application, we give some examples of projective general symplectic groups over finite fields which occur as quotients of the absolute Galois group of the field of rational numbers $\mathbb Q$.

For a compact simple Lie group $G$, we show that the element $[G, \mathcal{L}] \in \pi^S_*(S^0)$ represented by the pair $(G, \mathcal{L})$ is zero, where $\mathcal{L}$ denotes the left invariant framing of $G$. The proof relies on the method of E. Ossa [Topology, 21 (1982), 315–323].

We determine the indecomposable characters of several classes of infinite dimensional groups associated with operator algebras, including the unitary groups of arbitrary unital simple AF algebras and II₁ factors.

Motivated by the work of Baras–Goldstein (1984), we discuss when expectations of the Feynman–Kac type with singular potentials are divergent. Underlying processes are Brownian motion and $\alpha$-stable process. In connection with the work of Ishige–Ishiwata (2012) concerned with the heat equation in the half-space with a singular potential on the boundary, we also discuss the same problem in the half-space for the case of Brownian motion.

We use the convexity of a certain function discovered by W. Kendall on small metric balls in CAT(1)-spaces to show that any probability measure on a complete CAT(1)-space of small radius admits a unique barycenter. We also present various properties of barycenter on those spaces. This extends the results previously known for CAT(0)-spaces and CAT(1)-spaces of small diameter.

Let $D$ be a smooth divisor on a non singular surface $S$. We compute the Betti numbers of the Hilbert scheme of points of $S$ relative to $D$. In the case of $\mathbb{P}^2$ and a line in it, we give an explicit set of generators and relations for the corresponding cohomology groups.

In this note we establish the boundedness properties of local maximal operators $M_G$ on the fractional Sobolev spaces $W^{s,p}(G)$ whenever $G$ is an open set in ${\mathbb R}^n$, $0 \lt s \lt 1$ and $1 \lt p \lt \infty$. As an application, we characterize the fractional $(s,p)$-Hardy inequality on a bounded open set by a Maz'ya-type testing condition localized to Whitney cubes.