We consider random walks on random graphs determined by a some kind of continuum percolation on $\mathbf{R}$. The vertex set of the random graph is given by the Poisson points conditioned that all points of $\mathbf{Z}$ are contained. The edge set of the random graph is determined by the random radii of the spheres centered at each points. We give heat kernel estimates for the random walks under the condition on the moment of the random radii. We will also discuss random walks on continuum percolation clusters in $\mathbf{R}^d$, $d\ge 2$.

The purpose of this paper is to investigate the global topological monodromy of a certain fibration of the Fermat surface without using numerical analysis by computer.

We study an interplay between homoclinic behavior and singularities in surface endomorphisms. We show that appropriate rescalings near homoclinic orbits intersecting fold singularities yield families of non-invertible Hénon-like maps. Then we construct positive measure sets of parameters corresponding to maps which exhibit nonuniformly hyperbolic behavior. This implies an extension of the celebrated theorem of Benedicks and Carleson, and that of Mora and Viana to surface endomorphisms.

Let $M = 2m$ $(m = 1,2,\ldots)$. In [1] the best constant of $L^p$ Sobolev inequality $$ \sup_{-1\leq x\leq 1}\vert u(x)\vert\leq C\Biggl( \int_{-1}^{1}\vert u^{(M)}(x)\vert^{p}dx\Biggr)^{1/p} $$ was obtained for $u$ satisfying $u, u^{(M)} \in L^{p}(-1,1)$ and $u^{(2i)}(\pm 1) = 0$ $(0\leq i\leq [(M-1)/2])$. On the other hand, for the case $M$ is odd, up to now, only the case $M =1$ was treated for technical difficulty; see [2]. This paper treats the case $M =3$ with different two approach, one is based on the property of the function associated with certain Green function and another is on the property of function space. For the latter approach, symmetrizations of functions play an important role.

In this paper we shall prove that every Witten multiple zeta value of weight $w>3$ attached to $\mathfrak{sl}(4)$ at nonnegative integer arguments is a finite $\mathbb{Q}$-linear combination of MZVs of weight $w$ and depth three or less, except for the nine irregular cases where the Riemann zeta value $\zeta(w-2)$ and the double zeta values of weight $w-1$ and depth $<3$ are also needed.

It is well-known that any pair of closed orientable 3-manifolds are related by a finite sequence of Dehn surgeries on knots. Furthermore Kawauchi showed that such knots can be taken to be hyperbolic. In this article, we consider the minimal length of such sequences connecting a pair of 3-manifolds, in particular, a pair of lens spaces.

DS-diagram and flow spine are good tools for studying 3-manifolds ([5], [8]). In this paper, we introduce the concept of generalized DS-diagram and study its properties. We define two types of moves that change generalized DS-diagrams but do not change their associated manifolds. We prove that any two generalized DS-diagrams such that their associated manifolds are homeomorphic to each other can be deformed into each other by a finite sequence of moves of the types.

Let $S=\bigoplus_{n_1,\ldots, n_s \geq 0}S_{(n_1,\ldots, n_s)}$ be a finitely generated standard multi-graded algebra over a Noetherian local ring $A$. This paper investigates the positivity of mixed multiplicities of $S$ and characterizes them in terms of Hilbert-Samuel multiplicities. As an application, we get some results on the mixed multiplicities of ideals that covers the main results in [Vi] and [TV].

In this paper, we present the new proof of the Length Upper Bounds Theorem on curves in surfaces, which is crucial in the proof of Ending Lamination Conjecture by Minsky et al. Our proof is based on arguments in Bowditch [Bow2] but we use geometric limit arguments fully.

Let $I$ be the Stanley-Reisner ideal of pure simplicial complex $\Delta$ of dimension one. We shall give a formula for $S/I^{(r)}$ to be a $k$-Buchsbaum ring for each $r>0$, where $I^{(r)}$ is the $r$-th symbolic power of $I$. The main result is an improvement of the previous result in [MN] on the $k$-Buchsbaumness of $S/I^{(r)}$.

A real Bott manifold is obtained as the orbit space of the $n$-torus $T^n$ by a free action of an elementary abelian $2$-group $(\mathbb{Z}_2)^n$. This paper deals with the classification of 5-dimensional real Bott manifolds and studies certain types of $n$-dimensional real Bott manifolds ($n\geq 6$).

Let $\Gamma$ be a finitely generated discrete subgroup of $\mathrm{PGL}(4,\mathbf{C})$ acting on $\mathbf{P}^3$. Suppose that $\Gamma$ leaves invariant a surface in $\mathbf{P}^3$. Then, except for a few cases, we can find a plane which is invariant by a finite index subgroup of $\Gamma$. The exceptional cases will be determined explicitly.