A submanifold of a Riemannian symmetric space is called parallel if its second fundamental form is parallel. We classify parallel submanifolds of the Grassmannian $\mathrm{G}^{+}_{2}(\mathbb{R}^{n+2})$ which parameterizes the oriented 2-planes of the Euclidean space $\mathbb{R}^{n+2}$. Our main result states that every complete parallel submanifold of $\mathrm{G}^{+}_{2}(\mathbb{R}^{n+2})$, which is not a curve, is contained in some totally geodesic submanifold as a symmetric submanifold. The analogous result holds if the ambient space is the Riemannian product of two Euclidean spheres of equal curvature or the non-compact dual of one of the previously considered spaces. We also give a characterization of parallel submanifolds with curvature isotropic tangent spaces of maximal possible dimension in any symmetric space of compact or non-compact type.

In general, for a Markov process which does not have an invariant measure, it is possible to realize a stationary Markov process with the same transition probability by extending the probability space and by adding new paths which are born at random times. The distribution (which may not be a probability measure) is called a Kuznetsov measure. By using this measure we can construct a stationary Markov particle system, which is called an equilibrium process with immigration. This particle system can be decomposed as a sum of the original part and the immigration part (see [2]). In the present paper, we consider an absorbing stable motion on a half space $H$, i.e., a time-changed absorbing Brownian motion on $H$ by an increasing strictly stable process. We first give the martingale characterization of the particle system. Secondly, we discuss the finiteness of the number of particles near the boundary of the immigration part. (cf. [2], [3], [4].)

Barchini, Kable, and Zierau constructed a number of conformally invariant systems of differential operators associated to parabolic subalgebras of Heisenberg type. When they constructed such systems of operators, two constants, which play a role for the construction, were defined as the constants of proportionality between two expressions. In this paper we give concrete and uniform expressions for these constants. To do so we introduce a new constant inspired by a formula on the Dynkin index of a finite dimensional representation of a complex simple Lie algebra.

For a minimal diffusion process on $(a,b)$, any possible extension of it to a standard process on $[a,b]$ is characterized by the characteristic measures of excursions away from the boundary points $a$ and $b$. The generator of the extension is proved to be characterized by Feller's boundary condition.

In this paper, we introduce the notion of a connected sum $K_{1} \#^{Z} K_{2}$ of simplicial complexes $K_{1}$ and $K_{2}$, as well as define a strong connected sum. Geometrically, the connected sum is motivated by Lerman's symplectic cut applied to a toric orbifold, and algebraically, it is motivated by the connected sum of rings introduced by Ananthnarayan--Avramov--Moore [1]. We show that the Stanley--Reisner ring of a connected sum $K_{1} \#^{Z} K_{2}$ is the connected sum of the Stanley--Reisner rings of $K_{1}$ and $K_{2}$ along the Stanley--Reisner ring of $K_{1} \cap K_{2}$. The strong connected sum $K_{1} \#^{Z} K_{2}$ is defined in such a way that when $K_{1}$, $K_{2}$ are Gorenstein, and $Z$ is a suitable subset of $K_{1} \cap K_{2}$, then the Stanley--Reisner ring of $K_{1} \#^{Z} K_{2}$ is Gorenstein, by work appearing in [1]. We also show that cutting a simple polytope by a generic hyperplane produces strong connected sums. These algebraic computations can be interpreted in terms of the equivariant cohomology of moment angle complexes and toric orbifolds.

The Wirtinger integral is the uniformization to the upper half plane $H$ of the hypergeometric function defined on the complex projective line $\mathbb{P}^{1}$. In [5] we established the transformation formulas of the Wirtinger integral for the linear fractional transformations $\tau\to\tau+2$ and $\tau\to -1/\tau$ with the aide of the theory of theta functions. As a corollary we obtain the transformation formulas of the Wirtinger integral for the linear fractional transformations $\tau\to\tau+2$ and $\tau\to \tau/(-2\tau+1)$ which are identified with generators of the principal congruence subgroup $\varGamma(2)$ modulo center. These formulas correspond to the monodromy matrices of the hypergeometric function for generators of the fundamental group of $\mathbb{P}^{1}$ minus three points. The purpose of this paper is to generalize this result, that is, we establish the transformation formula of the Wirtinger integral for a general element $\left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right)$ of $\varGamma(2)$, which corresponds to a general monodromy matrix of the hypergeometric function.

An $\SH(3)$-move is an unknotting operation on oriented knots introduced by Hoste, Nakanishi and Taniyama. We consider some relationships to other local moves such as a band surgery, $\Gamma_{0}$-move, and $\Delta$-move, and give some criteria for estimating the $\SH(3)$-unknotting number using the Jones, HOMFLYPT, Q polynomials. We also show a table of $\SH(3)$-unknotting numbers for knots with up to 9 crossings.

In this paper we consider a symplectic basis of the first cohomology group and the sigma functions for algebraic curves expressed by a canonical form using a finite sequence $(a_{1}, \ldots, a_{t})$ of positive integers whose greatest common divisor is equal to one (Miura [13]). The idea is to express a non-singular algebraic curve by affine equations of $t$ variables whose orders at infinity are $(a_{1}, \ldots, a_{t})$. We construct a symplectic basis of the first cohomology group and the sigma functions for telescopic curves, i.e., the curves such that the number of defining equations is exactly $t-1$ in the Miura canonical form. The largest class of curves for which such construction has been obtained thus far is $(n, s)$-curves ([4] [15]), which are telescopic because they are expressed in the Miura canonical form with $t=2$, $a_{1}=n$, and $a_{2}=s$, and the number of defining equations is one.

Given a unicuspidal rational curve $C \subset \mathbb{P}^{2}$ with singular point $P$, we study the unique pencil $\Lambda_{C}$ on $\mathbb{P}^{2}$ satisfying $C \in \Lambda_{C}$ and $\mathrm{Bs}(\Lambda_{C})=\{P\}$. We show that the general member of $\Lambda_{C}$ is a rational curve if and only if $\tilde{\nu}(C) \ge 0$, where $\tilde{\nu}(C)$ denotes the self-intersection number of $C$ after the minimal resolution of singularities. We also show that if $\tilde{\nu}(C) \ge0$, then $\Lambda_{C}$ has a dicritical of degree $1$. Note that all currently known unicuspidal rational curves $C \subset \mathbb{P}^{2}$ satisfy $\tilde{\nu}(C) \ge 0$.

Let $p$ be an odd prime, and $k_{\infty}$ the cyclotomic $\mathbb{Z}_{p}$-extension of an abelian field $k$. For a finite set $S$ of rational primes which does not include $p$, we will consider the maximal $S$-ramified abelian pro-$p$ extension $M_{S} (k_{\infty})$ over $k_{\infty}$. We shall give a formula of the $\mathbb{Z}_{p}$-rank of $\mathrm{Gal}(M_{S} (k_{\infty})/k_{\infty})$. In the proof of this formula, we also show that $M_{\{q\}} (k_{\infty})/L(k_{\infty})$ is a finite extension for every real abelian field $k$ and every rational prime $q$ distinct from $p$, where $L(k_{\infty})$ is the maximal unramified abelian pro-$p$ extension over $k_{\infty}$.