We are interested in the statistical behavior of a certain continued fraction whose associated dynamical system has the infinite invariant measure. We show the growth rate of denominator $Q_n$ of the $n$-th convergent of negative continued fraction expansion of $x$ and the rate of approximation: $$ \frac{\log{n}}{n}\log{\left|x-\frac{P_n}{Q_n}\right|}\rightarrow -\frac{\pi^2}{3} \quad \text{in measure} $$ for $x$. In the course of the proof, we reprove related results that arithmetic mean of digits of negative continued fraction converges to 3 in measure, although the limit inferior is 2, and the limit superior is infinite almost everywhere.

Satoh and Taniguchi defined an invariant of virtual knots $J_n$ for a non-zero integer $n$. It is called the $n$-writhe. The $n$-writhes give the coefficients of some polynomial invariants for virtual knots including the index polynomial, the odd writhe polynomial and the affine index polynomial. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. The unknotting number by the virtualization is called the virtual unknotting number. Kishino's knot is a virtual unknotting number one knot which has the trivial $n$-writhe and the trivial Jones polynomial. In this paper, we construct infinitely many virtual knots which have the same properties as Kishino's knot.

There are not so many kinds of surface of revolution whose cut locus structure have been determined, although the cut locus structures of very familiar surfaces of revolution (in Euclidean space) such as ellipsoids, 2-sheeted hyperboloids, paraboloids and tori are now known.

Except for tori, the known cut locus structures are very simple, i.e., a single point or an arc. In this article, a new family $\{M_n\}_n$ of 2-spheres of revolution with simple cut locus structure is introduced. This family is also new in the sense that the number of points on each meridian which assume a local minimum or maximum of the Gaussian curvature function on the meridian goes to infinity as $n$ tends to infinity.

Thus, our family includes surfaces which have arbitrarily many bands of alternately increasing or decreasing Gaussian curvature, although each member of this family has a simple cut locus structure.

The celebrated Trotter approximation theorem provides a sufficient condition for the convergence of a sequence of operator semigroups in terms of the corresponding sequence of infinitesimal generators. There exist a few results on the rate of convergence in Trotter's theorem under some constraints. In the present paper, a new rate of convergence in Trotter's theorem in full generality is given. Moreover, we see that this rate of convergence works well to obtain quantitative estimates for some limit theorems in probability theory.

In this paper, we investigate properties of orbits of Hermann actions as submanifolds without assuming the commutability of involutions which define Hermann actions. In particular, we compute the second fundamental form of orbits of Hermann actions, and give a sufficient condition for orbits of Hermann actions to be weakly reflective submanifolds and arid submanifolds.

We determine the smooth affine $\mathbb{G}_m$-surfaces of logarithmic Kodaira dimension $\leq 0$ with finite Picard groups and trivial units defined over an algebraically closed field. Furthermore, we prove that a smooth factorial affine surface with trivial units is isomorphic to the affine plane if and only if it admits an effective $\mathbb{G}_m$-action.

We prove the nonabelian coefficient versions of proper base change theorem, log smooth base change theorem, and proper log smooth base change theorem for fs log analytic spaces. We also show that the real blowing-up of any log modification is a weak homotopy equivalence.

In this paper the predual spaces of weak Orlicz spaces are shown to exist. As an application, the Fefferman-Stein vector-valued maximal inequality is obtained for weak Orlicz spaces. In order to prove this inequality, Orlicz-Lorentz spaces are introduced and it is shown that the Hardy-Littlewood maximal operator is bounded on these spaces. As is proved in this paper, weak Orlicz spaces are not reflexive. This makes it difficult to use the duality argument for the proof of the boundedness of operators. To avoid this problem, predual spaces are used.

Let $K$ be an arbitrary field of characteristic $p>0$. Classifications of prime ideals and simple modules are obtained for the Weyl algebra $A_1=K\langle x,\partial \, : \, \partial x-x\partial =1\rangle$, the skew polynomial algebra $\mathbb{A} = K[h][x;\sigma ]$ and the skew Laurent polynomial algebra $\cal{A} := K[h][x^{\pm 1};\sigma ]$ where $\sigma (h) = h-1$. In particular, classifications of prime, completely prime, maximal and primitive ideals are obtained for the above algebras. The quotient ring (of fractions) of each prime factor algebra of $A_1$, $\mathbb{A}$ and $\cal{A}$ is described. It is either a matrix algebra over a field or else a cyclic algebra. These descriptions are a key fact in the classification of completely prime ideals and simple modules for the algebras above.

In the present paper, we discuss Monge-Ampère equations from the viewpoint of differential geometry. It is known that a Monge--Ampère equation corresponds to a special exterior differential system on a 1-jet space. In this paper, we generalize Monge--Ampère equations and prove that a $(k+1)$st order generalized Monge--Ampère equation corresponds to a special exterior differential system on a $k$-jet space and that its solution naturally corresponds to an integral manifold of the corresponding exterior differential system. Moreover, we show that the Korteweg-de Vries (KdV) equation and the Cauchy--Riemann equations are examples of our equations.

We enumerate and show tables of minimal diagrams for all prime knots up to the triple-crossing number equal to five. We derive a minimal generating set of oriented moves connecting triple-crossing diagrams of the same oriented knot. We also present a conjecture about a strict lower bound of the triple-crossing number of a knot related to the breadth of its Alexander polynomial.

The index of a meromorphic function $g$ on a compact Riemann surface is an invariant of $g$, which is defined as the number of negative eigenvalues of the differential operator $L:=-\Delta-|dG|^2$, where $\Delta$ is the Laplacian with respect to a conformal metric $ds^2$ on the Riemann surface and $G\colon M\to S^2$ is the holomorphic map corresponding to $g$. We consider the meromorphic function $w$ on the Riemann surface $M_a=\left\{(z,w)\in\widehat{\mathbb{C}}^2 \mid w^2=z(z-a)\left(z+\frac{1}{a}\right)\right\}(a\geqslant 1 )$ homeomorphic to a torus. We find $a_0>1$ and determine the index of $tw$ for all $a$ in the range $1\leqslant a\leqslant a_0$ and all $t>0$.

We characterize the Sobolev multiplier spaces and their preduals for the end point cases. It will be shown that under certain conditions the preduals coincide with the space of Choquet integrals. As an application, the boundedness of the local Hardy-Littlewood maximal function on the preduals will be obtained.