We study the geometry of the tangent bundle equipped with a two-parameter family of metrics, deforming the Sasaki and Cheeger-Gromoll metrics. After deriving the expression for the Levi-Civita connection, we compute the Riemann curvature tensor and the sectional, Ricci and scalar curvatures. We identify all metrics whose restrictions to the fibres have positive sectional curvature. When the base manifold is a space form, we characterise metrics with non-negative sectional curvature and show that one can always find parameters ensuring positive scalar curvature. This extends to compact manifolds and, under some curvature conditions, to general manifolds.

In this paper, we consider curves of degree 10 of torus type (2,5), $C:=\{f_5(x,y)^2+f_2(x,y)^5=0\}$. Assume that $f_2(0,0)=f_5(0,0)=0$. Then $O=(0,0)$ is a singular point of $C$ which is called an inner singularity. In this paper, we give a topological classification of singularities of $(C,O)$.

A generalization of the Schur Q-function is introduced. This generalization, called generalized Q-function, is indexed by any pair of strict partitions, and can be expressed by the Pfaffian. A connection to the theory of integrable systems is clarified. Firstly, the bilinear identities satisfied by the generalized Q-functions are given and proved to be equivalent to a system of partial differential equations of infinite order. This system is called the UC hierarchy of B-type (BUC hierarchy). Secondly, the algebraic structure of the BUC hierarchy is investigated from the representation theoretic viewpoint. Some new kind of the boson-fermion correspondence is established, and a representation of an infinite dimensional Lie algebra, denoted by $\mathfrak{go}_{2\infty}$, is obtained. The bilinear identities are translated to the language of neutral fermions, which turn out to characterize a $\boldsymbol{G}$-orbit of the vacuum vector, where $\boldsymbol{G}$ is the group corresponding to $\mathfrak{go}_{2\infty}$.

K. Habiro defined a $C_n$-move which is a local move on oriented links. He also proved that two knots are not distinguished by any Vassiliev invariants of order less than $n$ if and only if they are related by a finite sequence of $C_n$-moves. In the case of $n \geq 3$, it is known that the result does not hold for links. In this note we will introduce a special $C_n$-move and give a ``geometrical'' necessary and sufficient condition using the terms of $C_n$-moves for that two links are not distinguished by any Vassiliev invariants of order less than $3$ or $4$.

A local move called a $C_n$-move is related to Vassiliev invariants. It is known that two knots are related by $C_n$-moves if and only if they have the same values of Vassiliev invariants of order less than $n$. In the link case, it is shown that a $C_n$-move does not change the values of any Vassiliev invariants of order less than $n$. It is also known that, if two links can be transformed into each other by a $C_n$-move, then the $n$-th coefficients of the Conway polynomials of them, which are Vassiliev invariants of order $n$, are congruent to each other modulo $2$. An $SC_n$-move is defined as a special $C_n$-move. It is shown that an $SC_n$-move does not change the values of any Vassiliev invariants of links of order less than $n+1$. In this paper, we consider the difference of the $(n+1)$-st coefficients of the Conway polynomials of two links which can be transformed into each other by an $SC_n$-move.

It is known that there exists a time periodic solution of the Navier-Stokes equations with Dirichlet boundary conditions satisfying so-called stringent outflow condition (SOC). In this paper we will show the existence of periodic solutions of the Navier-Stokes equations with Dirichlet boundary conditions satisfying so-called general outflow condition (GOC).

The present paper is to investigate the existence of the scattering operators for the Klein-Gordon equations with a potential which depends on both the space and time variables, and admits a singularity in $L^q(\mathbb{R}^n)$ sense. The main tool is the Strichartz estimates of solutions.

We prove a generalization of a lemma by Schmitt and Vogel which will allow us to compute the arithmetical rank of new classes of monomial ideals.

For a compact Hausdorff space $X$, $C(X)$ denotes the ring of all complex-valued continuous functions on $X$. We say that $C(X)$ is \textit{algebraically closed} if every monic algebraic equation with $C(X)$-coefficients has a root in $C(X)$. Modifying the construction of [2], we show that, for each $m = 1,2, \cdots, \infty$, there exists an $m$-dimensional compact Hausdorff space $X(m)$ such that $C(X(m))$ is algebraically closed.

It is well known that 2-interval exchange transformation is isomorphic to the odometer transformation on some symbolic space and isomorphism is given by Ostrowski representation formula. The purpose of this paper is to extend these results to the case of 3-interval exchange transformation.

In the present paper, we investigate the predual of the Morrey spaces with non-doubling measures. We also study the modified maximal function, singular integrals and commutators on the predual spaces.

Let $(M=G/K, g, J)$ be a Kähler $C$-space with two isotropy summands. We classify all such spaces. Thus, by using previous work of O. Ikawa, we obtain a large class of examples where the differential equation $\nabla_{\dot{x}}\dot{x}=kJ\dot{x}$ of the motion of a charged particle under the electromagnetic field $kJ$ can be explicitly solved. In particular, geodesics curves in these spaces can also be described.

In this paper we will prove a maximum principle for the solutions of linear parabolic equation on complete non-compact manifolds with a time varying metric. We will prove the convergence of the Neumann Green function of the conjugate heat equation for the Ricci flow in $B_k\times (0,T)$ to the minimal fundamental solution of the conjugate heat equation as $k\to\infty$. We will prove the uniqueness of the fundamental solution under some exponential decay assumption on the fundamental solution. We will also give a detail proof of the convergence of the fundamental solutions of the conjugate heat equation for a sequence of pointed Ricci flow $(M_k\times (-\alpha,0],x_k,g_k)$ to the fundamental solution of the limit manifold as $k\to\infty$ which was used without proof by Perelman in his proof of the pseudolocality theorem for Ricci flow [P].

Using a ``complex Pisot number'' $\lambda \in \mathbb{C}$ with $|\lambda|>1$, the numerical expansion $\sum_{j=-k}^{\infty}{a_j}/{\lambda^j}$ of a complex number, where each digit $a_j$ is chosen from some finite set $\Gamma$ of $\mathbb{Z}[\lambda]$, was established recently as an analogue of $\beta$-numeration system $\sum_{j=-k}^{\infty}{b_j}/{\beta^j}$ of a real number, where $b_j \in \{0, 1, \cdots, \lfloor\beta\rfloor\}$. In this paper, we give a necessary and sufficient condition for a complex number to have eventually or purely periodic complex Pisot expansion.

Starting from a trivial pure state $\Psi^{0}$, we construct non-trivial tracial states $(\Psi^{i})$ on the $\theta$-deformed $2m$-plane $C^{alg}(\mathbb{R}_{\theta}^{2m})$. Furthermore we generalize $\Psi^{i}$ to another tracial state on $C^{alg}(\mathbb{R}_{\theta}^{2m})$. We study extreme points of the tracial state space of $C^{alg}(\mathbb{R}_{\theta}^{2m})$ in the case that deformation parameters are irrational numbers. Non-trivial pure states $(\Phi_{t}^{k})$ on $C^{alg}(\mathbb{R}_{\theta}^{2m})$ are also given.

Let $\Omega_n$ be the $n$-th layer of the cyclotomic $\mathbb{Z}_3$-extension of $\mathbb{Q}$ and $h_n$ the class number of $\Omega_n$. We claim that if $\ell$ is a prime number less than $10^4$, then $\ell$ does not divide $h_n$ for any positive integer $n$.