We compute the curvature tensor of the tangent bundle of a Riemannian manifold endowed with a natural metric and we get some relationships between the geometry of the base manifold and the geometry of the tangent bundle.

In this paper, we mainly investigate the global cotorsion dimension under the excellent extension of rings. We show that if $S$ is an excellent extension of $R$, then $\mathrm{cot.D}(S)=\mathrm{cot.D}(R)$. Furthermore, some known results, such as Corollary 3.8 and 3.12, can be also obtained as direct corollaries of our theorem.

We prove some generalization of a lemma by Schmitt and Vogel which yields the arithmetical rank in cases that could not be settled by the existing methods. Our results are based on divisibility conditions and exploit both combinatorial and linear algebraic considerations. They mainly apply to ideals generated by monomials.

In this paper we shall show the existence and the stability of almost periodic solutions of the boundary value problem to a nonlinear suspended string equation with a linear damping term and an almost periodic weakly nonlinear forcing term. We treat both weak solutions and strong solutions. Also we show the existence of time global solutions of the initial boundary value problem to the equation.

We consider the stationary Navier-Stokes equations with nonhomogeneous boundary condition in a domain with several boundary components. If the boundary value satisfies only the necessary flux condition (GOC), Leray's inequality does not holds true in general and we cannot prove the existence of a solution. But for a 2-D domain which is symmetric with respect to a line and where the data is also symmetric, C. Amick showed the existence of solutions by reduction to absurdity; later H. Fujita proved Leray-Fujita's inequality and hence the existence of symmetric solutions. In this paper we give a new short proof of Leray-Fujita's inequality and hence a proof of the existence of weak solutions.

Let $A$ and $B$ be subalgebras of $C(X)$ and $C(Y)$, respectively, for some topological spaces $X$ and $Y$. An arbitrary map $T: A\rightarrow B$ is said to be multiplicatively range-preserving if for every $f,g\in A$, $(fg)(X)=(TfTg)(Y)$, and $T$ is said to be separating if $TfTg=0$ whenever $fg=0$. For a given metric space $X$ and $\alpha\in (0,1]$, let Lip$_c(X,\alpha)$ be the algebra of all complex-valued functions on $X$ satisfying the Lipschitz condition of order $\alpha$ on each compact subset of $X$. In this note we first investigate the general form of multiplicatively range-preserving maps from $C(X)$ onto $C(Y)$ for realcompact spaces $X$ and $Y$ (not necessarily compact or locally compact) and then we consider such preserving maps from Lip$_c(X, \alpha)$ onto Lip$_c(Y,\beta)$ for metric spaces $X$ and $Y$ and $\alpha, \beta\in (0,1]$. We show that in both cases multiplicatively range-preserving maps are weighted composition operators which induce homeomorphisms between $X$ and $Y$. We also give a description of a linear separating map $T: A\rightarrow C(Y)$, where $A$ is either $C(X)$ for a normal space $X$ or Lip$_c(X,\alpha)$ for a metric space $X$ and $0<\alpha\le1$ and $Y$ is an arbitrary Hausdorff space.

We capture all the rational solutions of some difference Painlevé equations of P$_{\mathrm{I}}$ and P$_{\mathrm{II}}$ types. For non-autonomous cases, it is shown that all the rational solutions of the difference P$_{\mathrm{II}}$ are ones generated by successive application of auto-Bäcklund transformations to the seed solution vanishing identically, and that the other equations of P$_{\mathrm{I}}$ type admit no rational solutions. For autonomous cases, all the nontrivial rational solutions are obtained, and they exist under a certain condition on a fixed point of the equation. If such a condition is not satisfied, there exist solutions that are rational in an exponential function.

Primitive equations derived originally by Richardson in 1920's have been considered as the model equations describing the motion of atmosphere, ocean and coupled atmosphere and ocean. In this paper, we discuss the free boundary problem of the primitive equations for the ocean in three-dimensional strip with surface tension. Using the so-called $p$-coordinates and a coordinate transformation similar to that in [2] in order to fix the time-dependent domain, we prove temporally local existence of the unique strong solution to the transformed problem in Sobolev-Slobodetskiĭ spaces.

Given $F$ a real abelian field, $p$ an odd prime and $\chi$ any Dirichlet character of $F$, we give a method for computing the $\chi$-index $(H^1(G_S,\mathbf{Z}_p(r))^\chi: C^F(r)^\chi)$ where the Tate twist $r$ is an odd integer $r\geq 3$, the group $C^F(r)$ is the group of higher circular units, $G_S$ is the Galois group over $F$ of the maximal $S$ ramified algebraic extension of $F$, and $S$ is the set of places of $F$ dividing $p$. This $\chi$-index can now be computed in terms only of elementary arithmetic of finite fields $\mathbf{F}_\ell$. Our work generalizes previous results by Kurihara who used the assumption that the order of $\chi$ divides $p-1$.

We shall discuss Gauss sums on finite groups and give several examples including the case of the complex reflection groups $G(m,r,n)$, and hence finite symmetric groups, and also finite Weyl groups.

We consider the response of external field to the theory of liquid crystals. We treat the Landau-de Gennes functional with the Dirichlet boundary condition for the director field which may be non-constant. We show that there exist two families of critical points such that one carries out the superheating fields of superconductors and the other one carries out strong stability. We also show that under some conditions, strong field does not bring the pure nematic state which is different response from superconductors.

The main purpose of this article is to present a numerical data which shows relations between real quadratic fields of class number 1 and a mysterious behavior of the period of simple continued fraction expansion of certain quadratic irrationals. For that purpose, we define a class number, a fundamental unit,a discriminant and a Yokoi invariant for a non-square positive integer, and then see that a generalization of theorems of Siegel and of Yokoi holds. These and a theorem of Friesen and Halter-Koch imply several interesting conjectures for solving Gauss' class number problem for real quadratic fields.

Let $k$ be an integer. In [3, 4], Frankl, Ota and Tokushige proved that the maximum number of three-covers of a $k$-uniform intersecting family with covering number three is $k^3 - 3k^2 + 6k -4$ for $k=3$ or $k \ge 9$, but the case $4 \le k \le 8$ remained open. In this paper, we prove that the same holds for $k=4$, and show that a 4-uniform family with covering number three which has 36 three-covers is uniquely determined.