Let $p_k/q_k=[a_0;a_1,a_2,\cdots,a_k]$ be the $k$-th convergent of the continued fraction expansion of a real number $\alpha$. We shall show several interesting arithmetic properties concerning every third convergent of the continued fraction expansion of $e^{1/s}$ ($s\ge 1$).

In this paper we prove that the Landau-Kolmogorov inequality for functions on the half line holds for Lorentz spaces with the constants, which are best possible for $L_\infty$-space.

We are studying the action of Galois groups on the pro-$l$ completion of the fundamental group of $\mathbf{P}^1_{\overline{\mathbf{Q}(\mu_n)}} \backslash \{0,\mu_n,\infty\}$. If $n=2p$, where $p$ is an odd prime number then the Lie algebra of derivations associated to the image of $\text{Gal}(\overline{\mathbf{Q}} / \mathbf{Q}(\mu_{2p\cdot l^\infty}))$ has $\frac{p-1}{2}$ generators in each even degree and $\frac{p-1}{2}$ generators in each odd degree greater than $1.$ We shall show that generators in even degrees generate a free Lie algebra.

In this paper we shall give a proof of the Paley-Wiener theorem for hyperfunctions supported by a convex compact set by the heat kernel method.

Let $q$ with $|q|>1$ be an integer in an algebraic number field $\mathbf{K}$ given below, and $\{a_n\}$ a periodic sequence in $\mathbf{K}$ of period two, not identically zero. Let $f(q)=\sum_{n=1}^{\infty}{a_n}/(1-q^n).$ We prove that (i) If $\mathbf{K}$ is either the rational number field $\mathbb{Q}$ or an imaginary quadratic field, then $f(q)\notin \mathbf{K}$. (ii) For an algebraic integer $q$ such that $|q|>1$ and $|q^{\sigma}|<1$ for any $\sigma \in$ $\textrm{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$ with $q^\sigma\ne q$, if $k=\mathbb{Q}(q)$, then $f(q)\notin \mathbb{Q}(q)$. For example, the three numbers $$ 1\,,\quad \sum_{n=1}^{\infty}\frac{1}{q^n-1}\,, \quad \sum_{n=1}^{\infty}\frac{1}{q^n+1} $$ are linearly independent over $\mathbb{Q}$ for every $q\in\mathbb{Z}$ with $|q|\geq2.$ Further, irrationality results of the special values of the functions $$ \sum_{n=1}^{\infty} \frac{z^n}{R_{an+b}}\,,\qquad \sum_{n=1}^{\infty}\frac{z^n}{R_{an+b}R_{{a(n+1)}+b}}~~(z\in\mathbb{C}) $$ can be deduced, where $a>0$, $b\geq0$ are integers and $R_n$ is a certain binary recurrence.

In the present paper, we shall show a Koizumi-type interpolation theorem in order to show the relation between several Yano-type extrapolation estimates.

There is given a characterization of Noetherian local rings $A$ with $d = \dim A \geq 2$, in which the equality $(a_i \mid 1 \leq i \leq d)^n=\underset{\alpha}{\bigcap} (a_1^{\alpha_1}, a_2^{\alpha_2}, \cdots, a_d^{\alpha_d})$ holds true for all systems $a_1,a_2, \cdots, a_d$ of parameters and integers $n \geq 1$, where the suffix $\alpha$ runs over $\alpha = (\alpha _1,\alpha_2,\cdots,\alpha _d) \in \mathbf{Z}^d$ such that $\alpha _i \geq 1\ \text{for all} \ 1\leq i \leq d$ and $\sum_{i=1}^d \alpha _i =d+n-1$.

We consider Bianchi surfaces parametrized by a generalized Chebyshev net and show that such a surface with constant Chebyshev angle is a piece of a right helicoid.

We show a lower estimate of the Milnor number of an isolated hypersurface singularity, via its Newton number. We also obtain analogous estimate of the Milnor number of an isolated singularity of a similar complete intersection variety.

By using the Riemann-Liouville type fractional integral operators we shall reduce the complex Fourier-Jacobi transforms of even functions on $\mathbf{R}$ to the Euclidean Fourier transforms. As an application of the reduction formula, Parseval's formula and an inversion formula of the complex Jacobi transform are easily obtained. Moreover, we shall introduce a class of even functions, not $C^\infty$ and not compactly supported on $\mathbf{R},$ whose transforms have meromorphic extensions on the upper half plane.

In this note, we introduce a homological invariant for finitely generated modules over commutative noetherian local rings by slightly modifying the definition of complete intersection dimension defined by Avramov, Gasharov, and Peeva [4], and observe it from a relative point of view.

Our aim is to check that the notions of positive entropy, chaos in the sense of Devaney and $\omega$-chaos are equivalent for the graph maps.

An isolation theorem of the Bochner curvature tensor of a Kähler-Einstein manifold is given, when its $L^{n/2}$-norm is small. Similarly an isolation theorem of the contact Bochner curvature tensor for a Sasakian manifold is obtained. Those theorems are derived from the Weitzenböck formula which gives non-linearity constraint on the Bochner curvature tensors.

We compute explicitly the moduli space $\mathcal{M}_{g,1}^N$ of pointed algebraic curves of genus $g$ with a given numerical semigroup $N$ when $g$ is small ($2 \leq g \leq 5$). It is known that such a moduli space $\mathcal{M}_{g,1}^N$ is non-empty for $g \leq 7$. The main results obtained in this note are the irreducibility and the determination of the dimension of $\mathcal{M}_{g,1}^N$ for $g \leq 5$ except a few cases. In particular, it turns out that many of these moduli spaces are unirational.

For a real quadratic field $k$, we denote by $\lambda_2(k)$, $\mu_2(k)$ and $\nu_2(k)$ the Iwasawa $\lambda$-, $\mu$- and $\nu$-invariants of the cyclotomic $\mathbb Z_2$-extension of $k$, respectively. In this paper, we give certain families of real quadratic fields $k$ such that $\lambda_2(k)=\mu_2(k)=0$ and $\nu_2(k)=2$, by using Kuroda's class number formula.