In this paper, we prove new trigonometric identities, which are product-to-sum type formulas for the higher derivatives of the cotangent and cosecant functions. Furthermore, from specializations of our formulas, we derive various known and new reciprocity laws of generalized Dedekind sums.

Let $K$ be a cyclic quartic number field such that its 2-class group is of type $(2,4)$, $K_2^{(1)}$ be the Hilbert 2-class field of $K$, $K_2^{(2)}$ be the Hilbert 2-class field of $K_2^{(1)}$ and $G=\text{Gal}(K_2^{(2)}/K)$ be the Galois group of $K_2^{(2)}/K$. Our goal is to study the capitulation problem of 2-ideal classes of $K$ and to determine the structure of $G$.

The aim of this paper is to establish the uniqueness theorem for the Cauchy problem for the heat equation with the Tikhonov condition on the Heisenberg group. To do this, we give Green's formula and show the existence of a Lipschitz cut-off function on the Heisenberg group in accordance with the idea in [7].

In this paper we give a geometric characterization of non-Hopf hypersurfaces in the complex space form $M^2(c)$ under a condition on the shape operator. We also classify pseudo-parallel real hypersurfaces of $M^2(c)$.

Let $E(\alpha) \subset \mathbb{C}^{m+1}$ and $E(\beta) \subset \mathbb{C}^{n+1}$ be generalized pseudoellipsoids. Assume that the inequality $m<n$ holds. They are parametrized by $N$-tuples of positive integers $\alpha=(\alpha_1, \dots, \alpha_N)$ and $\beta=(\beta_1, \dots, \beta_N)$. (See introduction for the definition of a generalized pseudoellipsoid) Assume that there exists a proper holomorphic mapping between them. In this article, two facts are proved. Firstly, under the assumptions of the existence of such a mapping, certain nondegeneracy conditions of a submatrix of the Jacobian matrix and additional inequalities on dimensions, the parameters $(\alpha_1, \dots, \alpha_N)$ and $(\beta_1, \dots, \beta_N)$ coincide; $\alpha_1=\beta_1, \dots, \alpha_N=\beta_N$ after re-ordering if necessary. Secondly, such a proper holomorphic mapping is a linear embedding up to automorphisms of a source and a target domains.

Let $M$ be a non-compact complete Riemannian manifold of dimension two and $N$ a circle in $M$. We assume that $M$ is partitioned by $N$. We define a unital $C^{\ast}$-algebra $C_{b}^{\ast}(M)$, which is slightly larger than the Roe algebra of $M$. We also construct $[u_{\phi}]$ in $K_{1}(C_{b}^{\ast}(M))$, which is a counter part of Roe's odd index class. We prove that Connes' pairing of Roe's cyclic one-cocycle with $[u_{\phi}]$ is equal to the Fredholm index of a Toeplitz operator on $N$. It is a part of an extension of the Roe-Higson index theorem to even-dimensional partitioned manifolds.

We discuss the Hausdorff dimension of certain sets related to Diophantine approximations over an imaginary quadratic field $\mathbb{Q}(\sqrt{d})$. We show that, for an infinite subset $\mathcal{A}$ of $\mathbb{Z}[\omega] \backslash \{0\}$, the set of $z \in \mathbb{C}$ with $|z-a/r| < 1 / |r|^{1+\rho}$ having infinitely many solutions of $a \in \mathbb{Z}[\omega]$ and $r \in \mathcal{A}$ with some $\rho > 0$ has Hausdorff dimension $2(1+\gamma) / (1+\rho)$, where $\gamma$ is the sup of $h$ such that $\sum_{r \in \mathcal{A}} 1/(|r|^{2})^{h}$ diverges. This result is a version of a result by G. Harman for complex numbers without the coprime condition. In particular, this result implies a version of the classical Jarnik-Besicovitch result when we take $\mathcal{A} = \mathbb{Z}[\omega]\backslash\{0\}$. We also discuss the Hausdorff dimension of the set of complex numbers which have infinitely many solutions to the Diophantine inequality concerning the Duffin-Schaeffer conjecture over $\mathbb{Q}(\sqrt{d})$.

This paper investigates the families of smooth complete intersections containing $r$-planes in projective spaces. We are going in a primitive way to shed some light on a point and an $r$-plane containing the point in a complete intersection from the viewpoint of projective geometry.

Among the unitary reflection groups, the one on the title is singled out by its importance in, for example, coding theory and number theory. In this paper we examine the semi-simple structure of the centralizer algebra in the tensor representation, and show that the dimensions of the centralizers coincide with the numbers of some combinatorial objects.

We introduce $B_w^u$-function spaces which unify Lebesgue, Morrey-Campanato, Lipschitz, $B^p$, $\mathrm{CMO}$, local Morrey-type spaces, etc., and investigate the interpolation property of $B_w^u$-function spaces. We also apply it to the boundedness of linear and sublinear operators, for example, the Hardy-Littlewood maximal and fractional maximal operators, singular and fractional integral operators with rough kernel, the Littlewood-Paley operator, Marcinkiewicz operator, and so on.

We compute the Reidemeister torsion of the complement of a twist knot in $S^3$ and that of the 3-manifold obtained by a $\frac{1}{q}$-Dehn surgery on a twist knot.

In this paper, we introduce weighted backward shifts, which are asymptotically unweighted, and give several conditions for such operators on the classical $\ell^p$ spaces to be hypercyclic and chaotic.

In this paper, we give sufficient conditions for orbits of Hermann actions to be weakly reflective in terms of symmetric triads, that is a generalization of irreducible root systems. Using these sufficient conditions, we obtain new examples of weakly reflective submanifolds in compact symmetric spaces.

In his classical work, Pinkham discovered a beautiful theorem on the moduli space of pointed algebraic curves with a fixed Weierstrass gap sequence at the marked point. Namely, the complement of a Weierstrass gap sequence in the set of non-negative integers is a numerical semigroup, and he described such a moduli space in terms of the negative part of the miniversal deformation space of the monomial curve of this semigroup. Unfortunately, his theorem holds only in characteristic 0 and does not hold in positive characteristic in general. In this paper, we will study his theorem in positive characteristic, and give a fairly sharp condition for his theorem to hold in positive characteristic up to genus 4. As an application, we present a complete analysis of his theorem in positive characteristic in the low genus case.