We give another proof of $(1,1)$-periodicity of M.F. Atiyah's $KR$-theory and the characterization of the classifying space of $KR$-theory.

We give a conceptual clarification of Milnor's theorem, which tells us the Hopf algebra structure of the stable co-operations $H_{*}H$ in the odd primary ordinary cohomology. Directly connecting $H_{*}H$ with the quasi-strict automorphism group of some $1$-dimensional additive formal group law and modular invariants, we give a new proof of this theorem of Milnor.

Andreev's Theorem studies the existence of compact hyperbolic polyhedra of a given combinatorial type and given dihedral angles, all of them acute. In this paper we consider the same problem but without any restriction on the dihedral angles. We solve it for the descendants of the tetrahedron, i.e. those polyhedra that can be obtained from the tetrahedron by successively truncating vertices; for instance, the first of them is the triangular prism.

On the setting of the upper half-space $\mathbf{H}$ of the Euclidean $n$-space, we study weighted harmonic Bergman functions as follows. First, we define the fractional derivatives of some functions defined on $\mathbf{H}$. Next, we find the explicit formula for weighted Bergman kernel through the fractional derivative of the extended Poisson kernel and then we give the size estimates for derivatives of this kernel.

We study a factorization of bounded linear maps from an operator space $A$ to its dual space $A^{*}$. It is shown that $T: A \rightarrow A*$ factors through a pair of column Hilbert space ${\mathscr{H}}_c$ and its dual space if and only if $T$ is a bounded linear form on $A\otimes A$ by the canonical identification equipped with a numerical radius type Haagerup norm. As a consequence, we characterize a bounded linear map from a Banach space to its dual space, which factors through a pair of Hilbert spaces.

We consider the Cauchy problem for ${\partial }_t^{2}u-{\partial }_x^{2}u+u=-g({\partial }_{t}u{)}^3$ on the real line. It is shown that if $g>0$, the solution has an additional logarithmic time decay in comparison with the free evolution in the sense of $L^p$, $2\le p\le \mathrm{\infty }$. Moreover, the asymptotic profile of $u(t,x)$ as $t\to +\mathrm{\infty }$ is obtained. We also discuss a generalization. Consequently we see that the "null condition" in the sense of J.-M. Delort (Ann. Sci. École Norm. Sup., 34 (2001), 1--61) is not optimal for small data global existence for nonlinear Klein-Gordon equations.

For a based space $X$, we consider the group ${ℰ}_{\#n}\left(X\right)$ of all self homotopy classes $\alpha$ of $X$ such that ${\alpha }_{\#}=\mathrm{i}\mathrm{d}:{\pi }_i\left(X\right)\to {\pi }_i\left(X\right)$, for all $i\le n$, where $n\le \mathrm{\infty }$, and the group ${ℰ}_{\mathrm{\Omega }}\left(X\right)$ of all $\alpha$ such that $\mathrm{\Omega }\alpha =\mathrm{i}\mathrm{d}$. Analogously, we study the semigroups ${𝒵}_{\#n}\left(X\right)$ and ${𝒵}_{\mathrm{\Omega }}\left(X\right)$ defined by replacing '$\mathrm{i}\mathrm{d}$' by '$0$' above. There is a chain of containments of the $ℰ$-groups and the $𝒵$-semigroups, and we discuss examples for which the containment is proper. We then obtain various conditions on $X$ which ensure that the $ℰ$-groups and the $𝒵$-semigroups are equal. When $X$ is a group-like space, we derive lower bounds for the order of these groups and their localizations. In the last section we make specific calculations for the $ℰ$-groups and $𝒵$-groups of certain low dimensional Lie groups.

Rigidity theorems for equivariant elliptic genus and equivariant orbifold elliptic genus of almost complex orbifolds with compact connected group actions are formulated and proved.

We consider well-posedness of microhyperbolic Cauchy problems in the category of microlocal ultradistributions. For this purpose, we discuss about the expression of microdifferential operators, and define their irregularities. This enables us to give a general theory about the well-posedness.

Let $(X,d,\mu )$ be a metric measure space with a local regular Dirichlet form. We give necessary and sufficient conditions for a parabolic Harnack inequality with global space-time scaling exponent $\beta \ge 2$ to hold. We show that this parabolic Harnack inequality is stable under rough isometries. As a consequence, once such a Harnack inequality is established on a metric measure space, then it holds for any uniformly elliptic operator in divergence form on a manifold naturally defined from the graph approximation of the space.

The aim in this note is to discuss the behavior at infinity for superbiharmonic functions on ${\mathbf{R}}^n$ by use of spherical means.

In this paper, we study stability for harmonic foliations on locally conformal Kähler manifolds with complex leaves.

It is well known that the spectrum of Laplacian on a compact Riemannian manifold $M$ is an important analytic invariant and has important geometric meanings. There are many mathematicians to investigate properties of the spectrum of Laplacian and to estimate the spectrum in term of the other geometric quantities of $M$. When $M$ is a bounded domain in Euclidean spaces, a compact homogeneous Riemannian manifold, a bounded domain in the standard unit sphere or a compact minimal submanifold in the standard unit sphere, the estimates of the $k+1$-th eigenvalue were given by the first $k$ eigenvalues (see [9], [12], [19], [20], [22], [23], [24] and [25]). In this paper, we shall consider the eigenvalue problem of the Laplacian on compact Riemannian manifolds. First of all, we shall give a general inequality of eigenvalues. As its applications, we study the eigenvalue problem of the Laplacian on a bounded domain in the standard complex projective space ${\mathbf{C}\mathbf{P}}^n\left(4\right)$ and on a compact complex hypersurface without boundary in ${\mathbf{C}\mathbf{P}}^n\left(4\right)$. We shall give an explicit estimate of the $k+1$-th eigenvalue of Laplacian on such objects by its first $k$ eigenvalues.

We studied the distribution of units of an algebraic number field modulo prime ideals. Here we study the distribution of units of a cubic abelian field modulo rational prime numbers. For a decomposable prime number $p$, $2(p-1{)}^2$ is an upper bound of the order of the unit group modulo $p$, and we show that the conjectural density of primes which attain it is really positive.

We obtain a presentation of a certain orientation preserving periodic homeomorphism of a compact real surface of genus $g\ge 2$ by a product of right handed Dehn twists using a splitting family. It was expected that a presentation of a homeomorphism by right handed Dehn twists obtained from a splitting family is one of the shortest presentation. In this paper, we give a counter example of this conjecture.

Étant donnée une famille de fonctions analytiques en $0\in {\mathbf{C}}^n$ paramétrée par un espace lisse, nous étudions le polynôme de Bernstein de la fibre sur une variété irréductible $V$ de l'espace des paramètres et nous montrons qu'il est génériquement constant. Nous montrons que ce polynôme $b$ satisfait une équation fonctionnelle générique sur $V$ et l'on dérive une stratification constructible de l'espace des paramètres par le polynôme de Bernstein de la fibre. Lorsque l'hypersurface admet génériquement une singularité unique en $0\in {\mathbf{C}}^n$ nous montrons que $b$ est le polynôme de Bernstein générique au sens de Briançon-Geandier-Maisonobe. Les outils utilisés sont une généralisation formelle d'un algorithme de Oaku calculant le polynôme de Bernstein local et les bases standard génériques récemment étudiées par l'auteur.