We consider the intersection local time of Brownian motion without renormalization through Itô-Wiener expansions. In order to recognize the existence, we extend the Watanabe space. We also discuss how to substitute Wiener functionals for parameters of a generalized Wiener functional. As a consequence a relationship between the unrenormalized intersection local time and the local time is clarified.

We investigate the spectrum perturbations and spectrum localization of a class of operators on a tensor product of separable Hilbert spaces. In particular, estimates for the spectral radius and norm of the resolvent are derived. Applications to partial integral and integrodifferential operators are also discussed.

We shall study isothermal gas dynamics with spherical symmetry. In this case, existence theorems have been obtained outside a solid ball. However, little is known for the case including the origin, because the equation has a singularity there. In this paper, we will present discontinuous solutions for this case, by introducing a certain non-homogeneous conservation laws and using a modified Glimm’s scheme.

The purpose of this paper is to prove the truth of the conjecture in [12]: The suspension order of the real even dimensional projective space coincides with its stable order determined by Toda [21] (see Silberbush and Ucci [19]). We obtain the assertion by proving that the suspension order of the real 6-projective space is 8.

Let $\chi$ be the inverse of the Grötzsch modulus function and let $\sigma _{n}$ be the $n$-th iteration of the function $\sigma (r) = 2\sqrt{r}/(1 + r)$, $r > 0$. For a real constant $\beta \neq 0$ with $\beta >-2$, the difference $\chi (x)^{\beta}-\sigma _{n}(4e^{-2^{n}x})^{\beta}$ is estimated. In the particular case where $\beta =-2$ one has an approximation of the inverse $S$ of the Teichmüller modulus function, which is applied to improving the known upper and lower estimates concerning the error term of $\lambda (K) = \chi (\pi K/2)^{-2} -1$ from $16^{-1}e^{\pi K}-2^{-1}$ for the variable $K \geqslant 1$. Expressions of $\chi$ and $S$ in terms of theta functions are studied. Lipschitz continuity of $f$ or log $f$ for $f = \chi , S$, as well as other functions are proved.

Let $S$ be a Noetherian scheme, $\varphi : X \to Y$ a surjective $S$-morphism of $S$-schemes, with $X$ of finite type over $S$. We discuss what makes $Y$ of finite type.

First, we prove that if $S$ is excellent, $Y$ is reduced, and $\varphi$ is universally open, then $Y$ is of finite type. We apply this to understand Fogarty’s theorem in “Geometric quotients are algebraic schemes, Adv. Math. 48 (1983), 166-171” for the special case that the group scheme $G$ is flat over the Noetherian base scheme $S$ and that the quotient map is universally submersive. Namely, we prove that if $G$ is a flat $S$-group scheme of finite type acting on $X$ and $\varphi$ is its universal strict orbit space, then $Y$ is of finite type ($S$ need not be excellent. Geometric fibers of $G$ can be disconnected and non-reduced).

Utilizing the technique used there, we also prove that $Y$ is of finite type if $\varphi$ is flat. The same is true if $S$ is excellent, $\varphi$ is proper, and $Y$ is Noetherian.

In the previous paper [1], the author investigated the group structure of the homotopy set $[X,U(n)]$ with the pointwise multiplication, under the assumption that $X$ is a finite CW complex with its dimension $2n$ and $U(n)$ is the unitary group, and showed that $[X,U(n)]$ is an extension of $\Tilde{K}^{1}(X)$ by $N_{n}(X)$, where $N_{n}(X)$ is a group defined as the cokernel of a map $\Theta : \Tilde{K}^{0}(X)\to \mathrm{H}^{2n}(X;\mathbf{Z})$. In this paper, we offer another interpretation of $N_{n}(X)$ using Adams $e$-invariant and show that the extension $N_{n}(X) \to U_{n}(X) \to \Tilde{K}^{1}(X)$ is determined by some Toda brackets. Also we give some applications including the calculation of $[SO(4), U(3)]$.