The subject of this paper is to study the structure of the Eisenstein component of Hida’s universal ordinary $p$-adic Hecke algebra attached to modular forms (rather than cusp forms). We give a sufficient condition for such a ring to be Gorenstein in terms of companion forms in characteristic $p$; and also a numerical criterion which assures the validity of that condition. This type of result was already obtained in our previous work, in which two cases were left open. The purpose of this work is to extend our method to cover these remaining cases. New ingredients of the proof consist of: a new construction of a pairing between modular forms over a finite field; and a comparison result for ordinary modular forms of weight two with respect to ${\Gamma }_1(N)$ and ${\Gamma }_1(N)\bigcap {\Gamma }_0(p)$. We also describe the Iwasawa module attached to the cyclotomic $Z_p$-extension of an abelian number field in terms of the Eisenstein ideal, when an appropriate Eiesenstein component is Gorenstein.

For a semialgebraic mapping between semialgebraic sets, we consider the set of points at which the fibre is not smooth. In this paper we discuss whether the singular set is itself semialgebraic, when it has codimension bigger than or equal to 2 in the domain of $f$ and whether the mapping is semialgebraically trivial along the smooth part of the fibre, giving several examples which show optimality of those results. In addition, we give an example of a polynomial function $f$ such that even the $(a_f)$ condition in the weak sense fails in a neighbourhood of a smooth fibre, but $f$ is semialgebraically trivial along it.

We consider one-dimensional generalized diffusion processes (ODGDPs for brief), where both boundary points are accessible or asymptotically accessible. For such ODGDPs we consider stochastic processes induced by conditioning on hitting or asymptotical hitting the right boundary point before hitting or asymptotical hitting the left boundary point. The induced stochastic processes are again ODGDPs when the right boundary point is either accessible with the absorbing boundary condition or asymptotically accessible. However the probability distributions of the induced stochastic processes do not satisfy the Chapman-Kolmogorov equation when the right boundary point is accessible with the reflecting or elastic boundary condition.

The theory of exact $C^{*}$-algebras was introduced by Kirchberg and has been influential in recent development of $C^{*}$-algebras. A fundamental result on exact $C^{*}$-algebras is a local characterization of exactness. The notion of weakly exact von Neumann algebras was also introduced by Kirchberg. In this paper, we give a local characterization of weak exactness. As a corollary, we prove that a discrete group is exact if and only if its group von Neumann algebra is weakly exact. The proof naturally involves the operator space duality.

It is shown that for the inclusion of factors $(B\subseteq A):=(W^{*}(\mathit{S},\omega )\subseteq W^{*}(\mathit{R},\omega ))$ corresponding to an inclusion of ergodic discrete measured equivalence relations $\mathit{S}\subseteq \mathit{R}$, $\mathit{S}$ is normal in $\mathit{R}$ in the sense of Feldman-Sutherland-Zimmer ([9]) if and only if $A$ is generated by the normalizing groupoid of $B$. Though this fact has been already obtained in [3], we reprove it here by a quite different method.

We give some results about the product $supu\times infu$ for some elliptic operators of order 2 and 4. If we use those inequalities and the concentration phenomena, we can describe the asymptotic behavior of some nonlinear PDE of Yamabe type.

We show that if a group $G$ admits a finite dimensional contractible $G$-CW-complex $X$ then the vanishing of the $L^2$-Betti numbers for all stabilizers $G_\sigma$ of $X$ determines that of the $L^2$-Betti numbers for $G$. We also give a relation among the $L^2$-Euler characteristics for $X$ as a $G$-CW-complex and those for $X$ as a $G_\sigma$-CW-complex under certain assumptions. Finally, we present a new class of groups satisfying the Chatterji-Mislin conjecture which amounts to putting Brown's formula within the framework of $L^2$-homology.

Suppose that $\kappa$ is a regular uncountable cardinal. It has been known that the club filter on ${\mathit{P}}_{{\omega }_1}\kappa$ can be presaturated. In this paper we extend the result to the case of ${\mathit{P}}_{\mu }\kappa$, where $\mu$ is a regular uncountable cardinal $\le \kappa$. This involves suitably weakening the notion of presaturation. A new reflection principle for stationary sets in ${\mathit{P}}_{\kappa }\lambda$ plays a key role.

The first half of this paper (Section 1) deals with the structure of the twisted homology group associated to the Wirtinger integral. A basis of the first homology group is given, and the vanishing of the homology groups of the other dimensions is proved. The second half (Section 2) deals with the structure of the twisted cohomology groups associated to the Wirtinger integral. The isomorphism between the twisted cohomology groups and the cohomology groups associated to a certain subcomplex of the de Rham complex is established, and a basis of the first cohomology group of this subcomplex (therefore, of the first twisted cohomology group) is given. The vanishing of the cohomology groups of the other dimensions is also proved.

Let $G$ be an exponential solvable Lie group, and $\pi$ be an irreducible unitary representation of $G$. Then by induction from a unitary character of a connected subgroup, $\pi$ is realized in an $L^2$-space of functions on a homogeneous space. We are concerned with $C^\infty$vectors of $\pi$ from a viewpoint of rapidly decreasing properties. We show that the subspace $\mathscr{PE}$ consisting of vectors with a certain property of rapidly decreasing at infinity can be embedded as the space of the $C^\infty$vectors in an extension of $\pi$ to an exponential group including $G$. Using the space $\mathscr{PE}$, we also give a description of the space $\mathscr{APE}$ related to Fourier transforms of $L^1$-functions on $G$. We next obtain an explicit description of $C^\infty$vectors for a special case. Furthermore, we consider a space of functions on $G$ with a similar rapidly decreasing property and show that it is the space of the $C^\infty$vectors of an irreducible representation of a certain exponential solvable Lie group acting on $L^2(G)$.

In this paper: i) We compute the leafwise cohomology of a complete Riemannian Diophantine flow. ii) We solve explicitly the discrete cohomological equation for the Anosov diffeomorphism on the torus $T^n$ defined by a hyperbolic and diagonalizable matrix $A\in \mathit{SL}(n,Z)$ whose eigenvalues are all real positive numbers. We use this to solve the continuous cohomological equation of the Anosov flow $\mathit{F}$ on the hyperbolic torus $T_A^{n+1}$ obtained from $A$ by suspension. This enables us to compute some other geometrical objects associated to the diffeomorphism $A$ and the foliation $\mathit{F}$ like the invariant distributions and the leafwise cohomology.

We give a method of determining the centralizer of an elliptic element in a real semisimple Lie algebra $\mathfrak{g}$, in relation with the maximal compact subalgebra of $\mathfrak{g}$ and the compact dual of $\mathfrak{g}$. Moreover, we determine a special central element (called the $H$-element) of the isotropy subalgebra of each simple irreducible pseudo-Hermitian symmetric Lie algebra.

A generalized Calabi-Yau structure is a geometrical structure on a manifold which generalizes both the concept of the Calabi-Yau structure and that of the symplectic one. In view of a result of Lin and Tolman in generalized complex cases, we introduce in this paper the notion of a generalized moment map for a compact Lie group action on a generalized Calabi-Yau manifold and construct a reduced generalized Calabi-Yau structure on the reduced space. As an application, we show some relationship between generalized moment maps and the Bergman kernels, and prove the Duistermaat-Heckman formula for a torus action on a generalized Calabi-Yau manifold.

A word generated by coding of irrational rotation with respect to a general decomposition of the unit interval is shown to have an inverse limit structure directed by substitutions. We also characterize primitive substitutive rotation words, as those having quadratic parameters.