We construct approximately inner actions of discrete amenable groups on strongly amenable subfactors of type $I{I}_1$ with given invariants, and obtain classification results under some conditions. We also study the lifting of the relative $\chi$ group.

We consider the group of diffeomorphisms of a compact manifold $M$ which preserve a codimension one foliation $F$ on $M$. For the $C^2$ case if $F$ has compact leaves with nontrivial holonomy then at least one of these leaves is periodic. Our main result is proved in the context of diffeomorphisms which preserve commutative actions of finitely generated groups on $[0,1]$. Applying this result to foliations almost without holonomy we prove the periodicity of all compact leaves with nontrivial holonomy. We also study the codimension one foliation preserving diffeomorphisms that are $C^2$ close to the identity.

We show that a so called split Anosov $R^n$-action on a closed oriented $(2n+1)$-dimensional manifold is $C^{\infty }$ conjugate to the suspension of a split hyperbolic alfine representation of $Z^n$ on the $(n+1)$-dimensional torus.

We construct a model in which the least strongly compact cardinal $\kappa$ is also the least strong cardinaj $\kappa$ isn't $2^{\kappa }$ supercompact, and for any , if ${\delta }^{+\alpha }$ is regular, $\delta$ is ${\delta }^{+\alpha }$ strongly compact if and only if $\delta$ is $\delta +\alpha +1$ strong.

An exponential decay of a stochastic oscillatory integral with phase function determined as a stochastic line integral of a 1-form is studied. A sufficient condition for such an integral to decay exponentially fast is given in terms of the exterior derivative of the 1-form, i.e., the magnetic field.

Rankin obtained the asymptotic formula for the sum of coefficients of Rankin-Selberg $L$-series associated with a cusp form and a trivial character. $Ivi\acute{c}\text{-}$ Matsumoto-Tanigawa studied the error term in it by using a mean value formula which is yielded from the Voronoï formula of the Riesz mean. In this paper, we consider more general Rankin-Selberg $L$-series. It is associated with two cusp forms and a nontrivial character mod $d$. $Ivi\acute{c}$-Matsumoto-Tanigawa's method cannot be applied directly to our case. We consider the sum of coefficients of twisted Rankin-Selberg $L$-series by a modification of their method.

We prove that $b\le 0_2^{const}$ where $b$ is as usual the unbounding number, and $0_2^{const}$ is the constant prediction number, that is, the size of the least family $\Pi$ of functions $\pi$ : such that for each $x\in 2^{\omega }$ there are $\pi \in \Pi$ and $k$ such that for almost all intervals I of length $k$, one has $\pi (x\uparrow i)=x\left(i\right)$ for some $i\in I$. This answers a question of Kamo. We also include some related results.

In this paper, we consider Meyer's function of hyperelliptic mapping class groups of orientable closed surfaces and give certain explicit formulae for it. Moreover we study geometric aspects of Meyer's function, and relate it to the $\eta \text{-}$ invariant of the signature operator and Morita's homomorphism, which is the core of the Casson invariant of integral homology 3-spheres.

In the Jones Index theory, Longo's sector theory has been a powerful approach to the analysis for inclusions of factors and canonical endomorphisms have played an important role. In this paper, two topics on commuting canonical endomorphisms are studied: For a composition of two irreducible inclusions of depth 2 factors, the commutativity of corresponding canonical endomorphisms is shown to be the condition for the ambient irreducible inclusion to be of depth 2, that is, to give a finite dimensional Kac algebra. And an equivalent relation between the commuting $co$-commuting square condition and the existence of two simultaneous commuting canonical endomorphisms is discussed.

We study rationally connected (projective) manifolds $X$ via the concept of a model $(X,Y)$, where $Y$ is a smooth rational curve on $X$ having ample normal bundle. Models are regarded from the view point of Zariski equivalence, birational on $X$ and biregular around $Y$. Several numerical invariants of these objects are introduced and a notion of minimality is proposed for them. The important special case of models Zariski equivalent to $(P^n,line)$ is investigated more thoroughly. When the (ample) normal bundle of $Y$ in $X$ has minimal degree, new such models are constructed via special vector bundles on $X$. Moreover, the formal geometry of the embedding of $Y$ in $X$ is analysed for some non-trivial examples.

In this article we study the inverse of the period map for the family $F$ of complex algebraic curves of genus $6$ equipped with an automorphism of order $5$ having $5$ fixed points. This is a family with $2$ parameters, and is fibred over a Del Pezzo surface. Our period map is essentially same as the Schwarz map for the Appell hypergeometric differential equation $F_1(3/5,3/5,2/5,6/5)$.

This differential equation and the family $F$ are studied by G. Shimura (1964), T. Terada (1983, 1985), P. Deligne and G. D. Mostow (1986) and T. Yamazaki and M. Yoshida (1984). Based on their results we give a representation of the inverse of the period map in terms of Riemann theta constants. This is the first variant of the work of H. Shiga (1981) and K. Matsumoto (1989, 2000) to the co-compact case.

Let $\{{\psi }_r{\}}_{r>0}$ and $\{{\varphi }_r{\}}_{r>0}$ be the families of operator monotone functions on $[0,\infty )$ satisfying ${\psi }_r\left(x^{r}g\right(x\left)\right)=x^r,$${\varphi }_r\left(x^{r}g\right(x\left)\right)=x^{r}h\left(x\right)$, where $g$ and $h$ are continuous and $g$ is increasing. Suppose ${\sigma }_{{\psi }_a}$ and ${\sigma }_{{\varnothing }_r}$ are the corresponding operator connections. We will show that if A${\sigma }_{{\psi }_a}B\ge 1(a>0)$, then $A^r{\sigma }_{{\psi }_r}B$ and $A^r{\sigma }_{{\varnothing }_r}B$ are both increasing for $r\ge a$, and then we will apply this to the geometric operator means to get a simple assertion from which many operator inequalities follow.

In this paper we prove that the maximal operator $M_{\Omega }$, the singular integral operator $T_{\Omega }$, and the maximal singular integral operator $T_{\Omega }^{*}$ with rough kernels are all bounded operators from $L^p\left(v\right)$ to $L^p\left(u\right)$ for the weight functions pair $(u,v)$. Here the kernel function $\Omega$ satisfies a size condition only; that is, $\Omega \in L^q\left(S^{n-1}\right),$$q>1$, but has no smoothness on $S^{n-1}$.

We give a necessary and sufficient condition for a given point on the unit normal bundle of a closed submanifold $N$ of a 2-dimensional complete Riemannian manifold $M$ to be a differentiable point of the distance function to the cut locus of $N$.

Let $G$ be a compact semisimple Lie group, $g$ its Lie algebra, $(,)$ an $A{d}_G$-invariant inner product on $g$, and $M$ an adjoint orbit in 9. In this article, if $(M,(,{)}_{|M})$ is Kähler with respect to its canonical complex structure, then we give, for a closed minimal Lagrangian submanifold $L\subset M$, upper bounds on the first positive eigenvalue ${\lambda }_1\left(L\right)$ of the Laplacian ${\Delta }_L$, which acts on $C^{\infty }\left(L\right)$, and lower bounds on the volume of $L$. In particular, when $(M,(,{)}_{|M})$ is Kähler-Einstein, ($p=c\omega$, where $p$ and $\omega$ are Ricci form and Kähler form of $(M,(,{)}_{|M})$ with respect to the canonical complex structure respectively, and $c$ is a positive constant,) we prove ${\lambda }_1\left(L\right)\le c$. Combining with a result of Oh [5], we can see that $L$ is Hamiltonian stable if and only if ${\lambda }_1\left(L\right)=c$.

The purpose of this paper is to introduce mosaics of log-hyponormal operators and give a Helton-Howe type trace formula.

If we have a finite number of sections of a complex vector bundle $E$ over a manifold $M$, certain Chern classes of $E$ are localized at the singular set $S$, i.e., the set of points where the sections fail to be linearly independent. When $S$ is compact, the localizations define the residues at each connected component of $S$ by the Alexander duality. If $M$ itself is compact, the sum of the residues is equal to the Poincaré dual of the corresponding Chern class. This type of theory is also developed for vector bundles over a possibly singular subvariety in a complex manifold. Explicit formulas for the residues at an isolated singular point are also given, which express the residues in terms of Grothendieck residues relative to the subvariety.