Global existence of unique strong solutions is proved for the generalized complex Ginzburg-Landau equation. The proof is based on a new type perturbation theorem for $m$-accretive operators in complex Hilbert spaces.

Let $M$ be a von Neumann algebra, let $\alpha$ be a $*$-automorphism of $M$, and let $M{⋊}_{\alpha }Z$ be the crossed product determined by $M$ and $\alpha$. In this paper, considering the Cholesky decomposition for a positive operator in $M{⋊}_{\alpha }Z$, we give a factorization theorem for positive operators in $M{⋊}_{\alpha }Z$ with respect to analytic crossed product $M{⋊}_{\alpha }{Z}_{+}$ determined by $M$ and $\alpha$. And we give a necessary and sufficient condition that every positive operator in $M{⋊}_{\alpha }Z$ can be factored by the form $A^{*}$$A$, where $A$ belongs to $M{⋊}_{\alpha }{Z}_{+}\cap (M{⋊}_{\alpha }{Z}_{+}{)}^{-1}$.

We consider certain correspondences on disjoint unions $\Omega$ of circles which naturally give Hilbert $C^{*}$-bimodules $X$ over circle algebras $A$. The bimodules $X$ generate $C^{*}$-algebras $Ox$ which are isomorphic to a continuous version of Cuntz-Krieger algebras introduced by Deaconu using groupoid method. We study the simplicity and the ideal structure of the algebras under some conditions using (I)-freeness and $\left(II\right)\text{-}$ freeness previously discussed by the authors. More precisely, we have a bijective correspondence between the set of closed two sided ideals of ${\mathcal{O}}_X$ and saturated hereditary open subsets of $\Omega$. We also note that a formula of $K$-groups given by Deaconu is given without any minimality condition by just applying Pimsner's result.

A polygonal description of semisimple tensor categories is presented and the rigidity as well as the associated involutions are analysed in terms of this.

We give a strong unique continuation theorem for time harmonic Maxwell's equations in inhomogeneous anisotropic media of three dimensional space. The components of the matrices in the constitutive relations are supposed to be Hölder continuous. Furthermore, they are supposed to be differentiable except at one point where they may have critical singularities of Coulomb type. These assumptions prevent us to use the usual second order approach, so that we have to utilize a certain nice structure of Maxwell's equations as the first order system.

Menas [13] showed there exist normal ultrafilters on ${\mathcal{P}}_{\kappa }\lambda$ with the partition property if $\kappa$ is -supercompact. We first show that $\lambda$-supercompactness of $\kappa$ implies the existence of a normal ultrafilter on ${\mathcal{P}}_{\kappa }\lambda$ with the partition property. We also prove by a similar technic that part*$(\kappa ,$$\lambda )$ holds if $\kappa$ is $\lambda$-ineffable with $cf\left(\lambda \right)\ge \kappa$. Note that Magidor [11] showed $\kappa$ is $\lambda$-ineffable if part*$(\kappa ,$$\lambda )$ holds, and we proved the converse under some additional assumption in [7].

We compute the dimension group of the skew product extension of a Cantor minimal system associated with a finite group valued cocycle. Using it, we study finite subgroups in the commutant group of a Cantor minimal system and prove that a finite subgroup of the kernel of the mod map must be cyclic. Moreover, we give a certain obstruction for finite subgroups of commutant groups to have nonzero intersection to the kernel of mod maps. We also give a necessary and sufficient condition for dimension groups so that the kernel of the mod map can include a finite order element.

A. Némethi and A. Zaharia have defined the explicit set for a complex polynomial function $f$ : $C^n\to C$ and conjectured that the bifurcation set of the global fibration of $f$ is given by the union of the set of critical values and the explicit set of $f$. They have proved only the case $n=2$ and $f$ is Newton non-degenerate. In the present paper we will prove this for the case $n=2$, containing the Newton degenerate case, by using toric modifications and give an expression of the bifurcation set of $f$ in the words of Newton polygons.

In this note, we attack a question about the injectivity of the forgetful map posed ten years ago by Tsukiyama. We show that we can insert the forgetful map in an exact sequence and that the problem can be reduced to the computation of the sequence which turns out unexpectedly to be related to the phantom map problem and the famous Halperin conjecture in rational homotopy theory.

We give a general definition of the Radon-Penrose transform for a Zuckerman-Vogan derived functor module of a reductive Lie group $G$, which maps from the Dolbeault cohomology group over a pseudo-Kähler homogeneous manifold into the space of smooth sections of a vector bundle over a Riemannian symmetric space. Furthermore, we formulate a functorial property between two Penrose transforms in the context of the Kobayashi theory of discretely decomposable restrictions of unitary representations.

Based on this general theory, we study the Penrose transform for a family of singular unitary representations of $Sp(n,R)$ in details. We prove that the image of the Penrose transform is exactly the space of global holomorphic solutions of the system of partial differential equations of minor determinant type of odd degree over the bounded symmetric domain of type CI, which is biholomorphic to the Siegel upper half space. This system might be regarded as a generalization of the Gauss-Aomoto-Gelfand hypergeometric differential equations to higher order. We also find a new phenomenon that the kernel of the Penrose transform is non-zero, which we determine explicitly by means of representation theory.