Denote by $\mathcal{A}\left(n\right)$ the family of isometry classes of compact n-dimensional Alexandrov spaces with curvature $\ge -1$, and ${\lambda }_k\left(M\right)$ the $k^{th}$ eigenvalue of the Laplacian on $M\in \mathcal{A}\left(n\right)$. We prove the continuity of ${\lambda }_k:\mathcal{A}\left(n\right)$$\to R$ with respect to the GromovHausdorff topology for each $k,$$n\in N$, and moreover that the spectral topology introduced by Kasue-Kumura [7], [8] coincides with the Gromov-Hausdorff topology on $\mathcal{A}\left(n\right)$.

We consider quasilinear strongly resonant problems with discontinuous right hand side. To develop an existence theory we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points. We prove the existence of at least three nontrivial solutions. Our approach uses the nonsmooth critical point theory for locally Lipschitz functionals due to Chang and a generalized version of the Ekeland variational principle. At the end of the paper we also show that the nonsmooth (PS)-condition implies the coercivity of the functionaj extending this way a well known result of the "smooth" case.

We call a function from to $\omega$ a predictor. A function $f\in {\omega }^{\omega }$ is said to be constantly predicted by a predictor $\pi$, if there is an such that . Let ${\theta }_{\omega }$ denote the smallest size of a set $\Phi$ of predictors such that every $f\in {\omega }^{\omega }$ can be constantly predicted by some predictor in $\Phi$. In [7], we showed that ${\theta }_{\omega }$ may be greater than $cof\left(\mathcal{N}\right)$. In the present paper, we will prove that ${\theta }_{\omega }$ may be smaller than $d$.

We study convolutions with Oseen kernels (weakly singular and singular) in both two- and three-dimensional space. We give a detailed weighted $L^p$ theory for $p\in (1;\infty$] for anisotropic weights.

The aim of this note is to generalize the Liouville theorem to polyharmonic functions $u$ on $R^n$. We give a condition on spherical means to assure that $u$ is a polynomial.

We establish a uniform boundary Harnack principle for a uniform domain. As applications we study the Hölder continuity of the ratios of positive harmonic functions, the Martin boundary and the Fatou theorem for a uniform domain.

In this paper, we consider the realization problem of amenable bicategories by using AFD-bimodules. As a corollary, we can realize an amenable standard lattice as that of an AFD $I{I}_1$-subfactor. This gives another proof of S. Popa's theorem. Weak amenability of fusion algebras is also discussed.

This paper describes a generalisation of the methods of Iwasawa Theory to the field $F_{\infty }$ obtained by adjoining the field of definition of all the $p$-power torsion points on an elliptic curve, $E$, to a number field, $F$. Everything considered is essentially well-known in the case $E$ has complex multiplication, thus it is assumed throughout that $E$ has no complex multiplication. Let $G_{\infty }$ denote the Galois group of $F_{\infty }$ over $F$. Then the main focus of this paper is on the study of the $G_{\infty }$-cohomology of the $p^{\infty }\text{-}$ Selmer group of $E$ over $F_{\infty }$, and the calculation of its Euler characteristic, where possible. The paper also describes proposed natural analogues to this situation of the classical Iwasawa $\lambda$-invariant and the condition of having $\mu$-invariant equal to 0.

The final section illustrates the general theory by a detailed discussion of the three elliptic curves of conductor 11, at the prime $p=5$.

Let $f$ be a homogeneous polynomial in two variables such that on its graph $G_f$, the origin $0=(0,0,0)$ of $R^3$ is an isolated umbilical point. In this paper, the behavior of the principal distributions around $0$ is studied in relation to the existence of other umbilical points than $0$ and the behavior of the gradient vector field of $f$.