We explicitly describe a candidate for the regulator map from $H^1_{\M} \bigl(\Spec \C , \Z(n) \bigr)$ into $\R$ using analogues of polylogarithms. When $n=2$, the above procedure agrees with the one in the author's paper \cite{MS01}, which was shown to be compatible with Bloch's dilogarithm.

In this paper we establish a generalization of Toponogov's theorem for manifolds with asymptotically nonnegative sectional curvature, and we give a pinching condition under which asymptotically nonnegative curved manifolds are diffeomorphic to $\mathbb{R}^{n}$.

In this article we study the capacity of subanalytic sets. First, we show that a subanalytic set and its closure have the same capacity. Using this, we then prove that for subanalytic sets in ${\mathbb R}^2$ the capacity density exists, and for arbitrary dimension we give connections to certain volume densities. Finally, we connect volume densities with fine limit points of subanalytic sets.

For a stratified pseudomanifold $X$, we have the de Rham Theorem $ \lau{\IH}{*}{\per{p}}{X} = \lau{\IH}{\per{t} - \per{p}}{*}{X}, $ for a perversity $\per{p}$ verifying $\per{0} \leq \per{p} \leq \per{t}$, where $\per{t}$ denotes the top perversity. We extend this result to any perversity $\per{p}$. In the direction cohomology $\mapsto$ homology, we obtain the isomorphism

\[ \lau{\IH}{*}{\per{p}}{X} = \lau{\IH}{\per{t} -\per{p}}{*}{X,\ib{X}{\per{p}}}, \]

where

\[ \ib{X}{\per{p}} = \bigcup_{ S \preceq S_{1} \atop \per{p} (S_{1})< 0}S = \bigcup_{ \per{p} (S)< 0} \overline{S}. \]

In the direction homology $\mapsto$ cohomology, we obtain the isomorphism

\[ \lau{\IH}{\per{p}}{*}{X}=\lau{\IH}{*}{\max ( \per{0},\per{t} -\per{p})}{X}. \]

In our paper stratified pseudomanifolds with one-codimensional strata are allowed.

We introduce the notion of weak reduciblity for Dupin submanifolds with arbitrary codimension. We give a complete characterization of all weakly reducible Dupin submanifolds, as a consequence of a general result on a broader class of Euclidean submanifolds. As a main application, we derive an explicit recursive procedure to generate all holonomic Dupin submanifolds in terms of solutions of completely integrable systems of linear partial differential equations of first order. We obtain several additional results on Dupin submanifolds.

For hyperbolic metric spaces $X_1$, $X_2$ we define and study a one parameter family of ``hyperbolic products'' $Y_{\De}$, $\De \ge 0$, of $X_1$ and $X_2$. In particular, we investigate the relation between the boundaries at infinity of the factor spaces and the boundary at infinity of their hyperbolic products.

Boas' characterization of bounded domains for which the Bochner-Martinelli kernel is self-adjoint is extended to the case of a weighted measure. For strictly convex domains, this equivalently characterizes the ones whose Leray-Aĭzenberg kernel is self-adjoint with respect to weighted measure. In each case, the domains are complex linear images of a ball, and the measure is the Fefferman measure. The Leray-Aĭzenberg kernel for a strictly convex hypersurface in $\mathbb{C}^n$ is shown to be Möbius invariant when defined with respect to Fefferman measure.

For a closed $4$-manifold $X^4$ and closed $3$-manifold $M^3$ we investigate the smallest integer $n$ (perhaps $n=\infty$) such that $M^3$ embeds in $\#_nX^4$, the connected sum of $n$ copies of $X^4$. It is proven that any lens space (or homology lens space) embeds topologically locally flatly in $\#_2({\mathbf C}P^2\#\ \overline {{\mathbf C}P}^2)$, in $\#_4 S^2\times S^2$ and in $\#_8 \mathbf{C}P^2$.

For a Banach space $X$ with an (unconditional) basis, topological 0-1 law type dichotomies are stated for block-subspaces of $X$ as well as for subspaces of $X$ with a successive finite-dimensional decomposition on its basis. A uniformity principle for properties of block-sequences, results about block-homogeneity, and a possible method to construct a Banach space with an unconditional basis which has a complemented subspace without an unconditional basis, are deduced.

The set $V_n$ of $n$-vertices of a tile $T$ in $\R^d$ is the common intersection of $T$ with at least $n$ of its neighbors in a tiling determined by $T$. Motivated by the recent interest in the topological structure as well as the associated canonical number systems of self-similar tiles, we study the structure of $V_n$ for general and strictly self-similar tiles. We show that if $T$ is a general self-similar tile in $\R^2$ whose interior consists of finitely many components, then any tile in any self-similar tiling generated by $T$ has a finite number of vertices. This work is also motivated by the efforts to understand the structure of the well-known L\'evy dragon. In the case $T$ is a strictly self-similar tile or multitile in $\R^d$, we describe a method to compute the Hausdorff and box dimensions of $V_n$. By applying this method, we obtain the dimensions of the set of $n$-vertices of the L\'evy dragon for all $n\ge 1$.

We present a Radon-Nikodým theorem for vector measures of bounded variation that are absolutely continuous with respect to the Lebesgue measure on the unit interval. Traditional Radon-Nikodým derivatives are Banach space-valued Bochner integrable functions defined on the unit interval or some other measure space. The derivatives we construct are functions from $\ster[0,1]$, the nonstandard extension of the unit interval into a nonstandard hull of the Banach space $E$. For these generalized derivatives we have an integral that resembles the Bochner integral. Furthermore, we can standardize the generalized derivatives to produce the weak*-measurable $E''$-valued derivatives that Ionescu-Tulcea, Dinculeanu and others obtained in \cite{8} and \cite{5}.

We establish certain infinite product identities for Dirichlet series twisted by Dirichlet characters and give examples where the products have meromorphic continuation to the whole complex plane.

We prove that the integrated density of states (IDS) associated to a random Schrödinger operator is locally uniformly Hölder continuous as a function of the disorder parameter $\lambda$. In particular, we obtain convergence of the IDS, as $\lambda \rightarrow 0$, to the IDS for the unperturbed operator at all energies for which the IDS for the unperturbed operator is continuous in energy.

We give a necessary and sufficient condition on a sequence of convex sets in $\mathbb{R}^d$ for the corresponding sequence of measures to be a summing sequence.

We study the minimal graph equation in a Riemannian manifold. After explaining the geometric meaning of the solutions and giving some entire solutions of the minimal graph equation in Nil space and in a hyperbolic space we find a link among $p$-harmonicity, horizontal homothety, and the minimality of the vertical graphs of a submersion. We also study the transformation of the minimal graph equation under the conformal change of metrics. We prove that the foliation by the level hypersurfaces of a $p$-harmonic submersion is a minimal foliation with respect to a conformally deformed metric. This implies, in particular, that the graph of any harmonic function from a Euclidean space is a minimal hypersurface in a complete conformally flat space, thus providing an effective way to construct (foliations by) minimal hypersurfaces.

We show that the monotonic independence introduced by Muraki can also be used to define a multiplicative convolution. We also find a method for the calculation of this convolution based on an appropriate form of the Cauchy transform. Finally, we discuss infinite divisibility in the multiplicative monotonic context.

In this paper we introduce a new tool called ``Lipschitz triangulations'', which gives combinatorially all information about the metric type. We show the existence of such triangulations for semi-algebraic sets. As a consequence we obtain a bi-Lipschitz version of Hardt's theorem. Hardt's theorem states that, given a family definable in an o-minimal structure, there exists (generically) a trivialization which is definable in this o-minimal structure. We show that, for a polynomially bounded o-minimal structure, there exists such an isotopy which is bi-Lipschitz as well.

A four part dissection and rearrangement provides a new proof of the isoperimetric inequality in the plane as well as a new approach to Bonnesen-type error estimates for the isoperimetric deficit of compact convex sets and of star bodies that are centrally symmetric with respect to the origin.

In this note we extend the spectral theorem for bimodules to the higher rank graph $C^*$-algebra context. Under the assumption that the graph is row finite and has no sources, we show that a bimodule over a natural abelian subalgebra is determined by its spectrum iff it is generated by the Cuntz-Krieger partial isometries which it contains iff the bimodule is invariant under the gauge automorphisms. We also show that the natural abelian subalgebra is a masa iff the higher rank graph satisfies an aperiodicity condition.

A necessary condition is given for certain $L^2$ estimates to hold for the boundary Cauchy Riemann operator and the associated Kohn Laplacian on abstract $CR$ manifolds of arbitrary codimension. A circle of ideas is presented related to a priori estimates for linear partial differential operators.