The derived series for all co-compact non-perfect Fuchsian groups are investigated. These groups are residually finite and residually soluble. The intersection of the derived series for these groups is the identity. We will show that if $\Gamma$ is not perfect, then the number of terms in the derived series up to and including the first surface group cannot exceed $4$. We then use this result to compute the derived series of some important general triangle groups.

Let $f:M \to \mathbf{R}^2$ be a stable map of a closed surface $M$ into the plane and $\pi^2_2:\mathbf{R}^4 \to \mathbf{R}^2$ the orthogonal projection. In this paper, we will show that for any such $f$ there exists an embedding $F:M \to \mathbf{R}^4$ such that $f=\pi^2_2 \circ F$ is satisfied.

Minimization proofs using paired calibrations have in the past been done with vector fields of divergence zero. We generalize this method to find the shortest network connecting four points in the hyperbolic plane.

An upper bound for $\|D^{\beta}u\|_q$ in terms of other similar norms $\|D^{\alpha}u\|_p$ is derived for vector-valued test functions $u\in C_c^{\infty}(\mathbf{R}^n,X)$, where $X$ is a Banach space with the UMD property. This gives a new proof and an extension of a classical result of Besov-Il'in-Nikol'skiĭ for scalar functions.

A subset $\mathcal{P}$ of $\mathbb{N}^2$ is called Schur bounded if every infinite matrix with bounded scalar entries which is zero off of $\mathcal{P}$ yields a bounded Schur multiplier on $\mathcal{B}(\mathcal{H})$. Such sets are characterized as being the union of a subset with at most $k$ entries in each row with another that has at most $k$ entries in each column, for some finite $k$. If $k$ is optimal, there is a Schur multiplier supported on the pattern with norm $O(\sqrt k)$, which is sharp up to a constant. The same characterization also holds for operator-valued Schur multipliers in the cb-norm, i.e., every infinite matrix with bounded \emph{operator} entries which is zero off of $\mathcal{P}$ yields a completely bounded Schur multiplier. This result can be deduced from a theorem of Varopoulos on the projective tensor product of two copies of $l^\infty$. Our techniques give a new, more elementary proof of his result. We also consider the Schur multipliers for certain matrices which have a large symmetry group. In these examples, we are able to compute the Schur multiplier norm exactly. This is carried out in detail for a few examples including the Kneser graphs.

For every $1 < p < \infty$ we construct an asymptotic $\ell_{p}$ Banach space which is hereditarily indecomposable and such that its dual is asymptotic $\ell_{q}$ hereditarily indecomposable, where $q$ is the conjugate of $p$. We prove that $c_{0}$ is finitely representable in these spaces and that every bounded linear operator on these spaces is a strictly singular perturbation of a multiple of the identity.

A network is a countable, connected graph $X$ viewed as a one-complex, where each edge $[x,y]=[y,x]$ ($x,y\in X^0$, the vertex set) is a copy of the unit interval within the graph's one-skeleton $X^1$ and is assigned a positive conductance $\mathsf{c}(xy)$. A reference "Lebesgue" measure $X^1$ is built up by using Lebesgue measure with total mass $\mathsf{c}(xy)$ on each edge $xy$. There are three natural operators on $X$: the transition operator $P$ acting on functions on $X^0$ (the reversible Markov chain associated with $\mathsf{c}$), the averaging operator $A$ over spheres of radius~1 on $X^1$, and the Laplace operator $\Delta$ on $X^1$ (with Kirchhoff conditions weighted by $\mathsf{c}$ at the vertices). The relation between the $\ell^2$-spectrum of $P$ and the $H^2$-spectrum of~$\Delta$ was described by {Cattaneo} \cite{Cat}. In this note we describe the relation between the $\ell^2$-spectrum of $P$ and the $L^2$-spectrum of $A$.

We prove both local and global two-weight Poincaré inequalities for A-harmonic tensors on Riemannian manifolds. We also prove an analogue for the projection operator.

A Gotzmann monomial ideal of a polynomial ring is a monomial ideal which is generated in one degree and which satisfies Gotzmann's persistence theorem. Let $R=K[x_1,\dots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $M^d$ the set of monomials of $R$ of degree $d$. A subset $V\subset M^d$ is said to be a Gotzmann subset if the ideal generated by $V$ is a Gotzmann monomial ideal. In the present paper, we find all integers $a > 0$ such that every Gotzmann subset $V\subset M^d$ with $|V|=a$ is lexsegment (up to the permutations of the variables). In addition, we classify all Gotzmann subsets of $K[x_1,x_2,x_3]$.

We study an exact solution to a singular ordinary differential equation and use the solution to give a new estimate on the lower bound of the first non-zero eigenvalue of a closed Riemannian manifold with a negative lower bound on the Ricci curvature in terms of the lower bound on the Ricci curvature and the largest interior radius of the nodal domains of the eigenfunction. This provides a new way to estimate eigenvalues.

In this paper we give upper bounds for a certain terminating ${}_4F_3$ series. Our estimates confirm special cases of a conjecture of Kresch and Tamvakis. We also give asymptotic estimates when the parameters in the ${}_4F_3$ series are large, and they confirm the same conjecture.

We study the relations between boundaries for algebras of holomorphic functions on Banach spaces and complex convexity of their balls. In addition, we show that the Shilov boundary for algebras of holomorphic functions on an order continuous sequence space $X$ is the unit sphere $S_X$ if $X$ is locally c-convex. In particular, it is shown that the unit sphere of the Orlicz-Lorentz sequence space $\lambda_{\varphi, w}$ is the Shilov boundary for algebras of holomorphic functions on $\lambda_{\varphi, w}$ if $\varphi$ satisfies the $\delta_2$-condition.

Let $T$ be an $\R$-tree with a very small action of a free group $F_N$ which has dense orbits. Such a tree $T$ or its metric completion $\bar T$ are not locally compact. However, if one adds the Gromov boundary $\partial T$ to $\bar T$, then there is a coarser \emph{observers' topology} on the union $\bar T \cup \partial T$, and it is shown here that this union, provided with the observers' topology, is a compact space $\Tobs$.

To any $\R$-tree $T$ as above a \emph{dual lamination} $L^{2}(T)$ has been associated in \cite{chl1-II}. Here we prove that, if two such trees $T_{0}$ and $T_{1}$ have the same dual lamination $L^{2}(T_{0}) = L^{2}(T_{1})$, then with respect to the observers' topology the two trees have homeomorphic compactifications: $\Tobszero = \Tobsone$. Furthermore, if both $T_{0}$ and $T_{1}$, say with metrics $d_{0}$ and $d_{1}$, respectively, are minimal, this homeomorphism restricts to an $\FN$-equivariant bijection ${T_{0}} \to {T_{1}}$, so that on the identified set $ {T_{0}} = {T_{1}}$ one obtains a well defined family of metrics $\lambda d_{1}+(1-\lambda)d_{0}$. We show that for all $\lambda \in[0,1]$ the resulting metric space $T_{\lambda}$ is an $\R$-tree.

It is shown that the second eigenvalue of the Laplacian with either Dirichlet or Neumann boundary conditions in an annulus in a Euclidean space, or in a sphere, or in a hyperbolic space of dimension $n>1$ has multiplicity $n$.

If $A$ is an ordered Banach algebra ordered by an algebra cone $C$, then we reference the following problem as the `domination problem': If $0\leq a\leq b$ and $b$ has a certain property, then does $a$ inherit this property? We extend the analysis of this problem in the setting of radical elements and introduce it for inessential, rank one and finite elements. We also introduce the class of $r$-inessential operators on Banach lattices and prove that if $S$ and $T$ are operators on a Banach lattice $E$ such that $0\leq S\leq T$ and $T$ is $r$-inessential then $S$ is also $r$-inessential.

We consider two pairs, the standard unknotted $n$-sphere in $S^{n+2}$, and the product of two $p$-spheres trivially embedded in $S^{2p+2}$, and study orientation preserving diffeomorphisms of these pairs. Pseudo-isotopy classes of such diffeomorphisms form subgroups of the mapping class groups of $S^n$ and $S^p\times S^p$, respectively, and we determine the algebraic structure of such subgroups when $n>4$ and $p>1$.

The Cauchy problem $u_t - |D|^{\alpha}u_x + uu_x=0$ in $(-T,T) \times \mathbb{R}$, $u(0)=u_0$, is studied for $1< \alpha <2$. Using suitable spaces of Bourgain type, local well-posedness for initial data $u_0 \in H^s(\mathbb{R}) \cap \dot{H}^{-\omega}(\mathbb{R})$ for any $s > -\tfrac{3}{4}(\alpha-1)$, $\omega:=1/\alpha-1/2$ is shown. This includes existence, uniqueness, persistence, and analytic dependence on the initial data. These results are sharp with respect to the low frequency condition in the sense that if $\omega<1/\alpha-1/2$, then the flow map is not $C^2$ due to the counterexamples previously known. By using a conservation law, these results are extended to global well-posedness in $H^s(\mathbb{R}) \cap \dot{H}^{-\omega}(\mathbb{R})$ for $s \geq 0$, $\omega=1/\alpha-1/2$, and real valued initial data.

We introduce a class of Möbius invariant spaces of analytic functions in the unit disk, characterize these spaces in terms of Carleson type measures, and obtain a necessary and sufficient condition for a lacunary series to be in such a space. Special cases of this class include the Bloch space, the diagonal Besov spaces, BMOA, and the so-called $Q_p$ spaces that have attracted much attention lately.

This paper provides an elementary construction of the moduli space of stable maps $\overline{M}_{0,0}(\mathbb{P}^r,d)$ as a sequence of "weighted blow-ups along regular embeddings" of a projective variety. This is a corollary to a more general GIT construction of $\overline{M}_{0,n}(\mathbb{P}^r,d)$ that places stable maps, the Fulton-MacPherson space $\mathbb{P}^1[n]$, and curves $\overline{M}_{0,n}$ into a single context.

The Riemann Mapping Theorem asserts that there is an analytic function in the unit disk taking its boundary values on any given simple smooth curve in the plane. A theorem of Szegö says that an analytic function in the disk can have boundary values lying on prescribed circles about the origin. We shall prove that the circles of Szegö's theorem can be replaced by other curves, giving a common generalization of these theorems.

We point out two minor errors in our earlier paper.

In this short note we point out two errors in our paper "Covers of groups definable in o-minimal structures" [Illinois J. Math. 49 (2005), 99–120], and we show how to correct these errors.