In the ring $\mathbf{Z}[\frac{1+\sqrt{5}}{2}]$, there is a natural subdivision technique analogous to regular subdivision in rational algebraic rings like $\mathbf{Z}[\frac{1}{2}]$. The properties of this subdivision process are developed using the matrix associated to the Fibonacci substitution tiling. These properties are applied to prove some finiteness properties for a discrete group of piecewise-linear homeomorphisms.

We begin a systematic study of Euclidean piecewise isometric dynamical systems (p.i.d.s.) with a particular focus on the interplay between geometry, symbolic dynamics, and the group of isometries associated with p.i.d.s. We investigate various aspects of the dynamical information contained in the coding: symbolic growth and the periodic behavior of codings and cells. This theoretical investigation is motivated by the many examples of piecewise isometric dynamical systems found recently in the literature. Piecewise isometric dynamical systems are direct generalizations of interval exchange transformations to non-invertible, higher dimensional maps.

Let $M$ be a closed, oriented $4$-manifold with $b^{\pm}_{2} \gt 0$. In this paper we show that the space of transverse intrinsically harmonic $2$-forms in a fixed cohomology class is open in the space of closed $2$-forms, subject to a condition which arises from cohomological considerations of a singular differential ideal.

We consider operators $T^{\delta}$ associated with a localized height of cone multipliers. It is shown that $T^{\delta}$ is of weak type $(p,p)$ for the functions of the form $f(x,t)=g(|x|,t)$ if $p = 2n/(n +1+2\delta)$, $0 \lt \delta \lt (n- 1)/2$.

In [6], Shub and Sullivan posed the problem of finding complete measure theoretic invariants for analytic Lebesgue measure preserving expanding endomorphisms of $S^{1}$. In [2], the author gave necessary and sufficient conditions for two such endomorphisms to be isomorphic. These complete invariants were a mixture of a topological and measure-theoretic nature. Establishing them required finding a coboundary equation with no obvious method of construction. In this note we use a result by Arteaga to furnish a different set of complete isomorphism invariants, still of a mixed topological and measure-theoretic nature but far more easily checked than the ones established in [2].

Let $f$ be a transcendental meromorphic function and let $U$ be a wandering domain of $f$. Under some conditions, we prove that a finite limit function of $\{f^{n}\}$ on $U$ is in the derived set of the forward orbit of the set sing $(f^{-1})$ of singularities of the inverse function of $f$. The existence of $\{n_{k}\}$ such that $f^{n_k}}|_{U}$ tends to $\infty$ is also considered when $f$ is entire. If sing$(f^{-l})$ is bounded, however, we show that $\{f^{n}(z)\}_{n=o}^\infty$ in $F(f)$ does not tend to $\infty$.

We investigate the minimal free resolutions of cyclic modules $R/I$, where $I$ is an almost complete intersection in the local ring $R$. Our results concern various binomial lower bounds for the Betti numbers of the resolution. For example, we show that the sum of the Betti numbers is at least $2^{d}$ where $d$ is the dimension of $R$.

A conjecture of Fuglede states that a bounded measurable set $\Omega \subset \mathbb{R}^{d}$, of measure 1, can tile $\mathbb{R}^{d}$ by translations if and only if the Hilbert space $L^{2}(\Omega)$ has an orthonormal basis consisting of exponentials $e_{\lambda}(x)=\exp 2\pi i \langle \lambda,x \rangle$. If $\Omega$ has the latter property it is called spectral. We generalize a result of Fuglede, that a triangle in the plane is not spectral, proving that every non-symmetric convex domain in $\mathbb{R}^{d}$ is not spectral.

If $f$ is a canonical product with only real negative zeros and non-integral order $\rho,n(t,0)$ is the zero counting function, and $B(r,f)=\mathrm{sup}_{0 \lt \theta \lt \pi}|\log f(re^{i \theta})|$, then $$r^{-q-1}B(r,f) \leq \pi\{M \varphi(r) + MH\varphi(r)\}+\int_{0}^{\infty}{\frac{\varphi(t)dt}{t+r}},$$ where $\varphi(t) = t^{-q-1} n(t,0)$, $H$ is the Hilbert transform operator and $M$ is the Hardy-Littlewood maximal operator.

We establish global bounds for the heat kernel of Schrödinger operators $-\Delta + V$ where $V$ is a certain long range potential. As a consequence we find some conditions for the heat kernel to have global Gaussian lower and upper bound. Some of the conditions are sharp if the potential does not change sign. We also provide a generalized Liouville theorem for Schrödinger operators and a refined version of the trace formula of Sa Barreto and Zworski [SZ].

We give a necessary and sufficient condition for an integrable compactly supported function with mean value zero on the line to be in the Hardy space $H^{1}(\mathbf{R}^{1})$. As a corollary, we obtain a new characterization of $H^{1}(\mathbf{S}^{1})$ and $p$ independence of the spectrum of homogeneous Calderón-Zygmund operators.

In this paper we treat a linear differential equation with doubly periodic coefficients. We examine value distribution properties of meromorphic solutions. Some examples are presented to illustrate our results.

We study the possible dimensions of the groups of holomorphic automorphisms of hyperbolic Reinhardt domains. We are particularly interested in the problem of characterizing Reinhardt domains with automorphism group of prescribed dimension.

We say that a computably enumerable (c.e., for short) degree $\mathbf{a}$ is cuppable if there is a c.e. degree $\mathbf{b} \neq \mathbf{0}'$ such that $\mathbf{a} \vee \mathbf{b} \mathbf{0}'$. A c.e. degree $\mathbf{a}$ is called $low_{n}$-cuppable, $n \gt 0$, if there is a $\mathrm{low}_{n}$ c.e. degree $\mathbf{1}$ such that $\mathbf{a} \vee \mathbf{1} \mathbf{0}'$. Let $\mathbf{LC}_{n}$, be the set of all $\mathrm{low}_{n}$-cuppable c.e. degrees. In this paper, we show that $\mathbf{LC}_{1} \subset \mathbf{LC}_{2} \subseteq \mathbf{LC}_{3} \subseteq \cdots$, so giving a hierarchy for a class of cuppable degrees.

Let $f(x)$ and $g(x)$ be two relatively prime polynomials having integer coefficients with $g(0)\neq 0$. The authors show that there is an $N=N(f,g)$ such that if $n \geq N$, then the non-reciprocal part of the polynomial $f(x)x^{n}+g(x)$ is either irreducible or identically 1 or $-1$ with certain clear exceptions that arise from a theorem of Capelli. A version of this result is originally due to Andrzej Schinzel. The present paper gives a new approach that allows for an improved estimate on the value of $N$.

We introduce a homology theory with supports and with coefficients in a sheaf. It has a very explicit description of the chains in terms of a triangulation of an ambient space, making the theory useful for integration purposes. We prove a Poincaré Duality Theorem that states that our homology modules are isomorphic to the classical sheaf cohomology modules with supports. This theorem is a main ingredient in the proof of a criterion on the vanishing of real principal value integrals in terms of cohomology. We briefly explain how real principal value integrals appear as residues of poles of distributions $|f|^{s}$ and as coefficients of asymptotic expansions of oscillating integrals.

We study the a.e. convergence of the Cesàro-$(1+\alpha)$ ergodic averages and the a.e. existence in the Cesàro-$\alpha$ sense of the ergodic Hilbert transform associated with Cesàro bounded flow and $-1\lt \alpha \leq 0$.

We prove some geometric estimates for the equilibrium measure and the capacity of certain condensers. The proofs are based on the interpretation of the equilibrium measure as the distributional Laplacian of the corresponding potential, on a formula of T. Bagby, and on a method of Beurling and Nevanlinna that involves the transport of the Riesz mass of a superharmonic function.