Related papers: Quantitative bounds on integrality for post-critic…
Let $K$ be an algebraically closed field of characteristic zero, and for $c \in K$ and an integer $d \ge 2$, define $f_{d,c}(z) := z^d + c \in K[z]$. We consider the following question: If we fix $x \in K$ and integers $M \ge 0$, $N \ge 1$,…
We prove new parameterization theorems for sets definable in the structure $\mathbb{R}_{an}$ (i.e. for globally subanalytic sets) which are uniform for definable families of such sets. We treat both $C^r$-parameterization and (mild)…
Given a rational map $\phi: {\mathbb P}^1\to {\mathbb P}^1$ defined over a number field $K$, we prove a finiteness result for $\phi$-preperiodic points which are $S$-integral with respect to a non-preperiodic point $P$, provided $P$…
The order parameter of a critical system defined in a layered parallel plate geometry subject to Neumann boundary conditions at the limiting surfaces is studied. We utilize a one-particle irreducible vertex parts framework in order to study…
Let $E$ be an elliptic curve defined over a number field $K$. We say that a prime number $p$ is exceptional for $(E,K)$ if $E$ admits a $p$-isogeny defined over $K$. The so-called exceptional set of all such prime numbers is finite if and…
We study the quadratic integral points-that is, (S-)integral points defined over any extension of degree two of the base field-on a curve defined in P_3 by a system of two Pell equations. Such points belong to three families explicitly…
Let $f$ be a generically finite polynomial map $f: \mathbb{C}^n\to \mathbb{C}^m$ of algebraic degree $d$. Motivated by the study of the Jacobian Conjecture, we prove that the set $S_f$ of non-properness of $f$ is covered by parametric…
We obtain results on the geometry of D-semianalytic and subanalytic subsets over a complete, non-trivially valued non-Archimedean field K, which is not necessarily algebraically closed. Among the results are a parameterized smooth…
Lifshitz points are multicritical points at which a disordered phase, a homogeneous ordered phase, and a modulated ordered phase meet. Their bulk universality classes are described by natural generalizations of the standard $\phi^4$ model.…
The behavior under iteration of the critical points of polynomial maps plays an essential role in understanding its dynamics. We study the special case where the forward orbits of the critical points are finite. Thurston's theorem tells us…
Given a finite set of points $S\subset\mathbb{R}^d$, a $k$-set of $S$ is a subset $A \subset S$ of size $k$ which can be strictly separated from $S \setminus A $ by a hyperplane. Similarly, a $k$-facet of a point set $S$ in general position…
For a certain parametrized family of maps on the circle, with critical points and logarithmic singularities where derivatives blow up to infinity, a positive measure set of parameters was constructed in [19], corresponding to maps which…
We study the parameter space of a family of planar maps, which are linear on each of the right and left half-planes. We consider the set of parameters for which every orbit recurs to the boundary between half-planes. These parameters…
We present a semidefinite programming approach to bound the measures of cross-independent pairs in a bipartite graph. This can be viewed as a far-reaching extension of Hoffman's ratio bound on the independence number of a graph. As an…
We study the interplay between spectrum, geometry and boundary conditions for two distinguished self-adjoint realisations of the Laplacian on infinite metric graphs, the so-called riedrichs and Neumann extensions. We introduce a new…
If E is a locally convex topological vector space, let P(E) be the pre-ordered set of all continuous seminorms on E. We study, on the one hand, for g an infinite cardinal those locally convex spaces E which have the g-neighbourhood property…
We study a class of one-dimensional full branch maps admitting two indifferent fixed points as well as critical points and/or unbounded derivative. Under some mild assumptions we prove the existence of a unique invariant mixing absolutely…
In this paper we prove new multiplicity results for a critical growth anisotropic quasilinear elliptic system that is coupled through a subcritical perturbation term. We identify a certain scaling for the system and a parameter {\gamma}…
Given any postsingularly finite exponential function $p_\lambda(z) = \lambda \exp(z)$ where $\lambda \in \C^*$, we construct a sequence of postcritically finite unicritical polynomials $p_{d,\lambda_d}(z) = \lambda_d(1+\frac{z}{d})^d$ that…
The critical behavior of semi-infinite $d$-dimensional systems with $n$-component order parameter $\bm{\phi}$ and short-range interactions is investigated at an $m$-axial bulk Lifshitz point whose wave-vector instability is isotropic in an…