Rational self-affine tiles

An integral self-affine tile is the solution of a set equation $\mathbf{A} \mathcal{T} = \bigcup_{d \in \mathcal{D}} (\mathcal{T} + d)$, where $\mathbf{A}$ is an $n \times n$ integer matrix and $\mathcal{D}$ is a finite subset of $\mathbb{Z}^n$. In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices $\mathbf{A} \in \mathbb{Q}^{n \times n}$. We define rational self-affine tiles as compact subsets of the open subring $\mathbb{R}^n\times \prod_\mathfrak{p} K_\mathfrak{p}$ of the ad\'ele ring $\mathbb{A}_K$, where the factors of the (finite) product are certain $\mathfrak{p}$-adic completions of a number field $K$ that is defined in terms of the characteristic polynomial of $\mathbf{A}$. Employing methods from classical algebraic number theory, Fourier analysis in number fields, and results on zero sets of transfer operators, we establish a general tiling theorem for these tiles. We also associate a second kind of tiles with a rational matrix. These tiles are defined as the intersection of a (translation of a) rational self-affine tile with $\mathbb{R}^n \times \prod_\mathfrak{p} \{0\} \simeq \mathbb{R}^n$. Although these intersection tiles have a complicated structure and are no longer self-affine, we are able to prove a tiling theorem for these tiles as well. For particular choices of digit sets, intersection tiles are instances of tiles defined in terms of shift radix systems and canonical number systems. Therefore, we gain new results for tilings associated with numeration systems

Scaling of nascent nodes in extended s-wave superconductors

We analyze the low-energy properties of superconductors near the onset of accidental nodes, i.e. zeroes of the gap function not enforced by symmetry. The existence of such nodes has been motivated by recent experiments suggesting a transition between nodeless and nodal superconductivity in iron-based compounds. We find that the low-temperature behavior of the penetration depth, the specific heat, and the NMR spin-lattice relaxation rate are determined by the scaling properties of a quantum critical point associated with the nascent nodes. Although the power-law exponents are insensitive to weak short-range electronic interactions, they can be significantly altered by the curvature of the Fermi surface or by the three-dimensional character of the gap. Consequently, the behavior of macroscopic quantities near the onset of nodes can be used as a criterion to determine the nodal structure of the gap function.Comment: minor changes; version accepted for publicatio

Entropic Effects in the Very Low Temperature Regime of Diluted Ising Spin Glasses with Discrete Couplings

We study link-diluted $\pm J$ Ising spin glass models on the hierarchical lattice and on a three-dimensional lattice close to the percolation threshold. We show that previously computed zero temperature fixed points are unstable with respect to temperature perturbations and do not belong to any critical line in the dilution-temperature plane. We discuss implications of the presence of such spurious unstable fixed points on the use of optimization algorithms, and we show how entropic effects should be taken into account to obtain the right physical behavior and critical points.Comment: 4 pages, 4 figures. A major typo error in formula (8) has been correcte

Self-affine Manifolds

This paper studies closed 3-manifolds which are the attractors of a system of finitely many affine contractions that tile $\mathbb{R}^3$. Such attractors are called self-affine tiles. Effective characterization and recognition theorems for these 3-manifolds as well as theoretical generalizations of these results to higher dimensions are established. The methods developed build a bridge linking geometric topology with iterated function systems and their attractors. A method to model self-affine tiles by simple iterative systems is developed in order to study their topology. The model is functorial in the sense that there is an easily computable map that induces isomorphisms between the natural subdivisions of the attractor of the model and the self-affine tile. It has many beneficial qualities including ease of computation allowing one to determine topological properties of the attractor of the model such as connectedness and whether it is a manifold. The induced map between the attractor of the model and the self-affine tile is a quotient map and can be checked in certain cases to be monotone or cell-like. Deep theorems from geometric topology are applied to characterize and develop algorithms to recognize when a self-affine tile is a topological or generalized manifold in all dimensions. These new tools are used to check that several self-affine tiles in the literature are 3-balls. An example of a wild 3-dimensional self-affine tile is given whose boundary is a topological 2-sphere but which is not itself a 3-ball. The paper describes how any 3-dimensional handlebody can be given the structure of a self-affine 3-manifold. It is conjectured that every self-affine tile which is a manifold is a handlebody.Comment: 40 pages, 13 figures, 2 table

Merging Automobile Insurance Regulatory Bodies: The Case of Atlantic Canada

The recent automobile liability insurance crisis in Atlantic Canada has prompted the four provincial legislations (Newfoundland and Labrador, New Brunswick, Nova Scotia and Prince Edward Island) to setup a task force to redesign, if necessary, the personal automobile insurance system. After reviewing some of the most interesting new regulatory changes, our paper proposes a new area of discussion: The merger of the four provincial insurance regulatory bodies to combat insurance fraud. We base our paper on the principle that recent premium increases are mainly due to an increase in insurance fraud. We show that merging the regulatory bodies may reduce insurance fraud if the merger allows savings on the average audit cost and on the development of better fraud detection technology. Finally, we suggest a fraud reducing insurance taxation scheme to finance insurance fraud investigations. La récente crise dans le Canada Atlantique au sujet de l'assurance de la responsabilité civile des automobilistes a contraint les gouvernements des quatre provinces (Terre-Neuve et Labrador, Nouveau-Brunswick, Nouvelle-Écosse et l'Île-du-Prince-Édouard) à mettre en place une commission pour étudier la possibilité de réorganiser, si nécessaire, leur système d'assurance automobile. Après avoir analysé les plus intéressantes modifications à la réglementation mises en place dans ces provinces, notre étude offre un nouveau terrain de discussion, soit la fusion des quatre réglementations provinciales afin de combattre la fraude à l'assurance. Notre étude se base sur le principe que les récentes augmentations dans les primes sont attribuables à une augmentation de la fraude à l'assurance dans ces provinces. En fusionnant les réglementations des quatre provinces, nous montrons que la fraude à l'assurance peut diminuer si la fusion permet d'épargner une partie du coût de vérification encouru par les assureurs lors de réclamations douteuses. Également, nous montrons qu'une telle fusion peut permettre de développer une meilleure technologie de vérification des réclamations. Nous terminons le papier en suggérant un système de taxation particulier qui non seulement peut financer les investissements nécessaires pour combattre la fraude, mais peut également réduire la fraude directement.insurance fraud, asymmetric information, insurance taxation, public policy, fraude à l'assurance, information asymétrique, taxation à l'assurance, politique publique

Merging automobile regulatory bodies: The case of Atlantic Canada

The recent automobile liability insurance crisis in Atlantic Canada has prompted the four provincial legislations (Newfoundland and Labrador, New Brunswick, Nova Scotia and Prince Edward Island) to setup a task force to redesign, if necessary, the personal automobile insurance system. After reviewing some of the most interesting new regulatory changes, our paper proposes a new area of discussion: The merger of the four provincial insurance regulatory bodies to combat insurance fraud. We base our paper on the principle that recent premium increases are mainly due to an increase in insurance fraud. We show that merging the regulatory bodies may reduce insurance fraud if the merger allows savings on the average audit cost and on the development of better fraud detection technology. Finally, we suggest a fraud reducing insurance taxation scheme to finance insurance fraud investigations. --Insurance Fraud,Asymmetric Information,Insurance Taxation,Public Policy