Characterization for entropy of shifts of finite type on Cayley trees

The notion of tree-shifts constitutes an intermediate class in between one-sided shift spaces and multidimensional ones. This paper proposes an algorithm for computing of the entropy of a tree-shift of finite type. Meanwhile, the entropy of a tree-shift of finite type is $\dfrac{1}{p} \ln \lambda$ for some $p \in \mathbb{N}$, where $\lambda$ is a Perron number. This extends Lind's work on one-dimensional shifts of finite type. As an application, the entropy minimality problem is investigated, and we obtain the necessary and sufficient condition for a tree-shift of finite type being entropy minimal with some additional conditions

Electrical Resistivity of Equiatomic Rare-Earth-Noble-Metal Compounds

The electrical resistivity of twenty CsCl type intermediate phases of Gd, Tb, Dy, Ho, Er, and Tm with Cu, Ag, and Au and of Y and Nd with Ag were measured from 4.2°K to about 250°K. All of the resistivity-temperature curves, except that of YAg, show anomalies at temperatures which are in good agreement with the available antiferromagnetic-paramagnetic transition temperatures obtained by either neutron diffraction studies or magnetic susceptibility measurements. The spin disordering part of the resistivity of the CsCl phases were deduced and any attempt to apply the theory of de Gennes and Friedel which was used by Rocher for pure rare-earth elements, was not successful

Detecting Weakly Simple Polygons

A closed curve in the plane is weakly simple if it is the limit (in the Fr\'echet metric) of a sequence of simple closed curves. We describe an algorithm to determine whether a closed walk of length n in a simple plane graph is weakly simple in O(n log n) time, improving an earlier O(n^3)-time algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary, we obtain the first efficient algorithm to determine whether an arbitrary n-vertex polygon is weakly simple; our algorithm runs in O(n^2 log n) time. We also describe algorithms that detect weak simplicity in O(n log n) time for two interesting classes of polygons. Finally, we discuss subtle errors in several previously published definitions of weak simplicity.Comment: 25 pages and 13 figures, submitted to SODA 201