International audienceWe study the palindromic complexity of infinite words obtained by coding rotations on partitions of the unit circle by inspecting the return words. The main result is that every coding of rotations on two intervals is full, that is, it realizes the maximal palindromic complexity. As a byproduct, a slight improvement about return words in codings of rotations is obtained: every factor of a coding of rotations on two intervals has at most 4 complete return words, where the bound is realized only for a finite number of factors. We also provide a combinatorial proof for the special case of complementary-symmetric Rote sequences by considering both palindromes and antipalindromes occurring in it
It has been proved that, among the polyominoes that tile the plane by translation, the so-called squares tile the plane in at most two distinct ways. In this paper, we focus on double squares, that is, the polyominoes that tile the plane in exactly two distinct ways. Our approach is based on solving equations on words, which allows to exhibit properties about their shape. Moreover, we describe two infinite families of double squares. The first one is directly linked to Christoffel words and may be interpreted as segments of thick straight lines. The second one stems from the Fibonacci sequence and reveals some fractal features.
How many words—and which ones—are sufficient to define all other words? When dictionaries are analyzed as directed graphs with links from defining words to defined words, they reveal a latent structure. Recursively removing all words that are reachable by definition but that do not define any further words reduces the dictionary to a Kernel of about 10% of its size. This is still not the smallest number of words that can define all the rest. About 75% of the Kernel turns out to be its Core, a “Strongly Connected Subset” of words with a definitional path to and from any pair of its words and no word's definition depending on a word outside the set. But the Core cannot define all the rest of the dictionary. The 25% of the Kernel surrounding the Core consists of small strongly connected subsets of words: the Satellites. The size of the smallest set of words that can define all the rest—the graph's “minimum feedback vertex set” or MinSet—is about 1% of the dictionary, about 15% of the Kernel, and part‐Core/part‐Satellite. But every dictionary has a huge number of MinSets. The Core words are learned earlier, more frequent, and less concrete than the Satellites, which are in turn learned earlier, more frequent, but more concrete than the rest of the Dictionary. In principle, only one MinSet's words would need to be grounded through the sensorimotor capacity to recognize and categorize their referents. In a dual‐code sensorimotor/symbolic model of the mental lexicon, the symbolic code could do all the rest through recombinatory definition.
Let G be a simple graph on n vertices. We consider the problem LIS of deciding whether there exists an induced subtree with exactly i ≤ n vertices and leaves in G. We study the associated optimization problem, that consists in computing the maximal number of leaves, denoted by LG(i), realized by an induced subtree with i vertices, for 0 ≤ i ≤ n. We begin by proving that the LIS problem is NP-complete in general and then we compute the values of the map LG for some classical families of graphs and in particular for the d-dimensional hypercubic graphs Q d , for 2 ≤ d ≤ 6. We also describe a nontrivial branch and bound algorithm that computes the function LG for any simple graph G. In the special case where G is a tree of maximum degree ∆, we provide a O(n 3 ∆) time and O(n 2 ) space algorithm to compute the function LG.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.