Transition Property For Cube-Free Words

We study cube-free words over arbitrary non-unary finite alphabets and prove the following structural property: for every pair $(u,v)$ of $d$-ary cube-free words, if $u$ can be infinitely extended to the right and $v$ can be infinitely extended to the left respecting the cube-freeness property, then there exists a "transition" word $w$ over the same alphabet such that $uwv$ is cube free. The crucial case is the case of the binary alphabet, analyzed in the central part of the paper. The obtained "transition property", together with the developed technique, allowed us to solve cube-free versions of three old open problems by Restivo and Salemi. Besides, it has some further implications for combinatorics on words; e.g., it implies the existence of infinite cube-free words of very big subword (factor) complexity.Comment: 14 pages, 5 figure

Avoidability of formulas with two variables

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ of variables if there is no factor $f$ of $w$ such that $f=h(p)$ where $h:\Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. We consider the patterns such that at most two variables appear at least twice, or equivalently, the formulas with at most two variables. For each such formula, we determine whether it is $2$-avoidable, and if it is $2$-avoidable, we determine whether it is avoided by exponentially many binary words

On the intersections of Fibonacci, Pell, and Lucas numbers

We describe how to compute the intersection of two Lucas sequences of the forms $\{U_n(P,\pm 1) \}_{n=0}^{\infty}$ or $\{V_n(P,\pm 1) \}_{n=0}^{\infty}$ with $P\in\mathbb{Z}$ that includes sequences of Fibonacci, Pell, Lucas, and Lucas-Pell numbers. We prove that such an intersection is finite except for the case $U_n(1,-1)$ and $U_n(3,1)$ and the case of two $V$-sequences when the product of their discriminants is a perfect square. Moreover, the intersection in these cases also forms a Lucas sequence. Our approach relies on solving homogeneous quadratic Diophantine equations and Thue equations. In particular, we prove that 0, 1, 2, and 5 are the only numbers that are both Fibonacci and Pell, and list similar results for many other pairs of Lucas sequences. We further extend our results to Lucas sequences with arbitrary initial terms

A symmetry group of a Thue-Morse quasicrystal

We present a method of coding general self-similar structures. In particular, we construct a symmetry group of a one-dimensional Thue-Morse quasicrystal, i.e., of a nonperiodic ground state of a certain translation-invariant, exponentially decaying interaction.Comment: 6 pages, Late

Geometrical Frustration and Static Correlations in Hard-Sphere Glass Formers

We analytically and numerically characterize the structure of hard-sphere fluids in order to review various geometrical frustration scenarios of the glass transition. We find generalized polytetrahedral order to be correlated with increasing fluid packing fraction, but to become increasingly irrelevant with increasing dimension. We also find the growth in structural correlations to be modest in the dynamical regime accessible to computer simulations.Comment: 21 pages; part of the "Special Topic Issue on the Glass Transition

Dense Packings of Congruent Circles in Rectangles with a Variable Aspect Ratio

We use computational experiments to find the rectangles of minimum area into which a given number n of non-overlapping congruent circles can be packed. No assumption is made on the shape of the rectangles. Most of the packings found have the usual regular square or hexagonal pattern. However, for 1495 values of n in the tested range n =< 5000, specifically, for n = 49, 61, 79, 97, 107,... 4999, we prove that the optimum cannot possibly be achieved by such regular arrangements. The evidence suggests that the limiting height-to-width ratio of rectangles containing an optimal hexagonal packing of circles tends to 2-sqrt(3) as n tends to infinity, if the limit exists.Comment: 21 pages, 13 figure

Unambiguous 1-Uniform Morphisms

A morphism h is unambiguous with respect to a word w if there is no other morphism g that maps w to the same image as h. In the present paper we study the question of whether, for any given word, there exists an unambiguous 1-uniform morphism, i.e., a morphism that maps every letter in the word to an image of length 1.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Which game narratives do adolescents of different gameplay and sociodemographic backgrounds prefer? a mixed-methods analysis

OBJECTIVE: The aim of this study was to investigate which narrative elements of digital game narratives are preferred by the general adolescent population, and to examine associations with gender, socioeconomic status (SES), and gameplay frequency. Further, the study aims to discuss how results can be translated to serious digital games. MATERIALS AND METHODS: Adolescents were recruited through school to complete a survey on narrative preferences in digital games. The survey included questions on sociodemographic information, frequency of gameplay, and an open-ended question on what could be an appealing narrative for them. Data were analyzed in a mixed-methods approach, using thematic analysis and chi-square analyses to determine narrative preferences and the associations between game narrative elements and player characteristics (gender, SES, and frequency of gameplay). RESULTS: The sample consisted of 446 adolescents (12-15 years old) who described 30 narrative subthemes. Preferences included human characters as protagonists; nonhuman characters only as antagonists; realistic settings, such as public places or cities; and a strong conflict surrounding crime, catastrophe, or war. Girls more often than boys defined characters by their age, included avatars, located the narrative in private places, developed profession-related skills, and included a positive atmosphere. Adolescents of nonacademic education more often than adolescents of academic education defined characters by criminal actions. Infrequent players more often included human characters defined by their age than frequent players. After performing a Bonferroni correction, narrative preferences for several gender differences remained. CONCLUSION: Different narrative elements related to subgroups of adolescents by gender, SES, and frequency of gameplay. Customization of narratives in serious digital health games should be warranted for boys and girls; yet, further research is needed to specify how to address girls in particular

Receptor structure-based discovery of non-metabolite agonists for the succinate receptor GPR91

Besides functioning as an intracellular metabolite, succinate acts as a stress-induced extracellular signal through activation of GPR91 (SUCNR1) for which we lack suitable pharmacological tools.Here we first determined that the cis conformation of the succinate backbone is preferred and that certain backbone modifications are allowed for GPR91 activation. Through receptor modeling over the X-ray structure of the closely related P2Y1 receptor, we discovered that the binding pocket is partly occupied by a segment of an extracellular loop and that succinate therefore binds in a very different mode than generally believed. Importantly, an empty side-pocket is identified next to the succinate binding site. All this information formed the basis for a substructure-based search query, which, combined with molecular docking, was used in virtual screening of the ZINC database to pick two serial mini-libraries of a total of only 245 compounds from which sub-micromolar, selective GPR91 agonists of unique structures were identified. The best compounds were backbone-modified succinate analogs in which an amide-linked hydrophobic moiety docked into the side-pocket next to succinate as shown by both loss- and gain-of-function mutagenesis. These compounds displayed GPR91-dependent activity in altering cytokine expression in human M2 macrophages similar to succinate, and importantly were devoid of any effect on the major intracellular target, succinate dehydrogenase.These novel, synthetic non-metabolite GPR91 agonists will be valuable both as pharmacological tools to delineate the GPR91-mediated functions of succinate and as leads for the development of GPR91-targeted drugs to potentially treat low grade metabolic inflammation and diabetic complications such as retinopathy and nephropathy