On random primitive sets, directable NDFAs and the generation of slowly synchronizing DFAs

We tackle the problem of the randomized generation of slowly synchronizing deterministic automata (DFAs) by generating random primitive sets of matrices. We show that when the randomized procedure is too simple the exponent of the generated sets is O(n log n) with high probability, thus the procedure fails to return DFAs with large reset threshold. We extend this result to random nondeterministic automata (NDFAs) by showing, in particular, that a uniformly sampled NDFA has both a 2-directing word and a 3-directing word of length O(n log n) with high probability. We then present a more involved randomized algorithm that manages to generate DFAs with large reset threshold and we finally leverage this finding for exhibiting new families of DFAs with reset threshold of order $\Omega(n^2/4)$.Comment: 31 pages, 9 figures. arXiv admin note: text overlap with arXiv:1805.0672

On Randomized Generation of Slowly Synchronizing Automata

Motivated by the randomized generation of slowly synchronizing automata, we study automata made of permutation letters and a merging letter of rank n-1 . We present a constructive randomized procedure to generate synchronizing automata of that kind with (potentially) large alphabet size based on recent results on primitive sets of matrices. We report numerical results showing that our algorithm finds automata with much larger reset threshold than a mere uniform random generation and we present new families of automata with reset threshold of Omega(n^2/4) . We finally report theoretical results on randomized generation of primitive sets of matrices: a set of permutation matrices with a 0 entry changed into a 1 is primitive and has exponent of O(n log n) with high probability in case of uniform random distribution and the same holds for a random set of binary matrices where each entry is set, independently, equal to 1 with probability p and equal to 0 with probability 1-pwhen np-log n - > infty as n - > infty

A linear bound on the k-rendezvous time for primitive sets of NZ matrices

A set of nonnegative matrices is called primitive if there exists a product of these matrices that is entrywise positive. Motivated by recent results relating synchronizing automata and primitive sets, we study the length of the shortest product of a primitive set having a column or a row with k positive entries, called its k-rendezvous time (k-RT}), in the case of sets of matrices having no zero rows and no zero columns. We prove that the k-RT is at most linear w.r.t. the matrix size n for small k, while the problem is still open for synchronizing automata. We provide two upper bounds on the k-RT: the second is an improvement of the first one, although the latter can be written in closed form. We then report numerical results comparing our upper bounds on the k-RT with heuristic approximation methods.Comment: 27 pages, 10 figur

On a Centrality Maximization Game

The Bonacich centrality is a well-known measure of the relative importance of nodes in a network. This notion is, for example, at the core of Google's PageRank algorithm. In this paper we study a network formation game where each player corresponds to a node in the network to be formed and can decide how to rewire his m out-links aiming at maximizing his own Bonacich centrality, which is his utility function. We study the Nash equilibria (NE) and the best response dynamics of this game and we provide a complete classification of the set of NE when m=1 and a fairly complete classification of the NE when m=2. Our analysis shows that the centrality maximization performed by each node tends to create undirected and disconnected or loosely connected networks, namely 2-cliques for m=1 and rings or a special "Butterfly"-shaped graph when m=2. Our results build on locality property of the best response function in such game that we formalize and prove in the paper.Comment: 10 pages, 11 figure

The Synchronizing Probability Function for Primitive Sets of Matrices

Motivated by recent results relating synchronizing DFAs and primitive sets, we tackle the synchronization process and the related longstanding \v{C}ern\'{y} conjecture by studying the primitivity phenomenon for sets of nonnegative matrices having neither zero-rows nor zero-columns. We formulate the primitivity process in the setting of a two-player probabilistic game and we make use of convex optimization techniques to describe its behavior. We develop a tool for approximating and upper bounding the exponent of any primitive set and supported by numerical results we state a conjecture that, if true, would imply a quadratic upper bound on the reset threshold of a new class of automata.Comment: 24 pages, 9 figures. Submitted to DLT 2018 Special Issu

On a Network Centrality Maximization Game

We study a network formation game where $n$ players, identified with the nodes of a directed graph to be formed, choose where to wire their outgoing links in order to maximize their PageRank centrality. Specifically, the action of every player $i$ consists in the wiring of a predetermined number $d_i$ of directed out-links, and her utility is her own PageRank centrality in the network resulting from the actions of all players. We show that this is a potential game and that the best response correspondence always exhibits a local structure in that it is never convenient for a node $i$ to link to other nodes that are at incoming distance more than $d_i$ from her. We then study the equilibria of this game determining necessary conditions for a graph to be a (strict, recurrent) Nash equilibrium. Moreover, in the homogeneous case, where players all have the same number $d$ of out-links, we characterize the structure of the potential maximizing equilibria and, in the special cases $d=1$ and $d=2$, we provide a complete classification of the set of (strict, recurrent) Nash equilibria. Our analysis shows in particular that the considered formation mechanism leads to the emergence of undirected and disconnected or loosely connected networks.Comment: 42 pages, 11 figure

Molecular Insights into the Local Anesthetic Receptor within Voltage-Gated Sodium Channels Using Hydroxylated Analogs of Mexiletine

We previously showed that the β-adrenoceptor modulators, clenbuterol and propranolol, directly blocked voltage-gated sodium channels, whereas salbutamol and nadolol did not (Desaphy et al., 2003), suggesting the presence of two hydroxyl groups on the aromatic moiety of the drugs as a molecular requisite for impeding sodium channel block. To verify such an hypothesis, we synthesized five new mexiletine analogs by adding one or two hydroxyl groups to the aryloxy moiety of the sodium channel blocker and tested these compounds on hNav1.4 channels expressed in HEK293 cells. Concentration–response relationships were constructed using 25-ms-long depolarizing pulses at −30 mV applied from an holding potential of −120 mV at 0.1 Hz (tonic block) and 10 Hz (use-dependent block) stimulation frequencies. The half-maximum inhibitory concentrations (IC50) were linearly correlated to drug lipophilicity: the less lipophilic the drug, minor was the block. The same compounds were also tested on F1586C and Y1593C hNav1.4 channel mutants, to gain further information on the molecular interactions of mexiletine with its receptor within the sodium channel pore. In particular, replacement of Phe1586 and Tyr1593 by non-aromatic cysteine residues may help in the understanding of the role of π–π or π–cation interactions in mexiletine binding. Alteration of tonic block suggests that the aryloxy moiety of mexiletine may interact either directly or indirectly with Phe1586 in the closed sodium channel to produce low-affinity binding block, and that this interaction depends on the electrostatic potential of the drug aromatic tail. Alteration of use-dependent block suggests that addition of hydroxyl groups to the aryloxy moiety may modify high-affinity binding of the drug amine terminal to Phe1586 through cooperativity between the two pharmacophores, this effect being mainly related to drug lipophilicity. Mutation of Tyr1593 further impaired such cooperativity. In conclusion, these results confirm our former hypothesis by showing that the presence of hydroxyl groups to the aryloxy moiety of mexiletine greatly reduced sodium channel block, and provide molecular insights into the intimate interaction of local anesthetics with their receptor

Beta-Blocker Use in Older Hospitalized Patients Affected by Heart Failure and Chronic Obstructive Pulmonary Disease: An Italian Survey From the REPOSI Register

Beta (β)-blockers (BB) are useful in reducing morbidity and mortality in patients with heart failure (HF) and concomitant chronic obstructive pulmonary disease (COPD). Nevertheless, the use of BBs could induce bronchoconstriction due to β2-blockade. For this reason, both the ESC and GOLD guidelines strongly suggest the use of selective β1-BB in patients with HF and COPD. However, low adherence to guidelines was observed in multiple clinical settings. The aim of the study was to investigate the BBs use in older patients affected by HF and COPD, recorded in the REPOSI register. Of 942 patients affected by HF, 47.1% were treated with BBs. The use of BBs was significantly lower in patients with HF and COPD than in patients affected by HF alone, both at admission and at discharge (admission, 36.9% vs. 51.3%; discharge, 38.0% vs. 51.7%). In addition, no further BB users were found at discharge. The probability to being treated with a BB was significantly lower in patients with HF also affected by COPD (adj. OR, 95% CI: 0.50, 0.37-0.67), while the diagnosis of COPD was not associated with the choice of selective β1-BB (adj. OR, 95% CI: 1.33, 0.76-2.34). Despite clear recommendations by clinical guidelines, a significant underuse of BBs was also observed after hospital discharge. In COPD affected patients, physicians unreasonably reject BBs use, rather than choosing a β1-BB. The expected improvement of the BB prescriptions after hospitalization was not observed. A multidisciplinary approach among hospital physicians, general practitioners, and pharmacologists should be carried out for better drug management and adherence to guideline recommendations

The “Diabetes Comorbidome”: A Different Way for Health Professionals to Approach the Comorbidity Burden of Diabetes

(1) Background: The disease burden related to diabetes is increasing greatly, particularly in older subjects. A more comprehensive approach towards the assessment and management of diabetes’ comorbidities is necessary. The aim of this study was to implement our previous data identifying and representing the prevalence of the comorbidities, their association with mortality, and the strength of their relationship in hospitalized elderly patients with diabetes, developing, at the same time, a new graphic representation model of the comorbidome called “Diabetes Comorbidome”. (2) Methods: Data were collected from the RePoSi register. Comorbidities, socio-demographic data, severity and comorbidity indexes (Cumulative Illness rating Scale CIRS-SI and CIRS-CI), and functional status (Barthel Index), were recorded. Mortality rates were assessed in hospital and 3 and 12 months after discharge. (3) Results: Of the 4714 hospitalized elderly patients, 1378 had diabetes. The comorbidities distribution showed that arterial hypertension (57.1%), ischemic heart disease (31.4%), chronic renal failure (28.8%), atrial fibrillation (25.6%), and COPD (22.7%), were the more frequent in subjects with diabetes. The graphic comorbidome showed that the strongest predictors of death at in hospital and at the 3-month follow-up were dementia and cancer. At the 1-year follow-up, cancer was the first comorbidity independently associated with mortality. (4) Conclusions: The “Diabetes Comorbidome” represents the perfect instrument for determining the prevalence of comorbidities and the strength of their relationship with risk of death, as well as the need for an effective treatment for improving clinical outcomes