Abstract. We prove that every 3-regular, n-vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method provides is about 0.438n.Our method uses invariant Gaussian processes on the d-regular tree that satisfy the eigenvector equation at each vertex for a certain eigenvalue λ. We show that such processes can be approximated by i.i.d. factors provided that |λ| ≤ 2 √ d − 1. We then use these approximations for λ = −2 √ d − 1 to produce factor of i.i.d. independent sets on regular trees.

A set of nonnegative matrices M = {M 1 , M 2 , . . . , M k } is called primitive if there exist indices i 1 , i 2 , . . . , i m such that M i1 M i2 . . . M im is positive (i.e. has all its entries > 0). The length of the shortest such product is called the exponent of M. The concept of primitive sets of matrices comes up in a number of problems within control theory, non-homogeneous Markov chains, automata theory etc. Recently, connections between synchronizing automata and primitive sets of matrices were established. In the present paper, we significantly strengthen these links by providing equivalence results, both in terms of combinatorial characterization, and computational aspects.We study the maximal exponent among all primitive sets of n×n matrices, which we denote by exp(n). We prove that lim n→∞ log exp(n) n = log 3 3 , and moreover, we establish that this bound leads to a resolution of the Černý problem for carefully synchronizing automata. We also study the set of matrices with no zero rows and columns, denoted by NZ, due to its intriguing connections to the Černý conjecture and the recent generalization of Perron-Frobenius theory for this class. We characterize computational complexity of different problems related to the exponent of NZ matrix sets, and present a quadratic bound on the exponents of sets belonging to a special subclass. Namely, we show that the exponent of a set of matrices having total support is bounded by 2n 2 − 5n + 5.

Congenital heart defect (CHD) cases have been evaluated together as a group in some previous epidemiological studies. However, different CHD entities have different etiologies, and the underlying causes are unclear in the vast majority of patients. Thus the aim of this study was to analyze the possible association of different maternal diseases with the risk of four types of conotruncal defects (CTD), that is, truncus arteriosus, d-transposition of the great arteries, tetralogy of Fallot, and double-outlet right ventricle based on autopsy or surgical report diagnosis. Acute and chronic diseases with related drug treatments and peri-conceptual folic acid or multivitamin supplementations were compared in mothers of 598 CTD cases, of 902 matched controls, and 38,151 population controls without any defects, and with 20,896 malformed controls with other isolated non-cardiac defects in the population-based large dataset of the Hungarian Case-Control Surveillance of Congenital Abnormalities. Mothers who had medically recorded influenza and the common cold with secondary complications in the prenatal maternity logbook during the second and/or third gestational months were associated with a higher risk of CTD (OR with 95% CI: 2.22, 1.19-3.88). The common denominator of these maternal diseases may be high fever, which could be prevented by antifever therapies. On the other hand, high doses of medically recorded folic acid in early pregnancy were able to reduce the birth prevalence of CTD (OR with 95% CI: 0.54, 0.39-0.73), and this reduction was significant in transposition of the great arteries (0.46, 0.29-0.71) as well. In conclusion, high fever related maternal diseases may have a role in the origin of CTD, while high doses of folic acid in early pregnancy were able to reduce of CTD, particularly transposition of great vessels.

The push-sum algorithm allows distributed computing of the average on a directed graph, and is particularly relevant when one is restricted to one-way and/or asynchronous communications. We investigate its behavior in the presence of unreliable communication channels where messages can be lost. We show that exponential convergence still holds and deduce fundamental properties that implicitly describe the distribution of the final value obtained. We analyze the error of the final common value we get for the essential case of two nodes, both theoretically and numerically. We provide performance comparison with a standard consensus algorithm. * B. Gerencsér and J. M. Hendrickx are with ICTEAM Institute, Université catholique de Louvain, Belgium balazs.gerencser@uclouvain.be and julien.hendrickx@uclouvain.

We study the ergodic behaviour of a discrete-time process X which is a Markov chain in a stationary random environment. The laws of Xt are shown to converge to a limiting law in (weighted) total variation distance as t → ∞. Convergence speed is estimated and an ergodic theorem is established for functionals of X.Our hypotheses on X combine the standard "small set" and "drift" conditions for geometrically ergodic Markov chains with conditions on the growth rate of a certain "maximal process" of the random environment. We are able to cover a wide range of models that have heretofore been untractable. In particular, our results are pertinent to difference equations modulated by a stationary Gaussian process. Such equations arise in applications, for example, in discretized stochastic volatility models of mathematical finance.

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Abstract. Motivated by the Babai conjecture and theČerný conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with n states in this class, we prove that the reset thresholds are upperbounded by 2n 2 − 6n + 5 and can attain the value. In addition, we study diameters of the pair digraphs of permutation automata and construct n-state permutation automata with diameter Background and OverviewCompletely reachable automata, i.e., deterministic finite automata in which every non-empty subset of the state set occurs as the image of the whole state set under the action of a suitable input word, appeared in the study of descriptional complexity of formal languages [26] and in relation to theČerný conjecture [13]. In [6] an emphasis has been made on automata in this class with minimal transition monoid size. In the present paper we focus on automata being in a sense the extreme opposites of those studied in [6], namely, on automata of maximal transition monoid size. In other words, we consider automata with full transition monoid, i.e., transition monoid equal to the full monoid of transformations of the state set; clearly, automata with this property are completely reachable. There are several reasons justifying special attention to automata with full transition monoid. First, as observed in [6], the membership problem for this class of automata is decidable in polynomial time (of the size of the input automaton) while ⋆ Vladimir Gusev and Mikhail V.

The problems discussed in this paper are motivated by general ratio consensus algorithms, introduced by Kempe, Dobra, and Gehrke ( 2003) in a simple form as the push-sum algorithm, later extended by under the name weighted gossip algorithm. We consider a communication protocol described by a strictly stationary, ergodic, sequentially primitive sequence of non-negative matrices, applied iteratively to a pair of fixed initial vectors, the components of which are called values and weights defined at the nodes of a network. The subject of ratio consensus problems is to study the asymptotic properties of ratios of values and weights at each node, expecting convergence to the same limit for all nodes. The main results of the paper provide upper bounds for the rate of the almost sure exponential convergence in terms of the spectral gap associated with the given sequence of random matrices. It will be shown that these upper bounds are sharp. Our results complement previous results of Picci andTaylor (2013) andIutzeler, Ciblat andHachem (2013).