Algebraic synchronization criterion and computing reset words

We refine a uniform algebraic approach for deriving upper bounds on reset thresholds of synchronizing automata. We express the condition that an automaton is synchronizing in terms of linear algebra, and obtain upper bounds for the reset thresholds of automata with a short word of a small rank. The results are applied to make several improvements in the area. We improve the best general upper bound for reset thresholds of finite prefix codes (Huffman codes): we show that an $n$-state synchronizing decoder has a reset word of length at most $O(n \log^3 n)$. In addition to that, we prove that the expected reset threshold of a uniformly random synchronizing binary $n$-state decoder is at most $O(n \log n)$. We also show that for any non-unary alphabet there exist decoders whose reset threshold is in $\varTheta(n)$. We prove the \v{C}ern\'{y} conjecture for $n$-state automata with a letter of rank at most $\sqrt[3]{6n-6}$. In another corollary, based on the recent results of Nicaud, we show that the probability that the \v{C}ern\'y conjecture does not hold for a random synchronizing binary automaton is exponentially small in terms of the number of states, and also that the expected value of the reset threshold of an $n$-state random synchronizing binary automaton is at most $n^{3/2+o(1)}$. Moreover, reset words of lengths within all of our bounds are computable in polynomial time. We present suitable algorithms for this task for various classes of automata, such as (quasi-)one-cluster and (quasi-)Eulerian automata, for which our results can be applied.Comment: 18 pages, 2 figure

Echo-morphological correlates in atrioventricular valvar atresia

As we described in the previous review, double inlet ventricle is usually found with the atrial chambers connected to a dominant left ventricle, less frequently to a dominant right ventricle, and rarely to a solitary and indeterminate ventricle. As we have also discussed in this supplement,double inlet to the left ventricle was, for many years, considered the exemplar of so-called “single ventricle”, despite the fact that such patients unequivocally possess one big and one small ventricle. Echocardiographic interrogation has served to resolve this controversy, showing that such patients make up a significant proportion of those having functionally univentricular hearts. Such echocardiographic investigation has also served to resolve similar controversies regarding patients having tricuspid atresia. For some time, it was argued that patients with tricuspid atresia also had “univentricular hearts”, but the logic used to underscore this approach was just as flawed as that used to justify the use of “single ventricle” in patients with double inlet atrioventricular connection. The increasing use of the Fontan procedure has served to demonstrate that these patients, along with many having mitral atresia in the setting of hypoplastic left heart syndrome, also have functionally univentricular arrangements. As we will show in this review, however, the anatomical substrates found in patients with atrioventricular valvar atresia are much more complex than those seen in the setting of double inlet ventricle. This is because atrioventricular valvar atresia can be produced either by absence of one atrioventricular connection, or by presence of an imperforate valvar membranes closing completely one or other of the two normal atrioventricular junctions. This important difference, combined with multiple segmental combinations, produces a bewildering array of potential anatomical substrates, with the complications magnified by the fact that, when one atrioventricular connection is absent, the other atrioventricular junction can be shared between the two ventricles, the so-called uniatrial and biventricular arrangement. In our review, we will first describe the anatomical options, before concentrating our attention on the more frequent patterns seen in clinical practice

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 (the k-RT). We prove that this value is at most linear w.r.t. the matrix size n for small k, while the problem is still open for synchronizing automata. We then report numerical results comparing our upper bound on the k-RT with heuristic approximation methods