It has been suggested that infants resonate emotionally to others' positive and negative affect displays, and that these responses become stronger towards emotions with negative valence around the age of 12-months. In this study we measured 6- and 12-month-old infants' changes in pupil diameter when presented with the image and sound of peers experiencing happiness, distress and an emotionally neutral state. For all participants the perception of another's distress triggered larger pupil diameters. Perceiving other's happiness also induced larger pupil diameters but for shorter time intervals. Importantly, we also found evidence for an asymmetry in autonomous arousal towards positive versus negative emotional displays. Larger pupil sizes for another's distress compared to another's happiness were recorded shortly after stimulus onset for the older infants, and in a later time window for the 6-month-olds. These findings suggest that arousal responses for negative as well as for positive emotions are present in the second half of the first postnatal year. Importantly, an asymmetry with stronger responses for negative emotions seems to be already present at this age.

This paper concerns the general problem of classifying the finite deterministic automata that admit a synchronizing (or reset) word. (For our purposes it is irrelevant if the automata has initial or final states.) Our departure point is the study of the transition semigroup associated to the automaton, taking advantage of the enormous and very deep progresses made during the last decades on the theory of permutation groups, their geometry and their combinatorial structure.Let X be a finite set. We say that a primitive group G on X is synchronizing if G together with any non-invertible map on X generates a constant map. It is known (by some recent results proved by P. M. Neumann) that for some primitive groups G and for some singular transformations t of uniform kernel (that is, all blocks have the same number of elements), the semigroup G, t does not generate a constant map. Therefore the following concept is very natural: a primitive group G on X is said to be almost synchronizing if G together with any map of non-uniform kernel generates a constant map. In this paper we use two different methods to provide several infinite families of groups that are not synchronizing, but are almost synchronizing. The paper ends with a number of problems on synchronization likely to attract the attention of experts in computer science, combinatorics and geometry, groups and semigroups, linear algebra and matrix theory.

Let P be a partition of a finite set X. We say that a full transformation f : X −→ X preserves (or stabilizes) the partition P if for all P ∈ P there exists Q ∈ P such that P f ⊆ Q. Let T (X, P) denote the semigroup of all full transformations of X that preserve the partition P.In 2005 Huisheng found an upper bound for the minimum size of the generating sets of T (X, P), when P is a partition in which all of its parts have the same size. In addition, Huisheng conjectured that his bound was exact. In 2009 the first and last authors used representation theory to completely solve Hisheng's conjecture.The goal of this paper is to solve the much more complex problem of finding the minimum size of the generating sets of T (X, P), when P is an arbitrary partition. Again we use representation theory to find the minimum number of elements needed to generate the wreath product of finitely many symmetric groups, and then use this result to solve the problem.The paper ends with a number of problems for experts in group and semigroup theories.

Let Ω be a set of cardinality n, G be a permutation group on Ω and f : Ω → Ω be a map that is not a permutation. We say that G synchronizes f if the transformation semigroup G, f contains a constant map, and that G is a synchronizing group if G synchronizes every non-permutation.A synchronizing group is necessarily primitive, but there are primitive groups that are not synchronizing. Every non-synchronizing primitive group fails to synchronize at least one uniform transformation (that is, transformation whose kernel has parts of equal size), and it had previously been conjectured that this was essentially the only way in which a primitive group could fail to be synchronizing, in other words, that a primitive group synchronizes every non-uniform transformation.The first goal of this paper is to prove that this conjecture is false, by exhibiting primitive groups that fail to synchronize specific non-uniform transformations of ranks 5 and 6. As it has previously been shown that primitive groups synchronize every non-uniform transformation of rank at most 4, these examples are of the lowest possible rank. In addition, we produce graphs with primitive automorphism groups that have approximately √ n non-synchronizing ranks, thus refuting another conjecture on the number of non-synchronizing ranks of a primitive group.The second goal of this paper is to extend the spectrum of ranks for which it is known that primitive groups synchronize every non-uniform transformation of that rank. It has previously been shown that a primitive group of degree n synchronizes every non-uniform transformation of rank n − 1 and n − 2, and here this is extended to n − 3 and n − 4.In the process, we will obtain a purely graph-theoretical result showing that, with limited exceptions, in a vertex-primitive graph the union of neighbourhoods of a set of vertices A is bounded below by a function that is asymptotically |A|.Determining the exact spectrum of ranks for which there exist non-uniform transformations not synchronized by some primitive group is just one of several natural, but possibly difficult, problems on automata, primitive groups, graphs and computational algebra arising from this work; these are outlined in the final section.

We characterize the automorphism groups of circulant digraphs whose connection sets are relatively small, and of unit circulant digraphs. For each class, we either explicitly determine the automorphism group or we show that the graph is a "normal" circulant, so the automorphism group is contained in the normalizer of a cycle. Then we use these characterizations to prove results on the automorphisms of the endomorphism monoids of those digraphs. The paper ends with a list of open problems on graphs, number theory, groups and semigroups.