Fixed conjugacy classes of normal subgroups and the k(GV)-problem

We establish several new bounds for the number of conjugacy classes of a finite group, all of which involve the maximal number c of conjugacy classes of a normal subgroup fixed by some element of a suitable subset of the group. To apply these formulas effectively, the parameter c, which in general is hard to control, is studied in some important situations. These results are then used to provide a new, shorter proof of the most difficult case of the well-known k(GV)-problem, which occurs for p=5 and V induced from the natural module of a 5-complement of GL(2,5). We also show how, for large p, the new results reduce the k(GV)-problem to the primitive case, thereby improving previous work on this. Furthermore, we discuss how they can be used in tackling the imprimitive case of the as of yet unsolved noncoprime k(GV)-problem

Modeling Task Effects in Human Reading with Neural Attention

Humans read by making a sequence of fixations and saccades. They often skip words, without apparent detriment to understanding. We offer a novel explanation for skipping: readers optimize a tradeoff between performing a language-related task and fixating as few words as possible. We propose a neural architecture that combines an attention module (deciding whether to skip words) and a task module (memorizing the input). We show that our model predicts human skipping behavior, while also modeling reading times well, even though it skips 40% of the input. A key prediction of our model is that different reading tasks should result in different skipping behaviors. We confirm this prediction in an eye-tracking experiment in which participants answers questions about a text. We are able to capture these experimental results using the our model, replacing the memorization module with a task module that performs neural question answering