Humans are able to continuously monitor environmental situations and adjust their behavioral strategies to optimize performance. Here we investigate the behavioral and brain adjustments that occur when conflicting stimulus elements are, or are not, temporally predictable. Event-related potentials (ERPs) were collected while manual-response variants of the Stroop task were performed in which the stimulus onset asynchronies (SOAs) between the relevant-color and irrelevant-word stimulus components were either randomly intermixed, or held constant, within each experimental run. Results indicated that the size of both the neural and behavioral effects of stimulus incongruency varied with the temporal arrangement of the stimulus components, such that the random-SOA arrangements produced the greatest incongruency effects at the earliest irrelevant-first SOA (−200 ms) and the constant-SOA arrangements produced the greatest effects with simultaneous presentation. These differences in conflict processing were accompanied by rapid (~150 ms) modulations of the sensory ERPs to the irrelevant distracter components when they occurred consistently first. These effects suggest that individuals are able to strategically allocate attention in time to mitigate the influence of a temporally predictable distracter. As these adjustments are instantiated by the subjects without instruction, they reveal a form of rapid strategic learning for dealing with temporally predictable stimulus incongruency.

In this study we leveraged the high-temporal resolution of EEG to examine the neural mechanisms underlying the flexible regulation of cognitive control that unfolds over different timescales. We measured behavioral and neural effects of color-word incongruency as different groups of human participants performed three different versions of color-word ‘Stroop’ tasks in which the relative timing of the color and word features varied trial-to-trial. For this purpose we used a standard ‘Stroop’ color-identification task with equal congruent-to-incongruent proportions (50/50%), along with two versions of the ‘Reverse Stroop’ word-identification tasks, for which we manipulated the incongruency proportion (50/50% and 80/20%). Two canonical ERP markers of neural processing of stimulus incongruency, the fronto-central negative-polarity incongruency wave (NINC) and the late positive component (LPC), were evoked across the various conditions. Results indicated that color-word incongruency interacted with the relative feature timing, producing greater neural and behavioral effects when the task-irrelevant stimulus preceded the target, but still significant effects when it followed. Additionally, both behavioral and neural incongruency effects were reduced by nearly half in the word-identification task (ReverseStroop-50/50) relative to the color-identification task (Stroop-50/50), with these effects essentially fully recovering when incongruent trials appeared only infrequently (ReverseStroop-80/20). Across the conditions, NINC amplitudes closely paralleled reaction times, indicating this component is sensitive to the overall level of stimulus conflict. In contrast, LPC amplitudes were largest with infrequent incongruent trials, suggesting a possible readjustment role when proactive control is reduced. These findings thus unveil distinct control mechanisms that unfold over time in response to conflicting stimulus input under different contexts.

When A = k[x 1 , . . . , xn] and G is a small subgroup of GLn(k), Auslander's Theorem says that the skew group algebra A#G is isomorphic to End A G (A) as graded algebras. We prove a generalization of Auslander's Theorem for permutation actions on (−1)-skew polynomial rings, (−1)-quantum Weyl algebras, three-dimensional Sklyanin algebras, and a certain homogeneous down-up algebra. We also show that certain fixed rings A G are graded isolated singularities in the sense of Ueyama.

Let H be a semisimple Hopf algebra acting on an Artin-Schelter regular algebra A, homogeneously, inner-faithfully, preserving the grading on A, and so that A is an H-module algebra. When the fixed subring A H is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that H is a reflection Hopf algebra for A. We show that each of the semisimple Hopf algebras H 2n 2 of Pansera, and A 4m and B 4m of Masuoka is a reflection Hopf algebra for an AS regular algebra of dimension 2 or 3.2010 Mathematics Subject Classification. Primary 16T05, 16E65, 16G10.The Hopf algebras A 4m and B 4m can be viewed as deformations of kQ 4m (see [2]), and are extensions of the form:The examples of reflection Hopf algebras that we have computed indicate that there are an abundance of examples. The properties that characterize such a pair (A, H) are not clear, and invite further investigation. One obvious question is:Question 0.1. When is a bicrossed product H = K# τ σ H a reflection Hopf algebra for some AS regular algebra A?The method that is used in this paper is as follows. First, we compute the Grothendieck ring of finite-dimensional H-modules for each Hopf algebra H. The results are summarized in the following table.

A general criterion is given for when the center of a Taft algebra smash product is the fixed ring. This is applied to the study of the noncommutative discriminant. Our method relies on the Poisson methods of Nguyen, Trampel, and Yakimov, but also makes use of Poisson Ore extensions. Specifically, we fully determine the inner faithful actions of Taft algebras on quantum planes and quantum Weyl algebras.We compute the discriminant of the corresponding smash product and apply it to compute the Azumaya locus and restricted automorphism group.