Noncommutative lattices

The extended study of non-commutative lattices was begun in 1949 by Ernst Pascual Jordan, a theoretical and mathematical physicist and co-worker of Max Born and Werner Karl Heisenberg. Jordan introduced noncommutative lattices as algebraic structures potentially suitable to encompass the logic of the quantum world. The modern theory of noncommutative lattices began 40 years later with Jonathan Leech\u27s 1989 paper "Skew lattices in rings." Recently, noncommutative generalizations of lattices and related structures have seen an upsurge in interest, with new ideas and applications emerging, from quasilattices to skew Heyting algebras. Much of this activity is derived in some way from the initiation, over thirty years ago, of Jonathan Leech\u27s program of research that studied noncommutative variations of lattices. The present book consists of seven chapters, mainly covering skew lattices, quasilattices and paralattices, skew lattices of idempotents in rings and skew Boolean algebras. As such, it is the first research monograph covering major results due to the renewed study of noncommutative lattices. It will serve as a valuable graduate textbook on the subject, as well as handy reference to researchers of noncommutative algebras

Noncommutative Frames Revisited

In this note, we correct an error in arXiv:1702.04949 by adding an additional assumption of join completeness. We demonstrate with examples why this assumption is necessary, and discuss how join completeness relates to other properties of a skew lattice.Comment: 8 page

Reductions in task positive neural systems occur with the passage of time and are associated with changes in ongoing thought

Cognition is dynamic and involves both the maintenance of and transitions between neurocognitive states. While recent research has identified some of the neural systems involved in sustaining task states, it is less well understood how intrinsic influences on cognition emerge over time. The current study uses fMRI and Multi-Dimensional Experience Sampling (MDES) to chart how cognition changes over time from moments in time when external attention was established. We found that the passage of time was associated with brain regions associated with external attention decreasing in activity over time. Comparing this pattern of activity to defined functional hierarchies of brain organization, we found that it could be best understood as movement away from systems involved in task performance. Moments where the participants described their thoughts as off-task showed a significant similarity to the task-negative end of the same hierarchy. Finally, the greater the similarity of a participant's neural dynamics to this hierarchy the faster their rate of increasing off-task thought over time. These findings suggest topographical changes in neural processing that emerge over time and those seen during off-task thought can both be understood as a common shift away from neural motifs seen during complex task performance

Establishing brain states in neuroimaging data

The definition of a brain state remains elusive, with varying interpretations across different sub-fields of neuroscience-from the level of wakefulness in anaesthesia, to activity of individual neurons, voltage in EEG, and blood flow in fMRI. This lack of consensus presents a significant challenge to the development of accurate models of neural dynamics. However, at the foundation of dynamical systems theory lies a definition of what constitutes the 'state' of a system-i.e., a specification of the system's future. Here, we propose to adopt this definition to establish brain states in neuroimaging timeseries by applying Dynamic Causal Modelling (DCM) to low-dimensional embedding of resting and task condition fMRI data. We find that ~90% of subjects in resting conditions are better described by first-order models, whereas ~55% of subjects in task conditions are better described by second-order models. Our work calls into question the status quo of using first-order equations almost exclusively within computational neuroscience and provides a new way of establishing brain states, as well as their associated phase space representations, in neuroimaging datasets