On the Littlewood-Richardson rule for almost skew-shapes

We describe combinatorially the coefficients occurring in the irreducible decomposition of the Weyl module associated with an almost skew-shape belonging to the family .

The proof uses the fundamental exact sequence for almost skew-shapes to initiate an inductive procedure which ultimately reduces to the classical Littlewood-Richardson rule for skew partitions.

     

Homotopy equivalence of two families of complexes

An explicit homotopy equivalence is established between two families of complexes, both of which generalize the classical Koszul complex.

     

fMRI assessment of thalamocortical connectivity during attentional performance

Functional magnetic resonance imaging (fMRI) studies have shown dysfunction in key areas associated with the thalamocortical circuit in patients with schizophrenia. This study examined the functional connectivity involving the frontal-thalamic circuitry during a spatial focusing-of-attention task in 18 unmedicated patients with schizophrenia and 38 healthy controls. Functional connectivity was analyzed by assigning seed regions (in the thalamic nuclei (mediodorsal nucleus (MDN), pulvinar, anterior nucleus (AN)), the dorsolateral prefrontal cortex (Brodmann areas 9 and 46), and the caudate), and correlating their respective activity with that in the non-seed regions voxel-wise. Functional connectivity analysis demonstrated that functional connectivity was significantly impaired in patients, e.g., between the right pulvinar and regions such as the prefrontal and temporal cortices and the cerebellum. On the other hand, enhanced functional connectivity was found in patients e.g., between the AN and regions such as the prefrontal and temporal cortices. In addition, the patients had significantly lower task performance and less (but non-significant) brain activation than those of controls. These results revealed disturbed functional integration in schizophrenia, and suggested that the functional connectivity abnormalities in the thalamocortical circuitry, especially the frontal-thalamic circuitry, may underlie the attention deficits in schizophrenia patients. Further, this study suggested that functional connectivity analysis might be more sensitive than brain activation analysis in detecting the functional abnormalities in schizophrenia.