Comparing skew Schur functions: a quasisymmetric perspective

Reiner, Shaw and van Willigenburg showed that if two skew Schur functions sA and sB are equal, then the skew shapes $A$ and $B$ must have the same "row overlap partitions." Here we show that these row overlap equalities are also implied by a much weaker condition than Schur equality: that sA and sB have the same support when expanded in the fundamental quasisymmetric basis F. Surprisingly, there is significant evidence supporting a conjecture that the converse is also true. In fact, we work in terms of inequalities, showing that if the F-support of sA contains that of sB, then the row overlap partitions of A are dominated by those of B, and again conjecture that the converse also holds. Our evidence in favor of these conjectures includes their consistency with a complete determination of all F-support containment relations for F-multiplicity-free skew Schur functions. We conclude with a consideration of how some other quasisymmetric bases fit into our framework.

Sleep-wake disturbances in sporadic Creutzfeldt-Jakob disease

BACKGROUND: The prevalence and characteristics of sleep-wake disturbances in sporadic Creutzfeldt-Jakob disease (sCJD) are poorly understood. METHODS: Seven consecutive patients with definite sCJD underwent a systematic assessment of sleep-wake disturbances, including clinical history, video-polysomnography, and actigraphy. Extent and distribution of neurodegeneration was estimated by brain autopsy in six patients. Western blot analyses enabling classification and quantification of the protease-resistant isoform of the prion protein, PrPSc, in thalamus and occipital cortex was available in four patients. RESULTS: Sleep-wake symptoms were observed in all patients, and were prominent in four of them. All patients had severe sleep EEG abnormalities with loss of sleep spindles, very low sleep efficiency, and virtual absence of REM sleep. The correlation between different methods to assess sleep-wake functions (history, polysomnography, actigraphy, videography) was generally poor. Brain autopsy revealed prominent changes in cortical areas, but only mild changes in the thalamus. No mutation of the PRNP gene was found. CONCLUSIONS: This study demonstrates in sporadic Creutzfeldt-Jakob disease, first, the existence of sleep-wake disturbances similar to those reported in fatal familial insomnia in the absence of prominent and isolated thalamic neuronal loss, and second, the need of a multimodal approach for the unambiguous assessment of sleep-wake functions in these patients.