On eigenvalues of the Schr\"odinger operator with a complex-valued polynomial potential

In this paper, we generalize a recent result of A. Eremenko and A. Gabrielov on irreducibility of the spectral discriminant for the Schr\"odinger equation with quartic potentials. We consider the eigenvalue problem with a complex-valued polynomial potential of arbitrary degree d and show that the spectral determinant of this problem is connected and irreducible. In other words, every eigenvalue can be reached from any other by analytic continuation. We also prove connectedness of the parameter spaces of the potentials that admit eigenfunctions satisfying k>2 boundary conditions, except for the case d is even and k=d/2. In the latter case, connected components of the parameter space are distinguished by the number of zeros of the eigenfunctions.Comment: 23 page

On eigenvalues of the Schr\"odinger operator with an even complex-valued polynomial potential

In this paper, we generalize several results of the article "Analytic continuation of eigenvalues of a quartic oscillator" of A. Eremenko and A. Gabrielov. We consider a family of eigenvalue problems for a Schr\"odinger equation with even polynomial potentials of arbitrary degree d with complex coefficients, and k<(d+2)/2 boundary conditions. We show that the spectral determinant in this case consists of two components, containing even and odd eigenvalues respectively. In the case with k=(d+2)/2 boundary conditions, we show that the corresponding parameter space consists of infinitely many connected components

Production and characterization of biochar from biomasses

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The Course of Fatigue during the First 18 Months after First-Ever Stroke: A Longitudinal Study

Background. Little is known about the course of poststroke fatigue. Objectives. To describe the course of poststroke fatigue in relation to the patient's level of physical functioning, depressive symptoms, and self-reported history of prestroke fatigue. Methods. A longitudinal study using structured face-to-face interviews, questionnaires, and patients' medical records. Data were collected from 95 patients in Norway with first-ever stroke. Fatigue was measured with the Fatigue Severity Scale 7 item version and assessed for change between the acute phase, six, 12, and 18 months after stroke using 2-way ANOVA repeated-measures analyses. Results. The patients' level of fatigue did not change over time. However, those who reported prestroke fatigue showed a relatively high level of fatigue over time in the poststroke period, while patients with no history of pre-stroke fatigue showed a stable course of relatively low fatigue over time. Conclusion. Studies on poststroke fatigue should control for the patient's pre-stroke fatigue level

Sleep-Wake Patterns during the Acute Phase after First-Ever Stroke

This study describes the pattern of day and night sleep and explores relationships between these patterns and sociodemographic and clinical factors as well as sleep environmental context and the patient's subjective sleep quality. Data from 110 patients with first-ever stroke was collected by structured interview surveys, medical record, and objective estimated sleep data from wrist actigraphy. The variability in estimated sleep is large. Half the patients slept either <6 hours or >8 hours per night, and 78% had more than nine awakenings per night. Men slept less than women, and patients sleeping at home had fewer awakenings than those who slept in hospital. It was estimated sleep during daytime in all, except 4, patients. Longer stay in hospital was related to more daytime sleep, and the subjective sleep quality correlated with estimated sleep time, wake time, and wake percentage

Spatiotemporal dynamics of the postnatal developing primate brain transcriptome.

Developmental changes in the temporal and spatial regulation of gene expression drive the emergence of normal mature brain function, while disruptions in these processes underlie many neurodevelopmental abnormalities. To solidify our foundational knowledge of such changes in a primate brain with an extended period of postnatal maturation like in human, we investigated the whole-genome transcriptional profiles of rhesus monkey brains from birth to adulthood. We found that gene expression dynamics are largest from birth through infancy, after which gene expression profiles transition to a relatively stable state by young adulthood. Biological pathway enrichment analysis revealed that genes more highly expressed at birth are associated with cell adhesion and neuron differentiation, while genes more highly expressed in juveniles and adults are associated with cell death. Neocortex showed significantly greater differential expression over time than subcortical structures, and this trend likely reflects the protracted postnatal development of the cortex. Using network analysis, we identified 27 co-expression modules containing genes with highly correlated expression patterns that are associated with specific brain regions, ages or both. In particular, one module with high expression in neonatal cortex and striatum that decreases during infancy and juvenile development was significantly enriched for autism spectrum disorder (ASD)-related genes. This network was enriched for genes associated with axon guidance and interneuron differentiation, consistent with a disruption in the formation of functional cortical circuitry in ASD