From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond

The discovery of connections between the distribution of energy levels of heavy nuclei and spacings between prime numbers has been one of the most surprising and fruitful observations in the twentieth century. The connection between the two areas was first observed through Montgomery's work on the pair correlation of zeros of the Riemann zeta function. As its generalizations and consequences have motivated much of the following work, and to this day remains one of the most important outstanding conjectures in the field, it occupies a central role in our discussion below. We describe some of the many techniques and results from the past sixty years, especially the important roles played by numerical and experimental investigations, that led to the discovery of the connections and progress towards understanding the behaviors. In our survey of these two areas, we describe the common mathematics that explains the remarkable universality. We conclude with some thoughts on what might lie ahead in the pair correlation of zeros of the zeta function, and other similar quantities.Comment: Version 1.1, 50 pages, 6 figures. To appear in "Open Problems in Mathematics", Editors John Nash and Michael Th. Rassias. arXiv admin note: text overlap with arXiv:0909.491

Higher-Order Nuclear-Polarizability Corrections in Atomic Hydrogen

Nuclear-polarizability corrections that go beyond unretarded-dipole approximation are calculated analytically for hydrogenic (atomic) S-states. These retardation corrections are evaluated numerically for deuterium and contribute -0.68 kHz, for a total polarization correction of 18.58(7) kHz. Our results are in agreement with one previous numerical calculation, and the retardation corrections completely account for the difference between two previous calculations. The uncertainty in the deuterium polarizability correction is substantially reduced. At the level of 0.01 kHz for deuterium, only three primary nuclear observables contribute: the electric polarizability, $\alpha_E$, the paramagnetic susceptibility, $\beta_M$, and the third Zemach moment, $_{(2)}$. Cartesian multipole decomposition of the virtual Compton amplitude and its concomitant gauge sum rules are used in the analysis.Comment: 26 pages, latex, 1 figure -- Submitted to Phys. Rev. C -- epsfig.sty require

Effects of acute treatment with a tryptophan-rich protein hydrolysate on plasma amino acids, mood and emotional functioning in older women

RATIONALE: Effective functioning of the neurotransmitter serotonin is important for optimal cognitive and emotional function. Dietary supplements able to increase availability to the brain of the precursor amino acid, tryptophan (TRP), and thereby enhance serotonin synthesis, can have measurable impact on these psychological processes. OBJECTIVES: This study involves a randomised controlled trial of a TRP-rich egg-white protein hydrolysate (DSM Nutritional Products Ltd., Switzerland) on plasma amino acids, cognition, mood and emotional processing in older women. METHODS: Following a baseline test day without treatment, 60 healthy women aged 45–65 years received drinks containing either 2 or 4 g of TRP-rich protein hydrolysate product or 3.11 g casein hydrolysate as a control. One hour later, they undertook a 2-h battery of cognitive and emotional tests. RESULTS: The TRP-rich protein hydrolysate produced the expected dose-dependent increase in the ratio of plasma TRP to competing large neutral amino acids. TRP-rich protein hydrolysate (2 g only) prevented both the decline in wellbeing and increase in fatigue seen over the test session in the control group. This treatment dose resulted in a significant shift in emotional processing towards positive words and reduced negative bias in assessing negative facial expressions. Effects on cognition were small and not statistically reliable and are not reported here. However, there was no evidence for any adverse effects. CONCLUSIONS: Consumption of a low dose of TRP-rich protein hydrolysate may have beneficial effects on emotional function that could promote feelings of wellbeing, possibly conferring resistance to deterioration in mood in healthy subjects or depressive episodes

Functional connectivity underlying hedonic response to food in female adolescents with atypical AN: the role of somatosensory and salience networks.

Atypical anorexia nervosa (AN) usually occurs during adolescence. Patients are often in the normal-weight range at diagnosis; however, they often present with signs of medical complications and severe restraint over eating, body dissatisfaction, and low self-esteem. We investigated functional circuitry underlying the hedonic response in 28 female adolescent patients diagnosed with atypical AN and 33 healthy controls. Participants were shown images of food with high (HC) or low (LC) caloric content in alternating blocks during functional MRI. The HC > LC contrast was calculated. Based on the previous literature on full-threshold AN, we hypothesized that patients would exhibit increased connectivity in areas involved in sensory processing and bottom-up responses, coupled to increased connectivity from areas related to top-down inhibitory control, compared with controls. Patients showed increased connectivity in pathways related to multimodal somatosensory processing and memory retrieval. The connectivity was on the other hand decreased in patients in salience and attentional networks, and in a wide cerebello-occipital network. Our study was the first investigation of food-related neural response in atypical AN. Our findings support higher somatosensory processing in patients in response to HC food images compared with controls, however HC food was less efficient than LC food in engaging patients' bottom-up salient responses, and was not associated with connectivity increases in inhibitory control regions. These findings suggest that the psychopathological mechanisms underlying food restriction in atypical AN differ from full-threshold AN. Elucidating the mechanisms underlying the development and maintenance of eating behavior in atypical AN might help designing specific treatment strategies