Strong asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight

We study asymptotics of the recurrence coefficients of orthogonal polynomials associated to the generalized Jacobi weight, which is a weight function with a finite number of algebraic singularities on $[-1,1]$. The recurrence coefficients can be written in terms of the solution of the corresponding Riemann-Hilbert problem for orthogonal polynomials. Using the steepest descent method of Deift and Zhou, we analyze the Riemann-Hilbert problem, and obtain complete asymptotic expansions of the recurrence coefficients. We will determine explicitly the order $1/n$ terms in the expansions. A critical step in the analysis of the Riemann-Hilbert problem will be the local analysis around the algebraic singularities, for which we use Bessel functions of appropriate order.Comment: 31 pages, 6 figures, 21 reference

Universality for eigenvalue correlations at the origin of the spectrum

We establish universality of local eigenvalue correlations in unitary random matrix ensembles (1/Z_n) |\det M|^{2\alpha} e^{-n\tr V(M)} dM near the origin of the spectrum. If V is even, and if the recurrence coefficients of the orthogonal polynomials associated with |x|^{2\alpha} e^{-nV(x)} have a regular limiting behavior, then it is known from work of Akemann et al., and Kanzieper and Freilikher that the local eigenvalue correlations have universal behavior described in terms of Bessel functions. We extend this to a much wider class of confining potentials V. Our approach is based on the steepest descent method of Deift and Zhou for the asymptotic analysis of Riemann-Hilbert problems. This method was used by Deift et al. to establish universality in the bulk of the spectrum. A main part of the present work is devoted to the analysis of a local Riemann-Hilbert problem near the origin.Comment: 28 pages, 6 figures, technical problem in second version removed, to appear in Commun. Math. Phy

Feeling happy enhances early spatial encoding of peripheral information automatically: electrophysiological time-course and neural sources.

Previous research has shown that positive mood may broaden attention, although it remains unclear whether this effect has a perceptual or a postperceptual locus. In this study, we addressed this question using high-density event-related potential methods. We randomly assigned participants to a positive or a neutral mood condition. Then they performed a demanding oddball task at fixation (primary task ensuring fixation) and a localization task of peripheral stimuli shown at three positions in the upper visual field (secondary task) concurrently. While positive mood did not influence behavioral performance for the primary task, it did facilitate stimulus localization on the secondary task. At the electrophysiological level, we found that the amplitude of the C1 component (reflecting an early retinotopic encoding of the stimulus in V1) was enhanced in the positive, as compared with the neutral, mood group. Importantly, this effect appeared to be largely automatic, because it occurred regardless of the task relevance of the peripheral stimulus and prior to top-down gain control effects seen at the level of the subsequent P1 component. This early effect was also observed irrespective of a change of the target-related P300 component (primary task) by positive mood. These results suggest that positive mood can automatically boost the spatial encoding of peripheral stimuli early on following stimulus onset. This effect can eventually underlie the broadening of spatial attention, which has been associated with this specific mood state

Feeling happy when feeling down : the effectiveness of positive mental imagery in dysphoria

Background and objectives Mental imagery can evoke strong emotional responses, but imagery perspective can influence the response, with observer perspective reducing emotionality. This is important provided that positive imagery can be an effective mood repairing strategy in healthy individuals. However, (sub-clinical) depressed individuals tend to spontaneously adopt an observer perspective. We investigated whether positive imagery would result in a similar emotional response in dysphoric and non-dysphoric individuals when instructed and trained to use field perspective imagery. Additionally, we compared the emotional response in dysphoric individuals who received instructions to dysphoric individuals who received no instructions on processing mode during positive memory recall. Methods Dysphoric and non-dysphoric individuals completed a mood induction procedure imagining positive or neutral memories. They received instructions and practice in the use of field perspective imagery. An additional control group of dysphoric individuals recalled positive memories without receiving instructions on processing mode. Results Dysphoric and non-dysphoric individuals who received instructions on field perspective imagery reported similar use of field and observer perspective imagery, and a similar positive emotional response. Dysphoric individuals who did not receive specific instructions, as compared to those who did, reported greater use of observer perspective and lower levels of positive affect afterwards. Limitations A dysphoric sample limits generalization to clinically depressed individuals, although these individuals are at risk for developing depression. However, mental imagery used in relapse prevention is likely targeting sub-clinical populations. Conclusions Providing practice in field perspective imagery could potentially improve the effectiveness of positive memory recall as a mood repair strategy in (sub-clinically) depressed individuals, and may therefore have important therapeutic benefits.</p

Modulatory effects of positive mood on cognition : lessons from attention and error monitoring.

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