Well-Posedness for a System of Quadratic Derivative Nonlinear Schrodinger Equations with Radial Initial Data

Hirayama H, Kinoshita S, Okamoto M. Well-Posedness for a System of Quadratic Derivative Nonlinear Schrodinger Equations with Radial Initial Data. ANNALES HENRI POINCARE. 2020;21(8):2611-2636.In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrodinger equations. This system was introduced by Colin and Colin (Differ Integral Equ 17:297-330, 2004). The first and second authors obtained some well-posedness results in the Sobolev space H-s(R-d). We improve these results for conditional radial initial data by rewriting the system radial form

Decoupling inequality for paraboloid under shell type restriction and its application to the periodic Zakharov system

In this paper, we establish local well-posedness for the Zakharov system on $\mathbb{T}^d$, $d\ge3$ in a low regularity setting. Our result improves the work of Kishimoto. Moreover, the result is sharp up to $\varepsilon$-loss of regularity when $d=3$ and $d\ge5$ as long as one utilizes the iteration argument. We introduce ideas from recent developments of the Fourier restriction theory. The key element in the proof of our well-posedness result is a new trilinear discrete Fourier restriction estimate involving paraboloid and cone. We prove this trilinear estimate by improving Bourgain--Demeter's range of exponent for the linear decoupling inequality for paraboloid under the constraint that the input space-time function $f$ satisfies ${\rm supp}\, \hat{f} \subset \{ (\xi,\tau) \in \mathbb{R}^{d+1}: 1- \frac1N \le |\xi| \le 1 + \frac1N,\; |\tau - |\xi|^2| \le \frac1{N^{2}} \}$ for large $N\ge1$.Comment: Updated versio

Association between BDNF and clinical response in OCD

Aim: Brain-derived neurotrophic factor (BDNF) is a member of the neurotrophin family, and it promotes the development and function of dopaminergic and serotonergic neurons. The Met allele of the BDNF Val66Met polymorphism is associated with a decrease in activity-dependent secretion of BDNF compared with the Val allele, and a number of studies have provided evidence for the association between this polymorphism and obsessive-compulsive disorder (OCD). The purpose of this study was to investigate whether this functional variant of the BDNF gene is associated with OCD and treatment response in patients with OCD in the Japanese population. Methods: We first performed a case–control association study between the BDNF Val66Met polymorphism and OCD (175 cases and 2,027 controls). Then, we examined an association between this polymorphism and treatment response in 96 patients with OCD. Results: We found no significant association between the Met allele and OCD risk or between the Met allele and treatment responses to selective serotonin reuptake inhibitors or serotonin reuptake inhibitor with an atypical antipsychotic (P>0.05). Conclusion: Our results suggest that the BDNF Val66Met polymorphism may not be associated as a risk factor for developing OCD or with therapeutic response in patients with OCD in the Japanese population