Spectral square moments of a resonance sum for Maass forms

Let f be a Maass cusp form for Г0（N） with Fourier coefficients 1 k2. λf（n） and Laplace eigenvalue 1/4 ＋k2 For real α≠0 and β 〉 0, consider the sum Sx（f; α,β） = ∑n λf（n）e（αnβ）φ（n/X）, where φ is a smooth function of compact support. We prove bounds for the second spectral moment of Sx （f;α, β）, with the eigenvalue tending towards infinity. When the eigenvalue is sufficiently large, we obtain an average bound for this sum in terms of X. This implies that if f has its eigenvalue beyond X1/2＋ε, the standard resonance main term for Sx（f; ±2√q 1/2）, q ∈Z＋, cannot appear in general. The method is adopted from proofs of subconvexity bounds for Rankin-Selberg L-functions for GL（2） × GL（2）. It contains in particular a proof of an asymptotic expansion of a well-known oscillatory integral with an enlarged range of Kε≤ L≤ K1-ε. The same bounds can be proved in a similar way for holomorphie cusp forms.

Weighted stationary phase of higher orders

The subject matter of this paper is an integral with exponential oscillation of phase f（x） weighted by g（x） on a finite interval [α, β]. When the phase f（x） has a single stationary point in （α,β）, an nth-order asymptotic expansion of this integral is proved for n ≥ 2. This asymptotic expansion sharpens the classical result for n = 1 by M. N. Huxley. A similar asymptotic expansion was proved by V. Blomer, R. Khan and M. Young under the assumptions that f（x） and g（x） are smooth and g（x） is compactly supported on R. In the present paper, however, these functions are only assumed to be continuously differentiable on [α,β] 2n ＋ 3 and 2n ＋ 1 times, respectively. Because there are no requirements on the vanishing of g（x） and its derivatives at the endpoints α and β, the present asymptotic expansion contains explicit boundary terms in the main and error terms. The asymptotic expansion in this paper is thus applicable to a wider class of problems in analysis, analytic number theory, and other fields.

Intranasal administration of tetrabromobisphenol A bis(2-hydroxyethyl ether) induces neurobehavioral changes in neonatal Sprague Dawley rats

Tetrabromobisphenol A（TBBPA） and its derivatives are now being highly concerned due to their emerging environmental occurrence and deleterious effects on non-target organisms.Considering the potential neurotoxicity of TBBPA derivatives which has been demonstrated in vitro, what could happen in vivo is worthy of being studied. Tetrabromobisphenol A bis（2-hydroxyethyl ether）（TBBPA-BHEE）, a representative TBBPA derivative, was selected for a21-day exposure experiment on neonatal Sprague Dawley（SD） rats through intranasal administration. The neurobehavioral, histopathological changes, and differentially expressed genes based on RNA microarray were investigated to evaluate the neurological effects of this chemical. The results indicated that TBBPA-BHEE exposure significantly compromised the motor co-ordination performance and the locomotor activities（p 〈 0.05）. The neurobehavioral phenotype could be attributed to the obvious histopathological changes in both cerebrum and cerebellum, such as neural cell swelling, microglial activation and proliferation. A total of 911 genes were up-regulated, whereas 433 genes were down-regulated. Gene set enrichment analysis showed multiple signaling pathways, including ubiquitin-mediated proteolysis and wingless-int（Wnt） signaling pathway etc. were involved due to TBBPA-BHEE exposure. The gene ontology enrichment analysis showed the basic cellular function and the neurological processes like synaptic transmission were influenced. The toxicological effects of TBBPA-BHEE observed in this study suggested the potential neuronal threaten from unintended exposure,which would be of great value in the biosafety evaluation of TBBPA derivatives.