Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimension

Given a smoothly bounded domain $\Omega\Subset\mathbb{R}^n$ with $n\ge 1$ odd, we study the blow-up of bounded sequences $(u_k)\subset H^\frac{n}{2}_{00}(\Omega)$ of solutions to the non-local equation $$(-\Delta)^\frac n2 u_k=\lambda_k u_ke^{\frac n2 u_k^2}\quad \text{in }\Omega,$$ where $\lambda_k\to\lambda_\infty \in [0,\infty)$, and $H^{\frac n2}_{00}(\Omega)$ denotes the Lions-Magenes spaces of functions $u\in L^2(\mathbb{R}^n)$ which are supported in $\Omega$ and with $(-\Delta)^\frac{n}{4}u\in L^2(\mathbb{R}^n)$. Extending previous works of Druet, Robert-Struwe and the second author, we show that if the sequence $(u_k)$ is not bounded in $L^\infty(\Omega)$, a suitably rescaled subsequence $\eta_k$ converges to the function $\eta_0(x)=\log\left(\frac{2}{1+|x|^2}\right)$, which solves the prescribed non-local $Q$-curvature equation $$(-\Delta)^\frac n2 \eta =(n-1)!e^{n\eta}\quad \text{in }\mathbb{R}^n$$ recently studied by Da Lio-Martinazzi-Rivi\`ere when $n=1$, Jin-Maalaoui-Martinazzi-Xiong when $n=3$, and Hyder when $n\ge 5$ is odd. We infer that blow-up can occur only if $\Lambda:=\limsup_{k\to \infty}\|(-\Delta)^\frac n4 u_k\|_{L^2}^2\ge \Lambda_1:= (n-1)!|S^n|$

Prevalence of hypothyroidism amongst pregnant women: a study done in rural set up

Background: Thyroid disorders are the commonest endocrine disorders affecting women of reproductive age group. The most frequent thyroid disorder in pregnancy is maternal hypothyroidism. During early pregnancy the foetus is totally dependent on maternal thyroid hormone supply. Thyroid hormone is critical for foetal brain and intellectual development and some preventable conditions like abruption, pre-eclampsia etc. which produce morbidity and pose special risk for pregnancy and the developing foetus.Methods: All subjects enrolled in the study as per the inclusion criteria will be subjected to a detailed history and clinical examination using a predesigned proforma. A serum TSH value will be sent in 1st trimester between 6-10 weeks period of gestation for all pregnant women.Results: The prevalence of thyroid dysfunction in pregnancy was 10.8% with hypothyroidism being 9.2%, out of which 8.5% were cases of subclinical hypothyroidism and 0.7% were cases of overt hypothyroidism. On assessing the risk factors for developing thyroid dysfunction in pregnancy, increases in incidence were seen with maternal age and increasing BMI, both of which were statistically significant.Conclusions: This study showed an increased prevalence of hypothyroidism in pregnancy recommending a need for universal screening for all pregnant women in the first trimester itself. This study aims at validating the efficacy of the above-mentioned screening

Distribution of sound pressure around a singing cricket: radiation pattern and asymmetry in the sound field

Male field crickets generate calls to attract distant females through tegminal stridulation: the rubbing together of the overlying right wing which bears a file of cuticular teeth against the underlying left wing which carries a sclerotised scraper. During stridulation, specialised areas of membrane on both wings are set into oscillating vibrations to produce acoustic radiation. The location of females is unknown to the calling males and thus increasing effective signal range in all directions will maximise transmission effectiveness. However, producing an omnidirectional sound field of high sound pressure levels may be problematic due to the mechanical asymmetry found in this sound generation system. Mechanical asymmetry occurs by the right wing coming to partially cover the left wing during the closing stroke phase of stridulation. As such, it is hypothesised that the sound field on the left-wing side of the animal will contain lower sound pressure components than on the right-wing side as a result of this coverage. This hypothesis was tested using a novel method to accurately record a high resolution, three dimensional mapping of sound pressure levels around restrained Gryllus bimaculatus field crickets singing under pharmacological stimulation. The results indicate that a bilateral asymmetry is present across individuals, with greater amplitude components present in the right wing side of the animal. Individual variation in sound pressure to either the right or left-wing side is also observed. However, statistically significant differences in bilateral sound field asymmetry as presented here may not affect signalling in the field

Tagged tiger shark (Galeocerda cuvieri) landed atTuticorin

The shark was caught by a mechanised Tuticorin type boat using hooks and line . The fish was caught at a distance of 40 km from shore southeast of Tuticorin in the 26 fathom depth area The fish weighed 130 kg and was sold by auction for Rupees 916. Morphometric measurements (in mm) of tagged tiger shark Galeocerda cuvieri caught off Tuticorin are Total length 2,050, Fork length 1,680, and detailed morphometry were recorded

Quantum Cryptography Based on the Time--Energy Uncertainty Relation

A new cryptosystem based on the fundamental time--energy uncertainty relation is proposed. Such a cryptosystem can be implemented with both correlated photon pairs and single photon states.Comment: 5 pages, LaTex, no figure