Let Ω i , Ω o be bounded open connected subsets of R n that contain the origin. Let Ω(𝜖) ≡ Ω o ∖𝜖Ω i for small 𝜖 > 0. Then, we consider a linear transmission problem for the Helmholtz equation in the pair of domains 𝜖Ω i and Ω(𝜖) with Neumann boundary conditions on 𝜕Ω o . Under appropriate conditions on the wave numbers in 𝜖Ω i and Ω(𝜖) and on the parameters involved in the transmission conditions on 𝜖𝜕Ω i , the transmission problem has a unique solution (u i (𝜖, •), u o (𝜖, •)) for small values of 𝜖 > 0. Here, u i (𝜖, •) and u o (𝜖, •) solve the Helmholtz equation in 𝜖Ω i and Ω(𝜖), respectively. Then, we prove that ifx ∈ Ω o ∖ {0}, then u o (𝜖, x) can be expanded into a convergent power expansion of 𝜖, 𝜅 n 𝜖 log 𝜖, 𝛿 2,n log −1 𝜖 for 𝜖 small enough. Here, 𝜅 n = 1 if n is even and 𝜅 n = 0 if n is odd, and 𝛿 2, 2 ≡ 1 and 𝛿 2, n ≡ 0 if n ≥ 3.

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Objective: To investigate the association of perinatal risk factors including delivery mode with mortality in very low birthweight (VLBW) in a tertiary hospital setting. Methods: Medical records of 241 live-born VLBW infants (≤1500 g) were retrospectively reviewed. Details of maternal, obstetrical, perinatal risk factors and their associations with infant mortality were evaluated. Results: The overall infant mortality rate was 23.2%. Mortality was significantly higher for infants born at ≤27 gestational weeks and with a birthweight of ≤750 g (p = 0.000 and p = 0.000, respectively), showing a steep decrease thereafter. On ROC analysis, a cut off of 26.5 weeks was determined for mortality with a sensitivity of 57.1% and a specificity of 90.3% (area under the curve = 0.792, 95% CI: 0.719-0.866). On multivariate regression analysis, gestational week at birth, birthweight, antenatal steroid treatment and pathologic Doppler ultrasound findings were found as independent risk factors for mortality. Conclusions: Gestational week at birth, birthweight and antenatal steroid treatment remain the most important perinatal risk factors for infant mortality in VLBW infants. Mode of delivery does not seem to be associated with mortality when adjusted for other perinatal risk factors.

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Let f be a holomorphic function in the unit disc and |f(z)?1| < 1 for |z| < 1. We generalize the uniqueness portion of Schwarz?s lemma and provide sufficient conditions on the local behavior of f near a finite set of boundary points that needed for f to be a finite Blaschke product.