Let$S$denote the class of schlicht functions. D. Bertilsson proved recently that for$f∈S, p<0$and$1<-N<-2|p|+1$the modulus of the$N$th Taylor coefficient of ($f′$)^{$p$}takes its maximal value if$f$is the Koebe function. Here a short proof of a generalisation of this result is presented.

We study the spectral geometric properties of the scalar Laplace–Beltrami operator associated to the Weil–Petersson metric $g_{WP}$ on $\mathcal{M}_{\gamma}$ , the Riemann moduli space of surfaces of genus  $\gamma>1$. This space has a singular compactification with respect to $g_{WP}$, and this metric has crossing cusp-edge singularities along a finite collection of simple normal crossing divisors. We prove first that the scalar Laplacian is essentially self-adjoint, which then implies that its spectrum is discrete.

We show that the gradient mapping of the squared norm of FischerBurmeister function is globally Lipschitz continuous and semismooth, which provides a theoretical basis for solving nonlinear second-order cone complementarity problems via the conjugate gradient method and the semismooth Newtons method.

We study an efficient dynamic blind source separation algorithm of convolutive sound mixtures based on updating statistical information in the frequency domain, and minimizing the support of time domain demixing filters by a weighted least square method. The permutation and scaling indeterminacies of separation, and concatenations of signals in adjacent time frames are resolved with optimization of l 1 l norm on cross-correlation coefficients at multiple time lags. The algorithm is a direct method without iterations, and is adaptive to the environment. Computations on recorded benchmark mixtures of speech and music signals show excellent performance. The method in general separates a foreground source from a background of sounds as often encountered in realistic situations.

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