The Diffusion Kernel Filter

A particle filter method is presented for the discrete-time filtering problem with nonlinear ItA ` stochastic ordinary differential equations (SODE) with additive noise supposed to be analytically integrable as a function of the underlying vector-Wiener process and time. The Diffusion Kernel Filter is arrived at by a parametrization of small noise-driven state fluctuations within branches of prediction and a local use of this parametrization in the Bootstrap Filter. The method applies for small noise and short prediction steps. With explicit numerical integrators, the operations count in the Diffusion Kernel Filter is shown to be smaller than in the Bootstrap Filter whenever the initial state for the prediction step has sufficiently few moments. The established parametrization is a dual-formula for the analysis of sensitivity to gaussian-initial perturbations and the analysis of sensitivity to noise-perturbations, in deterministic models, showing in particular how the stability of a deterministic dynamics is modeled by noise on short times and how the diffusion matrix of an SODE should be modeled (i.e. defined) for a gaussian-initial deterministic problem to be cast into an SODE problem. From it, a novel definition of prediction may be proposed that coincides with the deterministic path within the branch of prediction whose information entropy at the end of the prediction step is closest to the average information entropy over all branches. Tests are made with the Lorenz-63 equations, showing good results both for the filter and the definition of prediction.

Mandibulofacial Dysostosis, Severe Lower Eyelid Coloboma, Cleft Palate, and Alopecia: A New Distinct Form of Mandibulofacial Dysostosis or a Severe Form of Johnson-McMillin Syndrome?

We describe a patient with a phenotype characterized by mandibulofacial dysostosis with severe lower eyelid coloboma, cleft palate, abnormal ears, alopecia, delayed eruption and crowded teeth, and sensorioneural hearing loss. The karyotype and the screening for mutations in the coding region of TCOF1 gene were normal. The clinical signs of our case overlap the new mandibulofacial dysostosis described by Stevenson et al. [2007] and the case with Johnson-McMillin syndrome described by Cushman et al. [2005]. The similar clinical signs, mainly, the severe facial involvement observed in these cases suggest that they can represent a new distinct form of mandibulofacial dysostosis or the end of the spectrum of Johnson McMillin syndrome. (C) 2010 Wiley-Liss, Inc.

The univalent polynomial of Suffridge as a summability kernel

A positive summability trigonometric kernel {K(n)(theta)}(infinity)(n=1) is generated through a sequence of univalent polynomials constructed by Suffridge. We prove that the convolution {K(n) * f} approximates every continuous 2 pi-periodic function f with the rate omega(f, 1/n), where omega(f, delta) denotes the modulus of continuity, and this provides a new proof of the classical Jackson`s theorem. Despite that it turns out that K(n)(theta) coincide with positive cosine polynomials generated by Fejer, our proof differs from others known in the literature.