Abstract. By extending the breadth rst search algorithm to any d-type critical or subcritical irreducible branching forest, we show that such forests may be encoded through d independent, integer valued, d-dimensional random walks. An application of this coding together with a multivariate extension of the Ballot Theorem which is proved here, allow us to give an explicit form of the law of the total progeny, jointly with the number of subtrees of each type, in terms of the ospring distribution of the branching process. We then apply these results to some enumeration formulas of multitype forests with given degrees and to a new proof of the Lagrange-Good inversion Theorem.
Sensor delay and observation uncertainty often occur in modern computerbased systems, e.g., when the measurement is transmitted to a remote controller through a network medium. In this paper, we revisit the Kalman filter design problem for a stochastic dynamic system with random one-step sensor delay, and derive the optimal unbiased state estimation algorithm. Both full-and reduced-order filters are studied, and the results compare favorably with those of the existing algorithms in examples via simulation.
Classical crystallography is based on the translational periodicity of crystals and the analysis of discrete Bragg reflections. However, it is inadequate for determining disordered structures, of which the diffuse scattering is vital to evaluate the disorder level. The correlated disorder of IM-18 presents as zigzag chains arranged in translational periodicity and the double four-ring units randomly distributed along two dimensions. Supercell models regulated by multiple probabilities were systematically built to simulate the single-crystal and powder X-ray diffraction patterns in order to ascertain the specific disorder configuration in the single-crystal or polycrystalline samples of IM-18. The presence of defects in the polycrystalline sample was proved by combining 29 Si magic angle spinning (MAS) NMR and 1 H-1 H double quantum MAS NMR spectra, and was quantitatively explored by the simulation method. The method could also elucidate other disordered structures in polycrystalline or singlecrystal samples, despite the presence of defects or multidimensional disorder.
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