The fundamental problem of nonlinear filtering theory is how to solve robust D-M-Z equation in real time and in memoryless manner. This paper describes a new real time algorithm which reduces the nonlinear filtering problem to off-line computations. Our algorithm gives convergent solutions in both pointwise sense and L/sup 2/ in case that the drift term and observation dynamic term have linear growths. The algorithm presented is slightly better than that given in our previous paper (2000).

The human cerebellum has recently been discovered to contribute to cognition and emotion beyond the planning and execution of movement, suggesting its functional heterogeneity. We aimed to identify the functional parcellation of the cerebellum using information from resting-state functional magnetic resonance imaging (rs-fMRI). For this, we introduced a new data-driven decomposition-based functional parcellation algorithm, called Sparse Dictionary Learning Clustering (SDLC). SDLC integrates dictionary learning, sparse representation of rs-fMRI, and k-means clustering into one optimization problem. The dictionary is comprised of an over-complete set of time course signals, with which a sparse representation of rs-fMRI signals can be constructed. Cerebellar functional regions were then identified using k-means clustering based on the sparse representation of rs-fMRI signals. We solved SDLC using a multi-block hybrid proximal alternating method that guarantees strong convergence. We evaluated the reliability of SDLC and benchmarked its classification accuracy against other clustering techniques using simulated data. We then demonstrated that SDLC can identify biologically reasonable functional regions of the cerebellum as estimated by their cerebello-cortical functional connectivity. We further provided new insights into the cerebello-cortical functional organization in children.

We give a definition of character sheaves on the group compactification which is equivalent to Lusztig's definition in [G. Lusztig, Parabolic character sheaves, II, Mosc. Math. J. 4 (4) (2004) 869896]. We also prove some properties of the character sheaves on the group compactification.

If M is a quaternionic manifold and P is an S 1-instanton over M, then Joyce constructed a hypercomplex manifold we call(M) over M. These hypercomplex manifolds admit a U (2)-action of a special type permuting the complex structures. We show that up to double covers, all such hypercomplex manifolds arise in this way. Examples, including that of a hypercomplex structure on SU (3), show the necessity of including double covers of(M).

We develop a specification test for the transition density of a discretely sampled continuous-time jump-diffusion process, based on a comparison of a nonparametric estimate of the transition density or distribution function with their corresponding parametric counterparts assumed by the null hypothesis. As a special case, our method applies to pure diffusions. We provide a direct comparison of the two densities for an arbitrary specification of the null parametric model using three different discrepancy measures between the null and alternative transition density and distribution functions. We establish the asymptotic null distributions of proposed test statistics and compute their power functions. We investigate the finite-sample properties through simulations and compare them with those of other tests. This article has supplementary material online.