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Let R = C[x1, x2, . . ., xn]/(f1, . . ., fm) be a positively graded Artinian algebra. A long-standing conjecture in algebraic geometry, differential geometry, and rational homotopy theory is the non-existence of negative weight derivations on R. Alexsandrov conjectured that there are no negative weight derivations when R is a complete intersection algebra, and Yau conjectured there are no negative weight derivations on R when R is the moduli algebra of a weighted homogeneous hypersurface singularity. This problem is also important in rational homotopy theory and differential geometry. In this paper we prove the non-existence of negative weight derivations on R when the degrees of f1, . . .,fm are bounded below by a constant C depending only on the weights of x1, . . ., xn. Moreover this bound C is improved in several special cases. [ABSTRACT FROM AUTHOR]

In this note, we consider the explicit solution of Duncan-Mortensen-Zakai (DMZ) equation for the finite-dimensional filtering system. We show that Yau filtering system (([[differential] [f.sub.i]/[differential] [x.sub.i]) - ([differential] [f.sub.i]/[differential][x.sub.j]) = [c.sub.ij] = constant for all (i, j) can be solved explicitly with an arbitrary initial condition by solving a system of ordinary differential equations and a Kolmogorov-type equation. Let n be the dimension of state space. We show that we need only n sufficient statistics in order to solve the DMZ equation. Index Terms--Duncan-Mortensen-Zakai (DMZ) equation, finite-dimensional filter, Kolmogorov equation, nonlinear drift.

Of central importance in nonlinear filtering theory is the classification of finite-dimensional estimation algebras, through which we can construct recursive filters. In this paper, we classify all t-dimensional estimation algebras, t [is less than or equal to] 4, for arbitrary state-space dimension. Index Terms--Estimation algebra, nonlinear filter, Wei-Norman approach, Yau filter.

For all known finite dimensional filters, one always assumes that the observation terms be degree one polynomials. However, in practice, the observation terms may be nonlinear, e.g. tracking problems. In this paper, we consider the Yau filtering system (is constant for alli,j) with nonlinear observation terms and arbitrary initial condition. The novelty of the paper lies in (i) the real time computation of the solution of the Duncan-Mortensen-Zakai (DMZ) equation is reduced to the computation of Kolmogorov equation. Based on Gaussian approximation of the initial condition, the Kolmogorov equation can be solved in terms of ordinary differential equations; (ii) For a given probability density function, we give a new and original approach to do Gaussian approximation which is very effective and simple. The direct method developed here can be easily implemented in a real time and memoryless way. Besides, we do not need the controllability and observability assumption. Compared to the extended Kalman filter, our method is much stable and has theoretical proof. The numerical experiments show that the proposed Gaussian approximation method is very effective and our method can track the states very well. [ABSTRACT FROM AUTHOR]

In this paper we study the structure of finite dimensional estimation algebras with state dimension 3 and rank 2 arising from a nonlinear filtering system by using the theories of the Euler operator and underdetermined partial differential equations. The structure of the Wong Ω-matrix is shown to be linear. The fundamental strategy we use in this paper to prove these results is to show that if they were not true, then infinite sequences could be constructed in the finite dimensional estimation algebra. [ABSTRACT FROM AUTHOR]