Let w be an Abelian differential on a compact Riemann surface of genus g ≥ 1. Then |w|2 defines a flat metric with conical singularities and trivial holonomy on the Riemann surface. We obtain an explicit holomorphic factorization formula for the ζ-regularized determinant of the Laplacian in the metric |w|2, generalizing the classical Ray-Singer result in g = 1.

Let $X$ be a smooth connected algebraic curve over an algebraically closed field, let $S$ be a finite closed subset in $X$, and let $\mathcal F_0$ be a lisse $\ell$-torsion sheaf on $X-S$. We study the deformation of $\mathcal F_0$. The universal deformation space is a formal scheme. Its generic fiber has a rigid analytic space structure. By studying this rigid analytic space, we prove a conjecture of Katz which says that if a lisse $\overline{\mathbb Q}_\ell$-sheaf $\mathcal F$ is irreducible and physically rigid, then it is cohomologically rigid, under the extra condition that $\mathcal F\mod \ell$ is absolutely irreducible or that $\mathcal F$ has finite monodromy.

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In this paper, we construct a new suboptimal filter by deriving the Ito's stochastic differential equations of the estimation of higher order central moments satisfy and imposing some conditions to form a closed system. The essentially infinite-dimensional cubic sensor problem has been investigated in detail numerically to illustrate the reasonableness of the imposed conditions, and the numerical experiments support our discussion. A 2-dimensional polynomial filtering problem has also been experimented.

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