In this paper, we construct a Dieudonné theory for $\pi$-divisible $\mathcal{O}$-modules over a perfect field of characteristic $p$. Applying this theory, we prove the existence of slope filtration of $\pi$-divisible $\mathcal{O}$-modules over an integral normal Noetherian base. We also study minimal $\pi$-divisible $\mathcal{O}$-modules over an algebraically closed field of characteristic $p$ and prove that their isomorphism classes are determined by their $\pi$-torsion parts by introducing Oort’s filtration. Moreover, after a detailed study of deformations of $\pi$-divisible $\mathcal{O}$-modules via displays, we prove the generalized Traverso’s isogeny conjecture.
For each odd integer $r$ greater than one and not divisible by three we give explicit examples of infinite families of simply and tangentially homotopy equivalent but pairwise nonhomeomorphic $5$-dimensional closed homogeneous spaces with fundamental group isomorphic to $\mathbb{Z}/r$. As an application we construct the first examples of manifolds which possess infinitely many metrics of nonnegative sectional curvature with pairwise non-homeomorphic homogeneous souls of codimension three with trivial normal bundle, such that their curvatures and the diameters of the souls are uniformly bounded. As a by-product, if $L$ is a smooth and closed manifold homotopy equivalent to the standard $3$-dimensional lens space then the moduli space of complete smooth metrics of nonnegative sectional curvature on $L \times S^2 \times \mathbb{R}$ has infinitely many components. These manifolds are the first examples of manifolds fulfilling such geometric conditions and they serve as solutions to a problem posed by I. Belegradek, S. Kwasik and R. Schultz.
We obtain an analogue of Nevanlinna theory of holomorphic mappings from a complete and stochastically complete Kähler manifold into a complex projective manifold. When certain curvature conditions are imposed, some Nevanlinna-type defect relations based on heat diffusion are established.
In this paper, we study the stability of feedback particle filter (FPF) for linear filtering systems with Gaussian noises. We first provide some local contraction estimates of the exact linear FPF, whose conditional distribution is exactly the posterior distribution of the state as long as their initial values are equal. Then we study the convergence of the linear FPF formed by $N$ particles, and prove that the mean squared errors between the actual moments $(m_t, P_t)$ and their approximations $(m^{(N)}_t , P^{(N)}_t)$ by FPF are of order $\mathcal{O}(1/N)$ and decay exponentially fast as time $t$ goes to infinity.
We introduce a new curvature constraint that provides an analog of the real bisectional curvature considered by Yang–Zheng [28] for the Aubin–Yau inequality. A unified perspective of the various forms of the Schwarz lemma is given, leading to novel Schwarz-type inequalities in both the Kähler and Hermitian categories.