The purpose of this paper is to investigate the k-nearest neighbor classification rule for spatially dependent data. Some spatial mixing conditions are considered, and under such spatial structures, the well known k-nearest neighbor rule is suggested to classify spatial data. We established consistency and strong consistency of the classifier under mild assumptions. Our main results extend the consistency result in the i.i.d. case to the spatial case.

We generalize the logarithmic decomposition theorem of Deligne–Illusie to a filtered version. There are two applications. The easier one provides a mod p proof for a vanishing theorem in characteristic zero. The deeper one gives rise to a positive characteristic analogue of a theorem of Deligne on the mixed Hodge structure attached to complex algebraic varieties.

A finite generating set of the centre of any quantum group is obtained, where the generators are given by an explicit formulae. For the slightly generalised version of the quantum group which we work with, we show that this set of generators is algebraically independent, thus the centre is isomorphic to a polynomial algebra.

This paper provides necessary as well as sufficient conditions on the Hurst parameters so that the continuous time parabolic Anderson model $\frac{\partial u}{\partial t}=\frac{1}{2}\Delta +u{\dot{W}}$ on $[0, \infty )\times {{\mathbb {R}}}^d $ with $d\ge 1$ has a unique random field solution, where W(t, x) is a fractional Brownian sheet on $[0, \infty )\times {{\mathbb {R}}}^d$ and formally $\dot{W} =\frac{\partial ^{d+1}}{\partial t \partial x_1 \cdots \partial x_d} W(t, x)$. When the noise W(t, x) is white in time, our condition is both necessary and sufficient when the initial data u(0, x) is bounded between two positive constants. When the noise is fractional in time with Hurst parameter $H_0>1/2$, our sufficient condition, which improves the known results in the literature, is different from the necessary one.

Mesh segmentation is a fundamental and critical task in mesh processing, and it has been studied extensively in computer graphics and geometric modeling communities. However, current methods are not well suited for segmenting large meshes which are now common in many applications. This paper proposes a new spectral segmentation method specifically designed for large meshes inspired by multi-resolution representations. Building on edge collapse operators and progressive mesh representations, we first devise a feature-aware simplification algorithm that can generate a coarse mesh which keeps the same topology as the input mesh and preserves as many features of the input mesh as possible. Then, using the spectral segmentation method proposed in Tong et al. (IEEE Trans Vis Comput Graph 26(4):1807–1820, 2020), we perform partition on the coarse mesh to obtain a coarse segmentation which mimics closely the desired segmentation of the input mesh. By reversing the simplification process through vertex split operators, we present a fast algorithm which maps the coarse segmentation to the input mesh and therefore obtain an initial segmentation of the input mesh. Finally, to smooth some jaggy boundaries between adjacent parts of the initial segmentation or align with the desired boundaries, we propose an efficient method to evolve those boundaries driven by geodesic curvature flows. As demonstrated by experimental results on a variety of large meshes, our method outperforms the state-of-the-art segmentation method in terms of not only speed but also usability.

We use reflecting Brownian motion (RBM) to prove the well-known Gauss–Bonnet–Chern theorem for a compact Riemannian manifold with boundary. The boundary integrand is obtained by carefully analyzing the asymptotic behavior of the boundary local time of RBM for small times.

We establish an identity for $Ef\left( \varvec{Y}\right) - Ef\left( \varvec{X}\right) $, when $\varvec{X}$ and $\varvec{Y}$ both have matrix variate skew-normal distributions and the function f satisfies some weak conditions. The characteristic function of matrix variate skew normal distribution is then derived. We then make use of it to derive some necessary and sufficient conditions for the comparison of matrix variate skew-normal distributions under six different orders, such as usual stochastic order, convex order, increasing convex order, upper orthant order, directionally convex order and supermodular order.

Let R be a commutative ring having nonzero identity and M be a unital R-module. Assume that $S\subseteq R$ is a multiplicatively closed subset of R. Then, M satisfies S-Noetherian spectrum condition if for each submodule N of M, there exist $s\in S$ and a finitely generated submodule $F\subseteq N$ such that $sN\subseteq \text {rad}_{M}(F)$, where $\text {rad}_{M}(F)$ is the prime radical of F in the sense (McCasland and Moore in Commun Algebra 19(5):1327–1341, 1991). Besides giving many properties and characterizations of S-Noetherian spectrum condition, we prove an analogous result to Cohen’s theorem for modules satisfying S-Noetherian spectrum condition. Moreover, we characterize modules having Noetherian spectrum in terms of modules satisfying the S-Noetherian spectrum condition.