Consider the following McKean–Vlasov SDE:

where $\mu _{X_t}$ stands for the distribution of $X_t$ and $K(t,x): {{\mathbb {R}}}_+\times {{\mathbb {R}}}^d\rightarrow {{\mathbb {R}}}^d$ is a time-dependent divergence free vector field. Under the assumption $K\in L^q_t({\widetilde{L}}_x^p)$ with $\frac{d}{p}+\frac{2}{q}<2$, where ${\widetilde{L}}^p_x$ stands for the localized $L^p$-space, we show the existence of weak solutions to the above SDE. As an application, we provide a new proof for the existence of weak solutions to 2D Navier–Stokes equations with measure as initial vorticity.

In this work, we aim to develop an effective fully discrete Spectral-Galerkin numerical scheme for the multi-vesicular phase-field model of lipid vesicles with adhesion potential. The essence of the scheme is to introduce several additional auxiliary variables and design some corresponding auxiliary ODEs to reformulate the system into an equivalent form so that the explicit discretization for the nonlinear terms can also achieve unconditional energy stability. Moreover, the scheme has a full decoupling structure and can avoid calculating variable-coefficient systems. The advantage of this scheme is its high efficiency and ease of implementation, that is, only by solving two independent linear biharmonic equations with constant coefficients for each phase-field variable, the scheme can achieve the second-order accuracy in time, spectral accuracy in space, and unconditional energy stability. We strictly prove that the fully discrete energy stability that the scheme holds and give a detailed step-by-step implementation process. Further, numerical experiments are carried out in 2D and 3D to verify the convergence rate, energy stability, and effectiveness of the developed algorithm.

Let G be a finite group and assume that a group of automorphisms A is acting on G such that A and G have coprime orders. We prove that the fact of imposing specific properties on the second maximal A-invariant subgroups of G determines that G is either soluble or isomorphic to a few non-soluble groups such as PSL(2, 5) or SL(2, 5).

In CNC machining, the tool path planning of the cutter plays an important role. In this paper, we generate a space-filling and continuous tool path for free-form surface represented by the triangular mesh with a confined scallop height. The tool path is constructed from connected Fermat spirals (CFS) but with fewer inflection points. Comparing with the newly developed CFS method, only about half of the number of inflection points are involved. Moreover, the kinematic constraints are simultaneously taken into account to increase the feedrates in machining. Finally, we use a micro-line trajectory technique to smooth the tool path. Experimental results and physical cutting tests are provided to illustrate and clarify our method.

A Frölicher-type inequality for Bott-Chern cohomology and its relation with

-lemma were introduced in [1]. In this paper, we generalize these results to the cohomology groups with coefficients in flat complex vector bundles.

We establish a global Torelli theorem for the complete family of Calabi-Yau threefolds arising from cyclic triple covers of ${{\mathbb {P}}}^3$ branched along six stable hyperplanes.

The generalized additive partial linear models (GAPLM) have been widely used for flexible modeling of various types of response. In practice, missing data usually occurs in studies of economics, medicine, and public health. We address the problem of identifying and estimating GAPLM when the response variable is nonignorably missing. Three types of monotone missing data mechanism are assumed, including logistic model, probit model and complementary log-log model. In this situation, likelihood based on observed data may not be identifiable. In this article, we show that the parameters of interest are identifiable under very mild conditions, and then construct the estimators of the unknown parameters and unknown functions based on a likelihood-based approach by expanding the unknown functions as a linear combination of polynomial spline functions. We establish asymptotic normality for the estimators of the parametric components. Simulation studies demonstrate that the proposed inference procedure performs well in many settings. We apply the proposed method to the household income dataset from the Chinese Household Income Project Survey 2013.

Positive data are very common in many scientific fields and applications; for these data, it is known that estimation and inference based on relative error criterion are superior to that of absolute error criterion. In prediction problems, conformal prediction provides a useful framework to construct flexible prediction intervals based on hypothesis testing, which has been actively studied in the past decade. In view of the advantages of the relative error criterion for regression problems with positive responses, in this paper, we combine the relative error criterion (REC) with conformal prediction to develop a novel REC-based predictive inference method to construct prediction intervals for the positive response. The proposed method satisfies the finite sample global coverage guarantee and to some extent achieves the local validity. We conduct extensive simulation studies and two real data analysis to demonstrate the competitiveness of the new proposed method.