The RNA velocity provides a new avenue to study the stemness and lineage of cells in the development in scRNA-seq data analysis. Some promising extensions of it are proposed and the community is experiencing a fast developing period. However, in this stage, it is of prime importance to revisit the whole process of RNA velocity analysis from the mathematical point of view, which will help to understand the rationale and drawbacks of different proposals. The current paper is devoted to this purpose. We present a thorough mathematical study on the RNA velocity model from dynamics to downstream data analysis. We derived the analytical solution of the RNA velocity model from both deterministic and stochastic point of view. We presented the parameter inference framework based on the maximum likelihood estimate. We also derived the continuum limit of different downstream analysis methods, which provides insights on the construction of transition probability matrix, root and ending-cells identification, and the development routes finding. The overall analysis aims at providing a mathematical basis for more advanced design and development of RNA velocity type methods in the future.

While social living is considered to be an indispensable part of human life in today's ever-connected world, social distancing has recently received much public attention on its importance since the outbreak of the coronavirus pandemic. In fact, social distancing has long been practiced in nature among solitary species, and been taken by human as an effective way of stopping or slowing down the spread of infectious diseases. Here we consider a social distancing problem for how a population, when in a world with a network of social sites, decides to visit or stay at some sites while avoiding or closing down some others so that the social contacts across the network can be minimized. We model this problem as a population game, where every individual tries to find some network sites to visit or stay so that he/she can minimize all his/her social contacts. In the end, an optimal strategy can be found for everyone, when the game reaches an equilibrium. We show that a large class of equilibrium strategies can be obtained by selecting a set of social sites that forms a so-called maximal $r$-regular subnetwork. The latter includes many well studied network types, which are easy to identify or construct, and can be completely disconnected (with $r=0$) for the most strict isolation, or allow certain degrees of connectivities (with $r>0$) for more flexible distancing. We derive the equilibrium conditions of these strategies, and analyze their rigidity and flexibility on different types of $r$-regular subnetworks. We also extend our model to weighted networks, when different contact values are assigned to different network sites.

In addition to measurement noises, real world data are often corrupted by unexpected internal or external errors. Corruption errors can be much larger than the standard noises and negatively affect data processing results. In this paper, we propose a method of identifying corrupted data in the context of function approximation. The method is a two-step procedure consisting of approximation stage and identification stage. In the approximation stage, we conduct straightforward function approximation to the entire data set for preliminary processing. In the identification stage, a clustering algorithm is applied to the processed data to identify the potentially corrupted data entries. In particular, we found $k$-means clustering algorithm to be highly effective. Our theoretical analysis reveals that under sufficient conditions the proposed method can exactly identify all corrupted data entries. Numerous examples are provided to verify our theoretical findings and demonstrate the effectiveness of the method.

The primal dual fixed point (PDFP) proposed in [7] was designed to solve convex composite optimization problems in imaging and data sciences. The algorithm was shown to have some advantages for simplicity and flexibility for divers applications. In this paper we study two modified schemes in order to accelerate its performance. The first one considered is an inertial variant of PDFP, namely inertial PDFP (iPDFP) and the second one is based on a prediction correction framework proposed in [20], namely Prediction Correction PDFP (PC-PDFP). Convergence analysis on both algorithms is provided. Numerical experiments on sparse signal recovery and CT image reconstruction using TV-$L_2$ model are presented to demonstrate the acceleration of the two proposed algorithms compared to the original PDFP algorithm.

In reflection seismology, the inversion of subsurface reflectivity from the observed seismic traces (super-resolution inversion) plays a crucial role in target detection. Since the seismic wavelet in reflection seismic data varies with the travel time, the reflection seismic trace is non-stationary. In this case, a relative amplitude-preserving super-resolution inversion has been a challenging problem. In this paper, we propose a super-resolution inversion method for the non-stationary reflection seismic traces. We assume that the amplitude spectrum of seismic wavelet is a smooth and unimodal function, and the reflection coefficient is an arbitrary random sequence with sparsity. The proposed method can obtain not only the relative amplitude-preserving reflectivity but also the seismic wavelet. In addition, as a by-product, a special $Q$ field can be obtained.
The proposed method consists of two steps. The first step devotes to making an approximate stabilization of non-stationary seismic traces. The key points include: firstly, dividing non-stationary seismic traces into several stationary segments, then extracting wavelet amplitude spectrum from each segment and calculating $Q$ value by the wavelet amplitude spectrum between adjacent segments; secondly, using the estimated $Q$ field to compensate for the attenuation of seismic signals in sparse domain to obtain approximate stationary seismic traces. The second step is the super-resolution inversion of stationary seismic traces. The key points include: firstly, constructing the objective function, where the approximation error is measured in $L_2$ space, and adding some constraints into reflectivity and seismic wavelet to solve ill-conditioned problems; secondly, applying a Hadamard product parametrization (HPP) to transform the non-convex problem based on the $L_p (0 < p < 1)$ constraint into a series of convex optimization problems in $L_2$ space, where the convex optimization problems are solved by the singular value decomposition (SVD) method and the regularization parameters are determined by the L-curve method in the case of single-variable inversion. In this paper, the effectiveness of the proposed method is demonstrated by both synthetic data and field data.

The paper aims to design a distributed algorithm for players in games such that the players can learn Nash equilibriums of non-cooperative games in finite time. We first consider the quadratic non-cooperative games and design estimate protocols for the players such that they can estimate all the other players' actions in distributed manners. In order to make the players track all the other players' real actions in finite time, a bounded gradient dynamics is designed for players to update their actions by using the estimate information. Then the algorithm is extended to more general non-cooperative games and it is proved that players' estimates can converge to all the other players' real actions in finite time and all players can learn the unique Nash equilibrium in finite time under mild assumptions. Finally, simulation examples are provided to verify the validity of the proposed finite-time distributed Nash equilibrium seeking algorithms.

The movement of dislocations and the corresponding crystal plastic deformation are highly influenced by the interaction between dislocations and nearby free surfaces. The boundary condition for inclination angle $θ$_inc which indicates the relation between a dislocation line and the surface is one of the key ingredients in the dislocation dynamic simulations. In this paper, we first present a systematical study on $θ$_inc by molecular static simulations in BCC-irons samples. We also study the inclination angle by using molecular dynamic simulations. A continuum description of inclination angle in both static and dynamic cases is derived based on Onsager's variational principle. We show that the results obtained from continuum description are in good agreement with the molecular simulations. These results can serve as boundary conditions for dislocation dynamics simulations.