Video deraining via nonlocal low-rank regularization

Outdoor videos captured in rainy weather may be significantly corrupted by the undesired rain streaks, which severely affect the video processing tasks in outdoor computer vision systems. In this paper, we propose a tensor-based video rain streaks removal method using the nonlocal low-rank regularization. Specifically, we first divide videos into overlapped spatial–temporal patches. Then for each patch, we group its nonlocal similar spatial–temporal patches to form a third-order tensor. To model the clean videos, we characterize the wealth redundancy by adopting the tensor nuclear norm to regularize the low-rankness of the third-order tensors formed by similar spatial–temporal patches of clean videos. We also consider the piecewise smoothness and the temporal continuity of clean videos and utilize the unidirectional total variation to enhance the smoothness and continuity. Moreover, as rain streaks are sparse and smooth along the rain direction, we model the rain streaks by employing an ℓ₁ norm and the unidirectional total variation penalty to boost the sparsity and directional smoothness, respectively. We develop an efficient alternating direction method of multipliers to solve the proposed model. Experimental results on both synthetic and real rainy videos show that our method outperforms the state-of-the-art methods quantitatively and qualitatively.     

Self-reciprocal and self-conjugate-reciprocal irreducible factors of xn − λ and their applications

In this paper, we present some necessary and sufficient conditions under which an irreducible polynomial is self-reciprocal (SR) or self-conjugate-reciprocal (SCR). By these characterizations, we obtain some enumeration formulas of SR and SCR irreducible factors of $x^n-\lambda$, $\lambda \in {\mathbb{F}}_q^{⁎}$, over ${\mathbb{F}}_q$, which are just open questions posed by Boripan et al. (2019). We also count the numbers of Euclidean and Hermitian LCD constacyclic codes and show some well-known results on Euclidean and Hermitian self-dual constacyclic codes in a simple and direct way.     

Uncertainty Relations Based on Modified Wigner-Yanase-Dyson Skew Information

Uncertainty relation is a core issue in quantum mechanics and quantum information theory. We introduce modified generalized Wigner-Yanase-Dyson (MGWYD) skew information and modified weighted generalized Wigner-Yanase-Dyson (MWGWYD) skew information, and establish new uncertainty relations in terms of the MGWYD skew information and MWGWYD skew information.

     

Component ordering strategies in assembly systems with uncertain capacity and random yield

Sourcing components in a complex global supplier network may lead to a high degree of supply uncertainty. Events, such as unexpected production defects or insufficient supplier capacity, can cause unexpected shortages of required components and halt the assembly of final products. Accordingly, the assembly enterprises must effectively manage various supply uncertainties in their component ordering decisions to avoid such component shortfalls. These issues have guided this research to investigate the optimal ordering strategies of an assembler facing the following two types of supply uncertainty: the uncertain production capacity of a standard component (component 1) and the random production yield of a core component (component 2). The assembler makes the component ordering decisions before these supply uncertainties are realized. We characterize the optimal ordering decision and find that the assembler should order components 1 and 2 according to a fixed ratio, which only depends on the random yield of component 2 and the production cost of component 1, but not on the uncertain capacity of component 1. A case study is presented to further explore the intertwined effects of these two uncertainties in an assembly system. Finally, the model is extended to consider a secondary option of buying additional component 1 s after observing some or all of the supply uncertainties, and this secondary option endows the firm with different capabilities in counteracting the supply uncertainties.     

More efficient time integration for Fourier pseudospectral DNS of incompressible turbulence

Time integration of Fourier pseudospectral DNS is usually performed using the classical fourth-order accurate Runge-Kutta method or other second- or third-order methods, with a fixed step size. We investigate the use of higher-order Runge-Kutta pairs and automatic step size control based on local error estimation. We find that the fifth-order accurate Runge-Kutta pair of Bogacki and Shampine gives much greater accuracy at a significantly reduced computational cost. Specifically, we demonstrate speedups of 2× to 10× for the same accuracy. Numerical tests (including the Taylor-Green vortex, Rayleigh-Taylor instability, and homogeneous isotropic turbulence) confirm the reliability and efficiency of the method. We also show that adaptive time stepping provides a significant computational advantage for some problems (like the development of a Rayleigh-Taylor instability) without compromising accuracy.     

Evaluation of singular integrals for anisotropic elastic boundary element analysis

To improve the numerical evaluation of weakly singular integrals appearing in the boundary element method, a logarithmic Gaussian quadrature formula is usually suggested in the literature. In this formula the singular function is expressed in terms of the distance between source point and field point, which is a real variable. When an anisotropic elastic solid is considered, most of the existing fundamental solutions are written in terms of complex variables. When the problems with holes, cracks, inclusions, or interfaces are considered, to suit for the shape of the boundaries usually a mapping function is introduced and then the solutions are expressed in terms of mapped complex variables. To deal with the trouble induced by the complex variables, in this study through proper change of variables we develop a simple way to improve the evaluation of weakly singular integrals, especially for the problems of anisotropic elastic solids containing holes, cracks, inclusions, or interfaces. By simple matrix expansion, the proposed method is extended to the problems with piezoelectric or magneto-electro-elastic solids. By using the dual reciprocity method, the proposed method employed for the elastostatic fundamental solution can also be applied to the elastodynamic analysis.     

The energy-critical nonlinear wave equation with an inverse-square potential

We study the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four. In the defocussing case, we prove that arbitrary initial data in the energy space lead to global solutions that scatter. In the focusing case, we prove scattering below the ground state threshold.