Some inequalities for the eigenvalues of the product of positive semidefinite Hermitian matrices

AbstractLet λ1(A)⩾⋯⩾λn(A) denote the eigenvalues of a Hermitian n by n matrix A, and let 1⩽i1< ⋯ <ik⩽n. Our main results are ∑t=1kλt(GH)⩽∑t=1kλit(G)λn−it+1(H) and ∑t=1kλit(GH)⩽∑t=1kλit(G)λn−t+1(H). Here G and H are n by n positive semidefinite Hermitian matrices. These results extend Marshall and Olkin's inequality ∑t=1kλt(GH)⩽∑t=1kλt(G)λn−t+1(H). We also present analogous results for singular values

Periodic solutions for a porous medium equation

In this paper, we study with a periodic porous medium equation with nonlinear convection terms and weakly nonlinear sources under Dirichlet boundary conditions. Based on the theory of Leray-Shauder fixed point theorem, we establish the existence of periodic solutions

Image Denoising via Nonlinear Hybrid Diffusion

A nonlinear anisotropic hybrid diffusion equation is discussed for image denoising, which is a combination of mean curvature smoothing and Gaussian heat diffusion. First, we propose a new edge detection indicator, that is, the diffusivity function. Based on this diffusivity function, the new diffusion is nonlinear anisotropic and forward-backward. Unlike the Perona-Malik (PM) diffusion, the new forward-backward diffusion is adjustable and under control. Then, the existence, uniqueness, and long-time behavior of the new regularization equation of the model are established. Finally, using the explicit difference scheme (PM scheme) and implicit difference scheme (AOS scheme), we do numerical experiments for different images, respectively. Experimental results illustrate the effectiveness of the new model with respect to other known models

Design and Development of Water-splitting Electrocatalysts Based on Conjugated Triazine Frameworks

Covalent triazine frameworks (CTFs) with rich nitrogen atoms and permanent porosity have been widely used in the field of opto/electronics as supports. In this study, two CTFs with different pore sizes (single pore and heteropore) were synthesized, after which Cu2+, Co2+, Ni2+, Pd2+, Pt2+, and the corresponding metal cluster were introduced into the CTFs as catalytic active sites through the confinement effect of the pores. Among a series of CTFs-based electrocatalysts, DCP-CTF-Pt2+ displays an outstanding electrocatalytic performance with an overpotential of 46 mV and a Tafel slope of 30.2 mV dec-1. Catalytic kinetics analysis indicates that electrocatalytic performance is closely relevant to hierarchical pore and metal size

Spectral Analysis of the Bounded Linear Operator in the Reproducing Kernel Space W

We first introduce some related definitions of the bounded linear operator L in the reproducing kernel space W2m(D). Then we show spectral analysis of L and derive several property theorems

A meshless method based on the Laplace transform for multi-term time-space fractional diffusion equation

Multi-term fractional diffusion equations can be regarded as a generalisation of fractional diffusion equations. In this paper, we develop an efficient meshless method for solving the multi-term time-space fractional diffusion equation. First, we use the Laplace transform method to deal with the multi-term time fractional operator, we transform the time into complex frequency domain by Laplace transform. The properties of the Laplace transform with respect to fractional-order operators are exploited to deal with multi-term time fractional-order operators, overcoming the dependence of fractional-order operators with respect to time and giving better results. Second, we proposed a meshless method to deal with space fractional operators on convex region based on quintic Hermite spline functions based on the theory of polynomial functions dense theorem. Meanwhile, the approximate solution of the equation is obtained through theory of the minimum residual approximate solution, and the error analysis are provided. Third, we obtain the numerical solution of the diffusion equation by inverse Laplace transform. Finally, we first experimented with a single space-time fractional-order diffusion equation to verify the validity of our method, and then experimented with a multi-term time equation with different parameters and regions and compared it with the previous method to illustrate the accuracy of our method

Influence of environmental parameters on the distribution of subfossil chironomids in surface sediments of Bosten lake (Xinjiang, China)

The distribution of larval subfossil chironomids in surface sediment samples obtained from Bosten lake was analysed, and ordination methods were used to identify the influences of physical and chemical parameters on the abundance and diversity of chironomids. A total of 18 chironomid taxa was identified across the 32 samples, 15 of which showed minimum abundance of >1% and were present in more than one site. Chironomus plumosus-type, Microchironomus and Tanytarsus glabrescens-type were the most abundant of the 18 chironomid taxa identified. The distribution of the main chironomid species within Bosten Lake exhibited spatial heterogeneity. T. glabrescens-type was found to dominate Huangshuiwan, in the northwest part of Bosten lake, which had higher salinity and organic content. Microchironomus was mainly distributed in the Kaidu river mouth, in the southwest part of Bosten lake, whereas C. plumosus-type was more abundant in the deep centre, suggesting the importance of water chemistry, water dynamics and water depth on chironomid distribution. The relationship of chironomid distribution with environmental parameters was assessed using redundancy analysis, and the results revealed that water depth, Secchi depth and salinity were the predominant factors influencing chironomid distribution; these factors could explain the chironomid variances of 18.3, 5.3 and 5.0%, respectively. Our surface sediment-derived data provide a baseline for the analysis of future environmental variations of the lakes in the arid regions of northwest China, which are threatened by impacts of global climate change and regional water management

Dust emission reduction enhanced gas-to-particle conversion of ammonia in the North China Plain

Liu et al. found that the formation rate of particulate ammonium is slower in the atmosphere than that observed in the laboratory, while it is sped up due to an increase in aerosol acidity driven by an emission reduction of dust in North China Plain.Ammonium salt is an important component of particulate matter with aerodynamic diameter less than 2.5 mu m (PM2.5) and has significant impacts on air quality, climate, and natural ecosystems. However, a fundamental understanding of the conversion kinetics from ammonia to ammonium in unique environments of high aerosol loading is lacking. Here, we report the uptake coefficient of ammonia (gamma(NH3)) on ambient PM2.5 varying from 2.2 x 10(-4) to 6.0 x 10(-4) in the North China Plain. It is significantly lower than those on the model particles under simple conditions reported in the literature. The probability-weighted gamma(NH3) increases obviously, which is well explained by the annual decrease in aerosol pH due to the significant decline in alkali and alkali earth metal contents from the emission source of dust. Our results elaborate on the complex interactions between primary emissions and the secondary formation of aerosols and the important role of dust in atmospheric chemistry.Peer reviewe