On the Number of Positive Solutions to a Class of Integral Equations

By using the complete discrimination system for polynomials, we study the number of positive solutions in {\small $C[0,1]$} to the integral equation {\small $\phi (x)=\int_0^1k(x,y)\phi ^n(y)dy$}, where {\small $k(x,y)=\phi_1(x)\phi_1(y)+\phi_2(x)\phi_2(y), \phi_i(x)>0, \phi_i(y)>0, 0<x,y<1, i=1,2,$} are continuous functions on {\small $[0,1]$}, {\small $n$} is a positive integer. We prove the following results: when {\small $n= 1$}, either there does not exist, or there exist infinitely many positive solutions in {\small $C[0,1]$}; when {\small $n\geq 2$}, there exist at least {\small 1}, at most {\small $n+1$} positive solutions in {\small $C[0,1]$}. Necessary and sufficient conditions are derived for the cases: 1) {\small $n= 1$}, there exist positive solutions; 2) {\small $n\geq 2$}, there exist exactly {\small $m(m\in \{1,2,...,n+1\})$} positive solutions. Our results generalize the existing results in the literature, and their usefulness is shown by examples presented in this paper.Comment: 9 page

Friedland–Hersonsky problem for matrix algebra

AbstractIn this paper we show that the answer to problem 3.9 in [Duke Math. J. 69 (1993) 593] is positive when n=2 and negative when n⩾3

THE SERVICE MODELS AND TECHNICAL MODELS OF AGRICULTURAL INFORMATIONIZATION

Types of rural-urban gaps exist and seem expand in China, although agriculture and rural lives have obtained significant enhancement since the opening and reforming policy started in 1980. Amongst these gaps, digital gap is critical itself, and may widen other gaps. How to promote agricultural infor-mationization (agri-informationization) effectively and efficiently is a great challenge to Chinese government and Chinese people nowadays. Meanwhile, the agri-informationization also brings opportunities with new markets, new technology and new rural culture, etc. Firstly, this article introduces the background about Chinese rural areas and rural economy in brief. Secondly, it gives focus on typical types of information service models and information technical models which appearing in the process of agri-informationization. Thirdly, the problems and challenges are analyzed. Finally, the article ends up with a conclusion section. We wish the information service models and technical models introduced here be meaningful references for multiple stakeholders, such as police-makers, basic-level officers, farmers, and information companies, in China and in other developing countries