On generalized quadratic equations in three prime variables

Leta ₁,a ₂ anda ₃ be any nonzero integers which are relatively prime and not all negative. In this paper, as a parallel problem of [11] for each integerk≥2, we consider the setE(X) of positive integersb≤X which satisfy the condition of congruent solubility and that the equation $$a_1 p_1^2 + a_2 p_2^2 + a_3 p_3^k = b$$ is insoluble in primesp _j. We obtain CardE(X)≤X ^1-ε. Our result extends the wellknown classical results (by Legendre and Gauss and byDavenport andHeilbronn [2]) on the equation $$x_1^2 + x_2^2 + x_3^k = b$$ in integral variablesx _j with the above bound for CardE(X) better than that in [2].     

Qualitative analysis for a mathematical model of aids

In this paper a mathematical model of AIDS is investigated. The conditions of the existence of equilibria and local stability of equilibria are given. The existences of transcritical bifurcation and Hopf bifurcation are also considered, in particular, the conditions for the existence of Hopf bifurcation can be given in terms of the coefficients of the characteristic equation. The method extends the application of the Hopf bifurcation theorem to higher differential equations which occur in biological models, chemical models, and epidemiological models etc.This project is supported by the National Science Foundation Tian Yuan Terms and LNM Institute of Mechanics Academy of Science.This project is supported by the National and Yunnan Province Natural Science Foundation of China.