Let (Mⁿ, g) be an n-dimensional complete noncompact Riemannian manifold with harmonic curvature and positive Sobolev constant. In this paper, by employing an elliptic estimation method, we show that (Mⁿ, g) is a space form if it has sufficiently small L^n/2-norms of trace-free curvature tensor and nonnegative scalar curvature. Moreover, we get a gap theorem for (Mⁿ, g) with positive scalar curvature.

This work is to analyze a spectral Jacobi-collocation approximation for Volterra integral equations with singular kernel ϕ(t, s) = (t − s)^−µ. In an earlier work of Y. Chen and T. Tang [J. Comput. Appl. Math., 2009, 233: 938–950], the error analysis for this approach is carried out for 0 < µ < 1/2 under the assumption that the underlying solution is smooth. It is noted that there is a technical problem to extend the result to the case of Abel-type, i.e., µ = 1/2. In this work, we will not only extend the convergence analysis by Chen and Tang to the Abel-type but also establish the error estimates under a more general regularity assumption on the exact solution.

This paper deals with the blow-up properties of the positive solutions to a degenerate parabolic system with localized sources and nonlocal boundary conditions. We investigate the influence of the reaction terms, the weight functions, local terms and localized source on the blow-up properties. We will show that the weight functions play the substantial roles in determining whether the solutions will blow-up or not, and obtain the blow-up conditions and its blow-up rate estimate.

This paper gives an analytic existence proof of the Schubart periodic orbit with arbitrary masses, a periodic orbit with singularities in the collinear three-body problem. A “turning point” technique is introduced to exclude the possibility of extra collisions and the existence of this orbit follows by a continuity argument on differential equations generated by the regularized Hamiltonian.

In this paper, we shall study the uniqueness problems on meromorphic functions sharing nonzero finite value or fixed point. We have answered some questions posed by Dyavanal. Our results improve or generalize a few of known results.