Construction of Gauss quadratures with prescribed knots via Jacobi matrices is extended to the case where not all orthogonal polynomials exist due to the weight function changing sign. An algorithm is described and is demonstrated by calculating the knots of Kronrod schemes and other Gauss quadratures with prescribed knots.

In this paper, the existence and multiplicity of positive solutions for a critical singular elliptic system with concave and convex nonlinearity and sign-changing weight function, are established. With the help of the Nehari manifold, we prove that the system has at least two positive solutions via variational methods.

In this work, we study C 1-regularity of solutions for one-dimensional p-Laplacian problems and systems with a singular weight which may not be in L 1. On the basis of the regularity result, we give an example to show the multiplicity of positive (or negative) solutions as well as sign-changing solutions especially when the nonlinear term is p-superlinear.

This paper studies a small neural network with three neurons. First, the activation function takes the sign function. Although the network is a simple hybrid systemwith all subsystems being exponentially stable, we find that it can exhibit very complex dynamics such as limit cycles and chaos. Since the sign function is a limit case of sigmoidal functions, we find that chaos robustly exists with...