We show some results concerning the number of solutions of the equation y+Ax=b (yTx=0, y0, x0) which plays a central role in the dc analysis of transistor circuits. In particular, we give sufficient conditions for the equation to possess exactly 2l (ln) solutions, where n is the dimension of the vector x.

The author once defined the Ω-matrix and showed that it played an important role for estimating the number of solutions of a resistive circuit containing active elements such as CCCS's. The Ω-matlix is a generalization of the wellknown P-matrix. This paper gives the necessary and sufficient conditions for the Ω-matrix.

The number of solutions of a nonlinear equation x = sgn(Wx) is discussed. The equation is derived for the determination of equilibrium points of a kind of Hopfield neural networks. We impose some conditions on W. The conditions correspond to the case where a Hopfield neural network has n neurons arranged on a ring, each neuron has connections only from k preceding neurons and the magnitude of k connections decrease as the distance between two neurons increases. We show that the maximum number of solutions for the above case is extremely few and is independent of the number of neurons, n, if k is less than or equal to 4. We also show that the number of solutions generally increases exponentially with n by considering the case where k = n-1.