Let $K$ be the quadratic field $\mathbf{Q}(\sqrt{m})$ with discirimant $d = pq$. Using Legendre's theorem on the solvability of the equation $ax^2 + by^2 = z^2$, we give necessary and sufficient conditions for the class number of $K$ in the narrow sense to be divisible by 8. The approach recovers known criteria but is simpler and can be extended to compute the sylow 2-subgroup of the ideal class group of quadratic fields.

We shall show a relation between Gauss' ternary quadratic form and an ideal of a quadratic field. Using this relation, we can compute rapidly the 2-part of ideal class group of a quadratic field in narrow sense and in wide sense.

We give a classification of $C^2$-regular projectively Anosov flows on closed three dimensional manifolds. More precisely, we show that if the manifold is connected then such a flow must be either an Anosov flow or represented as a finite union of $\mathbf{T}^2 \times I$-models.

In the following article we present the results on the existence of almost periodic solutions for impulsive neural networks. We also formulate several results on exponential stability of these equations.