The sign-pattern set of a sign stable switched system is an isogenous set.
We also say that two sign-patterns are isogenous with each other if they conform to the condition of Definition 8.
Considering an arbitrary isogenous sign-pattern set, we define the joining operation of all the sign-patterns to develop a special sign-pattern named the original sign-pattern of the set.
Let x [element of] F be a point on the line and denote by I(x) [subset] F the isogeny class of x (set of all y [element of] F isogenous to x).
Two points (by which we mean as usual "F-points") [x.sub.1],[x.sub.2] [element of] [A.sub.g,n] will be called isogenous if, the abelian F-varieties [A.sub.1], [A.sub.2] corresponding to these points are isogenous, by which we mean that there exists a surjective homomorphism with finite kernel [Pi] : [A.sub.1] [approaches] [A.sub.2].
The second Appendix will contain a "nondifferential" application of Theorem 0.1 providing a criterion for two isogenous elliptic curves to be isomorphic.
Moreover one can define a map [G.sup.0] : Y [right arrow] [A.sub.3] sending the curve C to a principally polarized abelian threefold isogenous to A.
In fact there are abelian threefolds with [End.sub.Q](A) [not equal to] Q just in codimension at least 2 and exactly 2 if and only if A is isogenous to a product of an abelian surface and an elliptic curve.
[MATHEMATICAL EXPRESSION OMITTED] has to be isogenous to a product of an abelian surface and an elliptic curve.