By Rogora E.

The 1st primary theorem of invariant conception for the motion of the particular orthogonal crew onm tuples of matrices via simultaneous conjugation is proved in [2]. during this paper, as a primary step towards developing the second one basic theorem, we learn a easy identification among SO(n, ok) invariants ofm matrices.

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A pseudo-gradient vector field). Hence, such a functional may exist but is not necessarily unique. 2. Since we require v to be locally Lipschitz continuous, even in a Hilbert space the gradient ∇ of a functional is not necessarily a pseudo-gradient vector field for . But if is C 2 , then ∇ is a pseudo-gradient vector field for . 2 A Technical Tool 37 This last point makes the following result due to Palais, which is true in general for Banach spaces, very interesting even in the case of a Hilbert space.

1, • the level set • while the set ε is connected has two components. 2. Now consider the function (x, y) = (x 2 + y 2 )2 − 2(x 2 + y 2 ) on R2 . 2). • If a < −1, then a is empty. • If −1 < a < 0, then a is a ring r 2 < x 2 + y 2 < R 2 . • If 0 < a, then a is a ball B(0, R). 2. Crossing irregular values breaks level sets topology (II). second one, it is more discrete; the number of components of a did not change, and it is always equal to 1. But in one situation the level set is simply connected whereas in the other it is not.

Nowadays, more and more authors take the following as an alternative definition of (PS): The functional is said to satisfy (PS) if and only if it satisfies (PS)c for all c ∈ R. 3. The condition (PS)c is a compactness condition on the functional the sense that the set Kc of critical points of at the level c, Kc = u ∈ X ; is compact. 2 Examples To develop some intuition about this compactness notion, we recall some illustrative examples from [956] stressing the fact that the (PS) condition does not have any influence on the size of the critical set of a given functional that can be either empty, finite, or infinite.

### A basic relation between invariants of matrices under the action of the special orthogonal group by Rogora E.

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