On Euler's criterion

On the Arnold Stability Criterion

The Arnold stability criterion suggests that a stationary ow of an ideal incompressible uid is stable if a certain quadratic form is deenite. We show that, in three or more dimensions, this quadratic form is never deenite. Typically the form is indeenite, and the spectrum of the associated Hermitian operator ranges from ?1 to 1. The exceptional case is where the velocity eld is harmonic (soleno...

Comments on "Routh Stability Criterion"

— In this note, we have shown special case on Routh stability criterion, which is not discussed, in previous literature. This idea can be useful in computer science applications.

On some criterion of conformality

In the paper it is shown that in order to prove the conformality of a given diffeomorphism between Riemannian manifolds it is enough to limit investigation to the conformal modules of some special families of curves. The main result (Theorem 1.1) asserts that a sufficient condition for the conformality of a diffeomorphism is the conservation of n-modules of a family of mutually orthogonal 1-dim...

On Prohorov's Criterion for Projective Limits

A note on the realignment criterion

For a quantum state in a bipartite system represented as a density matrix, researchers used the realignment matrix and functions on its singular values to study the separability of the quantum state. We obtain bounds for elementary symmetric functions of singular values of realignment matrices. This answers some open problems proposed by Lupo, Aniello, and Scardicchio. As a consequence, we show...