Let E be a n-dimensional vector space over a field k and ω a trivector of Λ3E. We can associate to the trivector ω several invariants either algebraic, arithmetic or geometric. In this paper we consider the following three invariants, the commutant C(ω), the complexity c(ω) and the automorphisms group Aut(ω). We show that there exists a vector space E and a trivector ω of Λ3E for which C(ω) is not a Frobenius algebra. We also show that the complexity c(ω) and the length l(ω) are equal. Finally, we prove the existence of a trivector ω such that Aut(ω) is not a FC-group.

Let E be a n-dimensional vector space over a field k and ω a trivector of Λ3E. We can associate to the trivector ω several invariants either algebraic, arithmetic or geometric. In this paper we consider the following three invariants, the commutant C(ω), the complexity c(ω) and the automorphisms group Aut(ω). We show that there exists a vector space E and a trivector ω of Λ3E for which C(ω) is not a Frobenius algebra. We also show that the complexity c(ω) and the length l(ω) are equal. Finally, we prove the existence of a trivector ω such that Aut(ω) is not a FC-group.

In the last decade the need for new and robust watermarking schemes has been increased because of the large illegal possession by not respecting the intellectual property rights in the multimedia in the internet. In this paper we introduce a novel blind robust watermarking scheme which exploits the positive circulant matrices in frequency domain which is the SVD, Different applications such as copyright protection, control and illicit distributions have been given. Simulation results indicate that the proposed method is robust against attacks as common digital processing: compression, blurring, dithering, printing and scanning, etc. and subterfuge attacks (collusion and forgery) also geometric distortions and transformations. Furthermore, good results of NC (normalized correlation) and PSNR (Peak signal-tonoise ratio) have been achieved while comparing with recent state of the art watermarking algorithms.