In this paper we consider the problem of the asymptotic expansion of double Laplace-type integrals, in the case when the set γ of points where the phase achieves its absolute minimum is a simple curve. It will be shown that the asymptotic behaviour of such integrals is governed by the order of degeneracy of normal derivatives of the phase with respect to the curve γ. Complete asymptotic expansions will be constructed if that order is constant along γ, and the first two coefficients will be explicitly computed. If not, a uniform asymptotic expansion method, involving special functions, is suggested.

In this paper we consider the problem of the asymptotic expansion of double Laplace-type integrals, in the case when the set γ of points where the phase achieves its absolute minimum is a simple curve. It will be shown that the asymptotic behaviour of such integrals is governed by the order of degeneracy of normal derivatives of the phase with respect to the curve γ. Complete asymptotic expansions will be constructed if that order is constant along γ, and the first two coefficients will be explicitly computed. If not, a uniform asymptotic expansion method, involving special functions, is suggested.

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.

Security is an important issue in image storage and communication, encryption is one of the most common ways to ensure security. Recently, many schemes based on chaotic map have been proposed, but most of this method suffers from small key space, which makes them vulnerable to brute forces attacks. In this study, we design a highly robust novel symmetric image encryption scheme which offer good confusion and diffusion qualities, and a large key space to ensure popular security factor and to overcome the weaknesses of the state of the art encryption schemes. In the security analysis section, we prove that our scheme can resist most known attacks, such as cipher image only attack, known and chosen plain image attacks, differential and exhaustive attacks. It is shown in this paper that the use of polar decomposition with chaotic map can gives a fast and secures encryption.

In this paper, we introduce the homogeneous weight and homogeneous Gray map over the ring R_q=F₂[u₁,u₂,…,u_q]/⟨u²_i=0,u_iu_j=u_ju_i⟩ for q≥1q≥1. We also consider the construction of simplex and MacDonald codes of types α and β over this ring and their covering radius.

In this paper, we define the simplex and MacDonald codes of types αα and β over Z₂Z₄. We also examine the covering radii of these codes. Further, we study the binary images of these codes and prove that the binary image of the simplex codes of type αα meets the Gilbert bound.

In this paper, we show that the asymptotic expansion of a double Laplace-type integral with a non-stationary minimum point, located on the boundary of the domain of integration, is governed by the order of contact between the boundary curve and the level curve of the phase through the minimum point. This achievement will enable us to construct complete asymptotic expansions in more general settings. Especially, the problem will be completely solved if the phase and the boundary curve of the domain of integration are analytic near the minimum point.

In this paper we are concerned with the exit time problem of a dynamic system perturbed by a white noise, in the case where the noiseless dynamics has a global attractor in a given domain. By the use of a singular perturbation technique and recent results on the asymptotic expansion of a class of Laplace-type integrals, new results concerning the problem under investigation will be presented.

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.