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.

Wireless capsule endoscopy (WCE) has several benefits over traditional endoscopy such as its portability and ease of usage, particularly for remote internet of things (IoT)-assisted healthcare services. During the WCE procedure, a significant amount of redundant video data is generated, the transmission of which to healthcare centers and gastroenterologists securely for analysis is challenging as well as wastage of several resources including energy, memory, computation, and bandwidth. In addition to this, it is inherently difficult and time consuming for gastroenterologists to analyze this huge volume of gastrointestinal video data for desired contents. To surmount these issues, we propose a secure video summarization framework for outdoor patients going through WCE procedure. In the proposed system, keyframes are extracted using a light-weighted video summarization scheme, making it more suitable for WCE. Next, a cryptosystem is presented for security of extracted keyframes based on 2D Zaslavsky chaotic map. Experimental results validate the performance of the proposed cryptosystem in terms of robustness and high-level security compared to other recent image encryption schemes during dissemination of important keyframes to healthcare centers and gastroenterologists for personalized WCE.

Wireless capsule endoscopy (WCE) has several benefits over traditional endoscopy such as its portability and ease of usage, particularly for remote internet of things (IoT)-assisted healthcare services. During the WCE procedure, a significant amount of redundant video data is generated, the transmission of which to healthcare centers and gastroenterologists securely for analysis is challenging as well as wastage of several resources including energy, memory, computation, and bandwidth. In addition to this, it is inherently difficult and time consuming for gastroenterologists to analyze this huge volume of gastrointestinal video data for desired contents. To surmount these issues, we propose a secure video summarization framework for outdoor patients going through WCE procedure. In the proposed system, keyframes are extracted using a light-weighted video summarization scheme, making it more suitable for WCE. Next, a cryptosystem is presented for security of extracted keyframes based on 2D Zaslavsky chaotic map. Experimental results validate the performance of the proposed cryptosystem in terms of robustness and high-level security compared to other recent image encryption schemes during dissemination of important keyframes to healthcare centers and gastroenterologists for personalized WCE.

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.