The algebraic structure of the rank two Racah algebra is studied in detail. We provide an automorphism group of this algebra, which is isomorphic to the permutation group of five elements. This group ...can be geometrically interpreted as the symmetry of a folded icosidodecahedron. It allows us to study a class of equivalent irreducible representations of this Racah algebra. They can be chosen symmetric so that their transition matrices are orthogonal. We show that their entries can be expressed in terms of Racah polynomials. This construction gives an alternative proof of the recurrence, difference and orthogonal relations satisfied by the Tratnik polynomials, as well as their expressions as a product of two monovariate Racah polynomials. Our construction provides a generalization of these bivariate polynomials together with their properties.
In this paper, we define some n x n Hessenberg matrices and then we obtain determinants and permanents of their matrices that give the odd and even terms of bivariate complex Perrin polynomials. ...Moreover, we use our results to apply the application cryptology area. We discuss the Affine-Hill method over complex numbers by improving our matrix as the key matrix and present an experimental example to show that our method can work for cryptography.
The maritime transportation industry is a prime target for cybersecurity attacks due to the inherent risks of transmitting sensitive information required for its smooth operation. The introduction of ...digital systems in maritime transport has made them susceptible to a range of cybersecurity threats. These attacks can lead to the unauthorized acquisition and misuse of vulnerable data, such as personal information of crew and passengers, vessel location, route, schedule, and other related information. To address this issue, a new lightweight authentication protocol has been proposed by utilizing drone technology in conjunction with the 5th generation mobile network (5G) communication. The effectiveness of the proposed protocol has been analyzed, which demonstrates its ability to withstand various security attacks while maintaining low communication and computation costs. Furthermore, it successfully meets the security and functionality requirements of anonymity and untraceability. A comprehensive simulation study and simulation using the network simulator (NS3) have been conducted to assess the impact on various network performance parameters for the proposed scheme. In addition, various experiments using machine-learning algorithms for Big data analytics show the efficiency of the proposed scheme in terms of accuracy, precision, recall and F1 score.
Quantum secret sharing is an important technique of quantum cryptography. A (
t
,
n
) threshold quantum secret sharing scheme with fairness is proposed. Firstly, the proposed scheme only requires ...the distributor to provide a share for each participant to achieve fairness. Secondly, the distributor interacts with participants only during the secret distribution phase. Another important point is that our scheme combines the privacy features of secure multi-party computing to ensure the reuse of participants’ secret shares. Finally, in our scheme, the correctness, security and fairness are discussed in detail.
This paper presents two efficient algorithms to determine whether a bivariate polynomial, possibly with complex coefficients, does not vanish in the cross product of two closed right-half planes (is ..."2-C stable"). A 2-C stable polynomial in the denominator of a two-dimensional analog filter has been proved (not long ago) to imply bounded-input bounded-output (BIBO) stability. The two algorithms are entirely different but both rely on a recently proposed fraction-free (FF) Routh test for complex polynomials in this transaction. The first algorithm tests the 2-C stability of a bivariate polynomial of degree <inline-formula> <tex-math notation="LaTeX">(n_{1},n_{2}) </tex-math></inline-formula> in order <inline-formula> <tex-math notation="LaTeX">n^{6} </tex-math></inline-formula> of elementary operations (when <inline-formula> <tex-math notation="LaTeX">n_{1}=n_{2}=n </tex-math></inline-formula>). It is a "tabular type" two-dimensional stability test that can be regarded as a "Routh table" whose scalar entries were replaced by univariate polynomials. The second 2-C stability test is obtained from the first by its telepolation. It carries out the 2-C stability test by a finite collection of FF Routh tests and requires only order <inline-formula> <tex-math notation="LaTeX">n^{4} </tex-math></inline-formula> elementary operations. Both algorithms possess an integer-preserving property that enhances them with additional merits including numerical error-free decision on 2-C stability.
Zero sets of bivariate Hermite polynomials Area, Iván; Dimitrov, Dimitar K.; Godoy, Eduardo
Journal of mathematical analysis and applications,
01/2015, Volume:
421, Issue:
1
Journal Article
Peer reviewed
Open access
We establish various properties for the zero sets of three families of bivariate Hermite polynomials. Special emphasis is given to those bivariate orthogonal polynomials introduced by Hermite by ...means of a Rodrigues type formula related to a general positive definite quadratic form. For this family we prove that the zero set of the polynomial of total degree n+m consists of exactly n+m disjoint branches and possesses n+m asymptotes. A natural extension of the notion of interlacing is introduced and it is proved that the zero sets of the family under discussion obey this property. The results show that the properties of the zero sets, considered as affine algebraic curves in R2, are completely different for the three families analyzed.
PDF
