Upper Bound for Lebesgue Constant of Bivariate Lagrange Interpolation Polynomial on the Second Kind Chebyshev Points

In the paper, we study the upper bound estimation of the Lebesgue constant of the bivariate Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the second kind on the square − 1,1 2 . And, we prove that the growth order of the Lebesgue constant is O n + 2 2 . This result is different from the Lebesgue constant of Lagrange interpolation polynomial on the unit disk, the growth order of which is O n . And, it is different from the Lebesgue constant of the Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the first kind on the square − 1,1 2 , the growth order of which is O ln n 2 .

More 1-cocycles for classical knots

Let [Formula: see text] be the topological moduli space of long knots up to regular isotopy, and for any natural number [Formula: see text] let [Formula: see text] be the moduli space of all [Formula: see text]-cables of framed long knots which are twisted by a string link to a knot in the solid torus [Formula: see text]. We upgrade the Vassiliev invariant [Formula: see text] of a knot to an integer valued combinatorial 1-cocycle for [Formula: see text] by a very simple formula. This 1-cocycle depends on a natural number [Formula: see text] with [Formula: see text] as a parameter and we obtain a polynomial-valued 1-cocycle by taking the Lagrange interpolation polynomial with respect to the parameter. We show that it induces a non-trivial pairing on [Formula: see text] already for [Formula: see text].

On Secret Sharing with Newton’s Polynomial for Multi-Factor Authentication

Security and access control aspects are becoming more and more essential to consider during the design of various systems and the tremendous growth of digitization. One of the related key building blocks in this regard is, essentially, the authentication process. Conventional schemes based on one or two authenticating factors can no longer provide the required levels of flexibility and pro-activity of the access procedures, thus, the concept of threshold-based multi-factor authentication (MFA) was introduced, in which some of the factors may be missing, but the access can still be granted. In turn, secret sharing is a crucial component of the MFA systems, with Shamir’s schema being the most widely known one historically and based on Lagrange interpolation polynomial. Interestingly, the older Newtonian approach to the same problem is almost left without attention. At the same time, it means that the coefficients of the existing secret polynomial do not need to be re-calculated while adding a new factor. Therefore, this paper investigates this known property of Newton’s interpolation formula, illustrating that, in specific MFA cases, the whole system may become more flexible and scalable, which is essential for future authentication systems.

A WSN Layer-Cluster Key Management Scheme Based on Quadratic Polynomial and Lagrange Interpolation Polynomial

Since current key management schemes are mainly designed for static and planar networks, they are not very suitable for the layer-cluster wireless sensor networks (WSNs), a WSN layer-cluster key management scheme based on quadratic polynomial and Lagrange interpolation polynomial is proposed, in which the main idea of this scheme along the research line of broadcast identity authentication, session key, group key, network key and personal key. Specifically, authentication key can be established on the basis of Fourier series for identity authentication; session key is established by a multiple asymmetric quadratic polynomial, in which session key information is encrypted by the authentication key to ensure the security of intermediate interactive information; based on the former two keys, group key is established on the basis of Lagrange interpolation polynomial, in which the nodes of the cluster are not directly involved; the generation and management of network key is similar to the group key, in which the establishment idea is to regard the BS and all cluster heads as a group; the generation and management of personal key is also similar to the group key, the difference is that the personal key can be obtained by cluster nodes through getting the Lagrange interpolation polynomial coefficients based on their own random key information. It is analyzed that the proposed layer-cluster key management scheme can guarantee the identity of network nodes firstly through forward authentication and reverse authentication, and session key, group key and network key will guarantee the independence of the keys’ management and avoids the problem of single point failure compared with LEAP protocol, and personal key will guarantee the privacy of network.

Fuzzy Volterra Integro-Differential Equations Using General Linear Method

In this paper, a fuzzy general linear method of order three for solving fuzzy Volterra integro-differential equations of second kind is proposed. The general linear method is operated using the both internal stages of Runge-Kutta method and multivalues of a multisteps method. The derivation of general linear method is based on the theory of B-series and rooted trees. Here, the fuzzy general linear method using the approach of generalized Hukuhara differentiability and combination of composite Simpson’s rules together with Lagrange interpolation polynomial is constructed for numerical solution of fuzzy volterra integro-differential equations. To illustrate the performance of the method, the numerical results are compared with some existing numerical methods.

Increasing signal/acoustic interference ratio in telecommunications audio exchange by adaptive filtering methods

The paper deals with the issues of increasing signal/noise ratio in telecommunication audio exchange systems. The study of characteristics of speech signals and acoustic noises, such as mathematical expectation, dispersion, relative intensity of acoustic speech signals and various types of acoustic noises and interference is carried out. It is shown that in the design of telecommunications systems, in particular loudspeaker systems operating under the influence of external acoustic noise of high intensity, it is necessary to solve the problem of developing algorithms to effectively suppress the above mentioned interference to ensure the necessary signal/noise ratio in communication systems. A mathematical model of the autocorrelation function of the speech signal by using the Lagrange interpolation polynomial of order 10, considered the creation of adaptive algorithms to suppress acoustic noise by linear filtering methods. Thus suppression of acoustic noises and hindrances is possible at the expense of operated change of area of a cutting in the interval from 0 Hz to 300-1000 Hz, depending on a hindrance conditions.