scholarly journals More 1-cocycles for classical knots

2021 ◽  
pp. 2150032
Author(s):  
Thomas Fiedler
Keyword(s):  
Natural Number ◽  
Moduli Space ◽  
Lagrange Interpolation ◽  
Simple Formula ◽  
Solid Torus ◽  
Interpolation Polynomial ◽  
Lagrange Interpolation Polynomial ◽  
Vassiliev Invariant ◽  
Number Formula

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].

scholarly journals Fuzzy Volterra Integro-Differential Equations Using General Linear Method

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 381 ◽  
Author(s):  
Zanariah Abdul Majid ◽  
Faranak Rabiei ◽  
Fatin Abd Hamid ◽  
Fudziah Ismail
Keyword(s):  
Differential Equations ◽  
Lagrange Interpolation ◽  
Linear Method ◽  
Interpolation Polynomial ◽  
General Linear ◽  
Kutta Method ◽  
Lagrange Interpolation Polynomial ◽  
Runge Kutta Method ◽  
General Linear Method ◽  
Generalized Hukuhara Differentiability

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.

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

Cryptography ◽  
2020 ◽  
Vol 4 (4) ◽  
pp. 34
Author(s):  
Sergey Bezzateev ◽  
Vadim Davydov ◽  
Aleksandr Ometov
Keyword(s):  
Access Control ◽  
Secret Sharing ◽  
Lagrange Interpolation ◽  
Interpolation Formula ◽  
Building Blocks ◽  
Interpolation Polynomial ◽  
Lagrange Interpolation Polynomial ◽  
Crucial Component

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.

Necessary and Sufficient Conditions for Mean Convergence of Lagrange Interpolation for Erdős Weights II

1996 ◽  
Vol 48 (4) ◽  
pp. 737-757 ◽  
Author(s):  
S. B. Damelin ◽  
D. S. Lubinsky
Keyword(s):  
Continuous Function ◽  
Lagrange Interpolation ◽  
Sufficient Conditions ◽  
Interpolation Polynomial ◽  
Weighting Factor ◽  
Lagrange Interpolation Polynomial ◽  
Mean Convergence ◽  
Turán Theorem ◽  
Image Position ◽  
Necessary And Sufficient

AbstractWe complete our investigations of mean convergence of Lagrange interpolation at the zeros of orthogonal polynomials pn(W2, x) for Erdős weights W2 = e-2Q. The archetypal example is Wk,α = exp(—Qk,α), whereα > 1, k ≥ 1, and is the k-th iterated exponential. Following is our main result: Let 1 < p < 4 and α ∊ ℝ Let Ln[f] denote the Lagrange interpolation polynomial to ƒ at the zeros of pn(W2, x) = pn(e-2Q, x). Then forto hold for every continuous function ƒ:ℝ. —> ℝ satisfyingit is necessary and sufficient that α > 1/p. This is, essentially, an extension of the Erdös-Turan theorem on L2 convergence. In an earlier paper, we analyzed convergence for all p > 1, showing the necessity and sufficiency of using the weighting factor 1 + Q for all p > 4. Our proofs of convergence are based on converse quadrature sum estimates, that are established using methods of H. König.

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