Binary sequences constructed by Legendre symbols are widely used in communication and cryptography since they have many good pseudo-random properties.In this paper,we determine the 2-adic complexity of the sum sequenc...Binary sequences constructed by Legendre symbols are widely used in communication and cryptography since they have many good pseudo-random properties.In this paper,we determine the 2-adic complexity of the sum sequence of any k many Legendre sequences and show that the 2-adic complexity of the sum sequences of any k many Legendre sequences reaches the maximum by proving the case of k=2 and 3,which implies that the sum sequences can resist the attack of rational approximation algorithm.展开更多
Legendre polynomial method is well-known in modeling acoustic wave characteristics.This method uses for the mechanical displacements a single polynomial expansion over the entire sandwich layers.This results in a limi...Legendre polynomial method is well-known in modeling acoustic wave characteristics.This method uses for the mechanical displacements a single polynomial expansion over the entire sandwich layers.This results in a limitation in the accuracy of the field profile restitution.Thus,it can deal with the guided waves in layered sandwich only when the material properties of adjacent layers do not change significantly.Despite the great efforts regarding this issue in the literature,there remain open questions.One of them is:“what is the exact threshold of contrasting material properties of adjacent layers for which this polynomial method cannot correctly restitute the roots of guided waves?”We investigated this numerical issue using the calculated guided phase velocities in 0°/φ/0°-carbon fibre reinforced plastics(CFRP)sandwich plates with gradually increasing angleφ.Then,we approached this numerical problem by varying the middle layer thickness h90°for the 0°/90°/0°-CFRP sandwich structure,and we proposed an exact thickness threshold of the middle layer for the Legendre polynomial method limitations.We showed that the polynomial method fails to calculate the quasi-symmetric Lamb mode in 0°/φ/0°-CFRP whenφ>25°.Moreover,we introduced a new Lamb mode so-called minimum-group-velocity that has never been addressed in literature.展开更多
We study the Legendre ellipsoid and the LYZ ellipsoid.First,we give a direct proof of the Cramer-Rao inequality for convex bodies.Second,we prove that origin-centered ellipsoids are the only convex bodies with identic...We study the Legendre ellipsoid and the LYZ ellipsoid.First,we give a direct proof of the Cramer-Rao inequality for convex bodies.Second,we prove that origin-centered ellipsoids are the only convex bodies with identical Löwner and Legendre ellipsoids.Finally,we establish a mean width inequality for convex bodies whose LYZ ellipsoids are the Euclidean unit ball.展开更多
The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and ...The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.展开更多
文摘Binary sequences constructed by Legendre symbols are widely used in communication and cryptography since they have many good pseudo-random properties.In this paper,we determine the 2-adic complexity of the sum sequence of any k many Legendre sequences and show that the 2-adic complexity of the sum sequences of any k many Legendre sequences reaches the maximum by proving the case of k=2 and 3,which implies that the sum sequences can resist the attack of rational approximation algorithm.
基金supported by the National Natural Science Foundation of China(Grant No.12102131).
文摘Legendre polynomial method is well-known in modeling acoustic wave characteristics.This method uses for the mechanical displacements a single polynomial expansion over the entire sandwich layers.This results in a limitation in the accuracy of the field profile restitution.Thus,it can deal with the guided waves in layered sandwich only when the material properties of adjacent layers do not change significantly.Despite the great efforts regarding this issue in the literature,there remain open questions.One of them is:“what is the exact threshold of contrasting material properties of adjacent layers for which this polynomial method cannot correctly restitute the roots of guided waves?”We investigated this numerical issue using the calculated guided phase velocities in 0°/φ/0°-carbon fibre reinforced plastics(CFRP)sandwich plates with gradually increasing angleφ.Then,we approached this numerical problem by varying the middle layer thickness h90°for the 0°/90°/0°-CFRP sandwich structure,and we proposed an exact thickness threshold of the middle layer for the Legendre polynomial method limitations.We showed that the polynomial method fails to calculate the quasi-symmetric Lamb mode in 0°/φ/0°-CFRP whenφ>25°.Moreover,we introduced a new Lamb mode so-called minimum-group-velocity that has never been addressed in literature.
文摘We study the Legendre ellipsoid and the LYZ ellipsoid.First,we give a direct proof of the Cramer-Rao inequality for convex bodies.Second,we prove that origin-centered ellipsoids are the only convex bodies with identical Löwner and Legendre ellipsoids.Finally,we establish a mean width inequality for convex bodies whose LYZ ellipsoids are the Euclidean unit ball.
文摘The numerical approach for finding the solution of fractional order systems of boundary value problems (BPVs) is derived in this paper. The implementation of the weighted residuals such as Galerkin, Least Square, and Collocation methods are included for solving fractional order differential equations, which is broadened to acquire the approximate solutions of fractional order systems with differentiable polynomials, namely Legendre polynomials, as basis functions. The algorithm of the residual formulations of matrix form can be coded efficiently. The interpretation of Caputo fractional derivatives is employed here. We have demonstrated these methods numerically through a few examples of linear and nonlinear BVPs. The results in absolute errors show that the present method efficiently finds the numerical solutions of fractional order systems of differential equations.
