A sound pulse is scattered by a sphere leading to an initial–boundary value problem for the wave equation. A method for solving this problem is developed using integral representations involving Legendre polynomials in a similarity variable and Volterra integral equations. The method is compared and contrasted with the classical method, which uses Laplace transforms in time combined with separa...

Most of the reconstruction-based robust adaptive beamforming (RAB) algorithms require covariance matrix reconstruction (CMR) by high-complexity integral computation. A Gauss-Legendre quadrature (GLQ) method with highest algebraic precision in interpolation-type is proposed to reduce complexity. The interference angular sector RAB regarded as GLQ range, and zeros three-order Legendre orthogonal ...

Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka in 1957. We develop a twovariable refinement WD(s,t) of the Jones polynomial that is invariant under symmetric Reidemeister moves. If D is a symmetric union diagram, representing a ribbon knot K, then the polynomial WD(s,t) nicely reflects their topological propertie...

The hyperbolic partial differential equation with an integral condition arises in many physical phenomena. In this research a numerical technique is developed for the one-dimensional hyperbolic equation that combine classical and integral boundary conditions. The proposedmethod is based on shifted Legendre tau technique. Illustrative examples are included to demonstrate the validity and applica...

In this paper, an Adomian decomposition method using Chebyshev orthogonal polynomials is proposed to solve a well-known class of weakly singular Volterra integral equations. Comparison with the collocation method using polynomial spline approximation with Legendre Radau points reveals that the Adomian decomposition method using Chebyshev orthogonal polynomials is of high accuracy and reduces th...

Difference sets are basic combinatorial structures that have applications in signal processing, coding theory, and cryptography. We consider the problem of identifying a shifted version of the characteristic function of a (known) difference set. We present a generic quantum algorithm that can be used to tackle any hidden shift problem for any difference set in any abelian group. We discuss spec...

Difference sets are basic combinatorial structures that have applications in signal processing, coding theory, and cryptography. We consider the problem of identifying a shifted version of the characteristic function of a (known) difference set and present a general algorithm that can be used to tackle any hidden shift problem for any difference set in any abelian group. We discuss special case...

We consider the sequences over the finite field of four elements obtained by inverse Gray mapping from a pair of binary sequences. We derive the linear complexity and the minimal polynomial of sequences constructed from Legendre sequences, Hall’s sextic sequences and twin-prime sequences using the technique proposed by Tang, Ding, Lim, Kim et al.