This work is to provide general spectral and pseudo-spectral Jacobi-PetrovGalerkin approaches for the second kind Volterra integro-differential equations. The Gauss-Legendre quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical implementation. For some spectral and pseudo-spectral ...

Legendre was the first to state the law of quadratic reciprocity in the form that we know it and he was able to prove it in some but not all cases, with the first complete proof being given by Gauss. In this paper we trace the evolution of Legendre’s work on quadratic reciprocity in his four great works on number theory. As is well known, Adrien-Marie Legendre (1752-1833) was the first to state...

Tensor products of Gauss-Lobatto quadrature points are frequently used as collocation points in spectral element methods. Unfortunately, it is not known if Gauss-Lobatto points exist in non-tensor-product domains like the simplex. In this work, we show that the n-dimensional tensor-product of Gauss-Lobatto quadrature points are also Fekete points. This suggests a way to generalize spectral meth...

Primes are of fundamental importance in number theory, and primality testing is one of the oldest problems in mathematics. Various algorithms have been presented over the past two millenia, ever since Eratosthenes detailed his eponymous sieve in 274 B.C. After important theoretical advances by Fermat, Euler, Legendre, and Gauss in the seventeeth and eighteenth centuries, the rise of computation...

We describe a technique to analytically compute the multipole moments of a charge distribution confined to a planar triangle, which may be useful in solving the Laplace equation using the fast multipole boundary element method (FMBEM) and for charged particle tracking. This algorithm proceeds by performing the necessary integration recursively within a specific coordinate system, and then trans...

In this thesis, an integrated approach is developed for the regional renement of global gravity eld solutions. The analysis concepts are tailored to the in-situ type character of the observations provided by the new satellite missions CHAMP, GRACE, and GOCE. They are able to evaluate data derived from short arcs of the satellite's orbit and, therefore, oer the opportunity to use regional satell...

This work presents a method to solve boundary value problems based on polynomial approximations and the application of the methods of moments and the Galerkin method. The weighted average residuals are evaluated by improved Gauss-Radau and Gauss-Lobatto quadratures, capable to exactly compute integrals of polynomials of degree 2n and 2n + 2 (where n is the number of internal quadrature points),...

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 ...

was surmised already by Legendre and Gauss. However, it took a hundred years before the first proofs appeared, one by Hadamard and one by de la Vall~e Poussin (1896). Their and all but one of the subsequent proofs make heavy use of the Riemann zeta function. (The one exception is the long so-called elementary proof by Selberg [11] and Erd6s [41.) For Re s > 1 the zeta function is given by the D...