In this paper, we develop an efficient spectral method for numerically solving the nonlinear Volterra integral equation with weak singularity and delays. Based on symmetric collocation points, is illustrated, convergence results are obtained. end, two numerical experiments carried out to confirm theoretical results.

The spectral methods seek the numerical solutions by a set of known polynomials. The main advantage of using spectral methods for solving atmospheric problems is the high efficiency and conservations of important quadratic quantities such as kinetic energy and enstrophy. Namely, we can get very high accuracy through the exponential convergence. The conservation of the quadratic quantities are i...

In the framework of structural representations for applications in image understanding, we establish links between similarities, hypergraph theory and mathematical morphology. We propose new similarity measures and pseudo-metrics on lattices of hypergraphs based on morphological operators. New forms of these operators on hypergraphs are introduced as well. Some examples based on various dilatio...

We present an efficient preconditioner for the orbital minimization method when the Hamiltonian is discretized using planewaves (i.e., pseudospectral method). This novel preconditioner is based on an approximate Fermi operator projection by pole expansion, combined with the sparsifying preconditioner to efficiently evaluate the pole expansion for a wide range of Hamiltonian operators. Numerical...

Computer-produced summaries have traditionally been evaluated by comparing them with human-produced summaries using the Fmeasure. However, the F-measure is not appropriate when alternative sentences are possible in a human-produced extract. In this paper, we examine some evaluation methods devised to overcome the problem, including utility-based evaluation. By giving scores for moderately impor...

The high-order accuracy of Fourier method makes it the method of choice in many large scale simulations. We discuss here the stability of Fourier method for nonlinear evolution problems, focusing on the two prototypical cases of the inviscid Burgers’ equation and the multi-dimensional incompressible Euler equations. The Fourier method for such problems with quadratic nonlinearities comes in two...

A numerical study is given on the spectral methods and the high order WENO finite difference scheme for the solution of linear and nonlinear hyperbolic partial differential equations with stationary and non-stationary singular sources. The singular source term is represented by the δ-function. For the approximation of the δ-function, the direct projection method is used that has been proposed i...