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Creating a mesh is the first step in a wide range of applications, including scientific computing and computer graphics. An unstructured simplex mesh requires a choice of meshpoints (vertex nodes) and a triangulation. We want to offer a short and simple MATLAB code, described in more detail than usual, so the reader can experiment (and add to the code) knowing the underlying principles. We find the node locations by solving for equilibrium in a truss structure (using piecewise linear force-displacement relations) and we reset the topology by the Delaunay algorithm.The geometry is described implicitly by its distance function. In addition to being much shorter and simpler than other meshing techniques, our algorithm typically produces meshes of very high quality. We discuss ways to improve the robustness and the performance, but our aim here is simplicity. Readers can download (and edit) the codes from

No abstract

Wavelets are new families of basis functions that yield the representation f(x) = C b,kW(2Jx-k). Their construction begins with the solution $(x) to a dilation equation with coefficients ck. Then W comes from @, and the basis comes by translation and dilation of W. It is shown in Part 1 how conditions on the ck lead to approximation properties and orthogonality properties of the wavelets. Part 2 describes the recursive algorithms (also based on the ck) that decompose and reconstruct f: The object of wavelets is to localize as far as possible in both time and frequency, with efficient algorithms.

Abstract. Each discrete cosine transform (DCT) uses N real basis vectors whose components are cosines. In the DCT-4, for example, the jth component of v k is cos(j + 1 2. These basis vectors are orthogonal and the transform is extremely useful in image processing. If the vector x gives the intensities along a row of pixels, its cosine series c k v k has the coefficients c k = (x, v k )/N . They are quickly computed from a Fast Fourier Transform. But a direct proof of orthogonality, by calculating inner products, does not reveal how natural these cosine vectors are.We prove orthogonality in a different way. Each DCT basis contains the eigenvectors of a symmetric "second difference" matrix. By varying the boundary conditions we get the established transforms DCT-1 through DCT-4. Other combinations lead to four additional cosine transforms. The type of boundary condition (Dirichlet or Neumann, centered at a meshpoint or a midpoint) determines the applications that are appropriate for each transform. The centering also determines the period: N − 1 or N in the established transforms, N − in the other four. The key point is that all these "eigenvectors of cosines" come from simple and familiar matrices.