mathematics
Approximation for weighted residuals
Intro We ended up the article about weighted residuals pointing out three possible approaches to the problem. We expand them here.
Read Full Post | Make a Comment ( None so far )The Rayleigh-Ritz method
Intro This method starts straight from the weak form of a differential equation. It follows by taking the same functional space for the solution functions and the weight functions. This makes it similar to the straight Galerkin method – indeed it is often equivalent, as we will discuss in a future article.
Read Full Post | Make a Comment ( 1 so far )The weak form
Intro One can write a general differential equation as: An example, which we will often consider is ex. to model e.g. heat conduction with a space-dependent heat conduction coefficient and a heat production term . (Notice the “ex” when we are talking about a particular example.) This is pretty technical, so please stop reading here [...]
Read Full Post | Make a Comment ( None so far )The method of weighted residuals
Intro One can write a general differential equation as: An example, which we will often consider is ex. to model e.g. heat conduction with a space-dependent heat conduction coefficient and a heat production term . (Notice the “ex” when we are talking about a particular example.) This is pretty technical, so please stop reading here [...]
Read Full Post | Make a Comment ( None so far )Math fonts for (La)TeX
How could I have overlooked this Survey of Free Math Fonts for TeX and LaTeX! An excellent read, it shows the author has himself been involved in font design. As I mentioned in my entry on XeTeX, it is great to have many fonts for text (and XeTeX really takes a leap forward in this [...]
Read Full Post | Make a Comment ( None so far )Natural coordinates
Here is an interesting application of Voronoi tesselations / Delaunay triangulations (see previous post The alpha shapes for another one.) Suppose you have a set of points carrying some information, let’s call them particles; the simplest case is just a scalar number representing some field whose value is only known for the particles. Then, you [...]
Read Full Post | Make a Comment ( None so far )The alpha shapes
Typical problem in computational geometry: given a set of points, what is its shape? This may often be intuitive for a human being, but try to define this concept mathematically, then implement it numerically. One of the proposals is called the “α shape”, and is closely related to the Delaunay triangulation. More details below. See [...]
Read Full Post | Make a Comment ( 3 so far )The CGAL project
I have been following and using this project for some years now, on and off. In my opinion, a very remarkable cooperative effort of several research centers, coordinated at INRIA Sophia Antipolis. They provide very robust implementations of algorithms on computational geometry: Delaunay triangulations, Voronoi diagrams ski sums, meshes, alpha shapes, convex hulls, all that. [...]
Read Full Post | Make a Comment ( 1 so far )Insane mathematical databases
A brief post, on two notable mathematical databases I have recently come across: On-Line Encyclopedia of Integer Sequences, maintained by N. J. A. Sloane (for that “next term” in a series that puzzles you. Encyclopedia of Triangle Centers by Clark Kimberling. The amount of work the maintenance of these pages must demand (not to talk [...]
Read Full Post | Make a Comment ( None so far )Computer math systems
A brief survey of the ones I have tried, with the student version price (as of mid 2008). Computer algebra systems (see wikipedia comparison) derive. Very simple, very easy. Unfortunately, it seems discontinued. Mathematica. Incredibly powerful. But: crazy syntax, annoying graphic interface (it used to be impossible to rotate 3D objects, I don’t know if [...]
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