# Category Archives: Lecture

Wave equation Let $M$ be a surface and $f : \mathbb{R} \times M \rightarrow \mathbb{R}$ a function. The wave equation is given by: $\ddot{f} = \Delta f.$ After the space discretization we have $f:\mathbb{R} \times V \rightarrow \mathbb{R},$ $p:= \left(\begin{matrix} … Continue reading Posted in Lecture | Comments Off on Wave- and heat-equation on surfaces ## Partial differential equations involving time Let$\Omega \subset \mathbb{R}^n$be a domain, we will consider$\Omegato be the “space” and functions: \begin{align} f : \mathbb{R} \times \Omega \rightarrow \mathbb{R}, \\ (t,x) \mapsto f(t,x),\end{align} will be view as time dependent functions \begin{align} & f_t : … Continue reading Posted in Lecture | Comments Off on Partial differential equations involving time ## Laplace operator 2 LetM= \left(V, E, F \right)$be an oriented triangulated surface without boundary and$p:V \rightarrow \mathbb{R}^3$a realization. In earlier lectures we considered the space of piecewise linear functions on$M: W_{PL}:=\left\{ \tilde{f} :M\rightarrow \mathbb{R} \, \big \vert \,\left. … Continue reading Posted in Lecture | Comments Off on Laplace operator 2 ## Triangulated surfaces with metric and the Plateau problem Let \Sigma = (V,E,F) be a triangulated surface (with boundary). A realization of the surface in \mathbb{R}^3 is given by a map p:V \rightarrow \mathbb{R}^3 such that p_i,p_j,p_k form a non degenerated triangle in \mathbb{R}^3 for all \{i,j,k\} \in \Sigma, … Continue reading Posted in Lecture | Comments Off on Triangulated surfaces with metric and the Plateau problem ## Dirichlet energy 2 Let M = (V,E,F) be a triangulated domain in the plane where V denotes the set of vertices, E the set of edges and F the set of triangles. We consider the set of functions on the vertices : \begin{align*} … Continue reading Posted in Lecture | Comments Off on Dirichlet energy 2 ## Gradient and Dirichlet energy on triangulated domains. Let M be a triangulated domain with the functionspace: \[W_{PL}:=\bigl\{f:M\rightarrow\mathbb{R}\,\bigl\vert\bigr.\,\,\,\left. f\right|_{T_{\sigma}} \mbox{ is affine for all } \sigma \in \Sigma_2 \bigr\}. On the interior of each triangleT_{\sigma}$in$M$the gradient of a function$g \in W$is well … Continue reading Posted in Lecture | Comments Off on Gradient and Dirichlet energy on triangulated domains. ## Triangulated surfaces and domains We want to derive a discrete version of the laplace operator defined on triangulated surfaces. At first we will define what a triangulated surface with and without boundary is, and consider triangulated domains of$\mathbb{R}^2$as an important example. Let … Continue reading Posted in Lecture | Comments Off on Triangulated surfaces and domains ## Laplace operator 1 Let$M \subset \mathbb{R}^2$be a domain with smooth boundary$\partial M$and outpointing normal vector field$N$. For a smooth function$f \in C^{\infty}(M,\mathbb{R})$the gradient vector field$\mbox{grad} \, f :M \rightarrow \mathbb{R}^2\$ is defined as : \[ … Continue reading

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## Random Fourier polynomials

What is a typical function? The answer of this question certainly depends on the branch of mathematics you are into – while functions in differential geometry are usually smooth, the wiggling graphs appearing at the stock market are far from … Continue reading

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## The 3-Sphere

So far we have seen geometries in 2D and 3D, which are the dimensions we are familiar with.  But mathematician have found interesting geometries in higher dimensions and it would be great if we could visualize them.  In particular a huge … Continue reading

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