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I want to solve the Laplace equation in 2 different medium in 2D, each with its own dielectric constant. This is the extremely simplified geometry. The problem is the same error (which you will find out in the end!) for my more complex geometry in 3D. I start with defining the region: Needs[“NDSolve`FEM`”] bmesh =Read more

A “set” like …{{}}… Where the dots indicate infinitely many braces is not a set. There is no outer layer of braces to find it’s elements in, so its elements are undefined. I am asking if von Neumann’s ordinals are generated out to some aleph, do pathologies like this appear in the ordinals created? ARead more

Let $ M$ be a von Neumann sub-algebra of $ B(H)$ where $ H$ is a separable Hilbert space. Let us consider the normal functional $ \omega=\sum_1^n \zeta_i\otimes\zeta_i$ on $ M$ given by $ x\to \sum_1^n\langle x\zeta_i,\zeta_i\rangle$ . Let $ \epsilon>0$ . Does there exists a projection $ q\in M$ with $ p\leq q$ suchRead more

Let $ \Omega$ be an open bounded domain with a boundary $ \partial\Omega$ . Consider the following Neumann eigenvalue problem for Laplacian: find $ (\phi_n,\lambda_n)\in H^1(\Omega)\times \mathbb{R}$ \begin{align*} -\Delta \phi_n& = \lambda_n \phi_n\quad \mbox{in }\Omega,\ \frac{\partial \phi_n}{\partial \nu} & = 0 \quad \mbox{on }\partial\Omega. \end{align*} Now by spectral theory (see the notes at here https://faculty.math.illinois.edu/~laugesen/595Lectures.pdf),Read more

Let $ H$ be a Hilbert space and let $ M$ be a densely defined operator $ D(M) \subset H \to H$ . It is closable iff its adjoint $ M^{\star}$ is densely defined, and then its closure $ \overline{M}$ is $ M^{\star \star}$ . Let $ \mathcal{M}$ be the smallest von Neumann algebra thatRead more

I have the following code to solve 2 separate laplacian equations in one dimensions: op1 = D[u[x], {x, 2}]; op2 = D[v[x], {x, 2}]; x0 = 1; x2 = 3; x1 = 1.75; DSolve[{op1 == 0, op2 == 2 x, u[x0] == 0, v [x2] == 2, u[x1] == v[x1], (D[u[x], x] /. x ->Read more

I want to prove that if the constant $ k>0$ is sufficiently large, then for every $ f\in L^2(0,1)$ , there exists a unigue $ u\in H^2(0,1) $ satisfying $ $ -u”+ku=f \; on \;(0,1) $ $ $ $ u'(0)=0,\;\; u'(1)=u(1)$ $ for this problem, i want to find appropriate Bilinear form B to satisfyRead more