# Questions tagged [laplacian]

The Laplacian matrix is the representation of a graph in matrix form.

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### Quasimode construction on a compact Riemannian manifold

Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...
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### Algorithm for efficiently calculating $(A+\sum_{i=1}^n B_i)^{-1}$ where $A^{-1}\in\mathbb S^n_+$ is known and $B_i$ are sparse matrices

Let $A\in\mathbb R^{n\times n}$ be a symmetric positive-definite matrix and $A^{-1}$ is already known. Now I want to compute the matrix $(A+\sum_{i=1}^n B_i)^{-1}$ where each $B_i$ is a sparse ...
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### Laplacian on sphere after stereographic projection

How to derive that after stereo-graphical projection, $\Delta u$ in $\mathbb{R}^n$ is transformed to $$\Delta_{\mathbb{S}^n}u - \frac{n(n-2)}{4}u\ \text{in}\ \mathbb{S}^n.$$ To be more precise, in ...
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### Surprising symmetry in the Ramanujan bound

The condition for a connected $(q+1)$-regular graph to be Ramanujan is that every nonzero eigenvalue $\lambda$ of the graph Laplacian satisfy $$q+1-2\sqrt{q}\le \lambda\le q+1+2\sqrt{q}.$$ With a ...
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### Explicit expression for the Poisson kernel solving the Dirichlet problem for geodesic balls

Let $X$ be a Riemannian manifold with the Laplace-Beltrami operator denoted by $\mathscr L$ and we look at its geodesic balls say $B$. Let $u$ be a continuous function on the geodesic sphere which is ...
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### First eigenvalue of the Laplacian on the traceless-transverse 2-forms

Let $(S^3/\Gamma, g)$ be a spherical space form with constant sectional curvature $1$, where $\Gamma$ is a finite subgroup of $SO(4)$ acting freely on $S^3$. Consider the first nonzero eigenvalue ...
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Let $S^3/\Gamma$ be a spherical space form where $\Gamma$ is a finite subgroup of $O(4)$ acting freely on $S^3$. If $\Gamma$ is trivial, it is well-known that the spectra of the Laplacian operator on $... • 849 3 votes 0 answers 194 views ### Are smooth functions with compact support a core for the Laplacian on compact manifolds with boundary? If$M$is a complete Riemannian manifold and$L$is the Friedrichs extension of the Laplacian$-\Delta$, then it is known (first proven by Gaffney in the '50) that$C_0 ^\infty (M)$is a core for$L$. ... • 5,187 2 votes 1 answer 200 views ### Regularity bound For$\Delta f_g = g$, can we prove that$f_g \in L^\infty_{\text{per}}(\Omega)and \begin{align*} \|f_g\|_{L^\infty_{\rm per}} \le c \|g\|_{L^\infty_{\rm per}} \end{align*} wherec$does not ... 5 votes 1 answer 261 views ### Smallest eigenvalue of Laplacian of periodic lattice after removing a vertex Consider a 4-regular graph with$N^2$vertices, which can be interpreted as a$N\times N$lattice with periodic boundary conditions so that every vertex has degree 4. For an unweighted and undirected ... • 97 5 votes 1 answer 318 views ### Lower bound on the first eigenvalue of the Lichnerowicz Laplacian on positive Einstein manifolds Suppose$(M^n,g)$is an$n$-dimensional Einstein manifold with$Ric=(n-1)g$. Let$\lambda$be the minimal eigenvalue of the Lichnerowicz Laplacian$\Delta_L$defined on all transverse-traceless ... • 849 0 votes 1 answer 109 views ### Inequality involving the fractional Laplacian Let$s \in \mathbb{R}$such that$0<s<1$. Consider the fractional Laplacian$(-\Delta)^s$in the real line defined via Fourier series as follows: if$f:[-\pi,\pi] \subset \mathbb{R} \...
I was wondering what we can say about the eigenvectors of a matrix $A$ fullfilling $Ax =0$ where $A$ is symmetric with a diagonal equal to one and every row sums up to 0. Obviously this is a ...