# Questions tagged [laplacian]

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

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### Relation between expected values of eigenvalues of Laplacian matrix of a graph and eigenvalues of expected Laplacian matrix of that graph?

Particularly, I am dealing with Erdős–Rényi random 𝐺(𝑛,𝑝), so the expected Laplacian matrix of 𝐺(𝑛,𝑝) is 𝑝(𝐽𝑛−𝐼𝑛), where 𝐽𝑛 and 𝐼𝑛 are one and identity matrices, respectively. In ...
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### Growth of Laplacian eigenvalues on a compact domain?

Let $\mathcal{M}$ be a compact Riemannian manifold and let $\Delta$ be the (scalar) Laplace-Beltrami operator on $\mathcal{M}$. Then $\Delta$ has a discrete spectrum and if we order its distinct ...
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### What is a random eigenfunction on the hyperbolic plane?

Is there an (invariant under isometries) notion of a random eigenfunction on the hyperbolic plane, for a given eigenvalue? It is a reference request because the answer is probably positive and I even ...
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### Laplacian of Fourier-like function

The question that follows has to do with the effects of a turbulent atmosphere on wave propagation. The structure function, $D(\vec{r})$, which is defined as, \begin{equation} D(\vec{r}) = \left\...
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### How to understand the combinatorial Laplacian $\Delta$ which is defined on the graph?

I have a question about the combinatorial Laplacian $\Delta$ which is defined by $$\Delta(u,v)=c(u)1_{u=v}-c(u,v)$$ where $u, v$ are some vertices in the graph $G=(V, E)$, and $c(u,v)$ is a ...
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### From one eigenvector to many, in a very local graph?

Let $\Gamma$ be an undirected graph of bounded degree $d$ with $V = \{1,2,\dotsc,N\}$ as its set of vertices, and edges only between vertices that are at a distance $\leq M$ apart (where $M$ is much ...
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### Approach to solve a coupled system of PDE (heat transfer in cylindrical coordinates)

I have the following two PDEs, which describe steady-state coupled heat transport between an externally heated axisymmetric solid body (Eq. 1, $T(r,z)$) and a fluid (Eq. 2, $t(z)$) flowing inside it: ...
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### Normalized Laplacian matrix versus walk Laplacian matrix (or normalized adjacency matrix versus walk adjacency matrix)

In graphs, found that two different normalization matrices exist for Laplacian and adiacency matrix. I will ask about the adjacency matrix (for the Laplacian matrix the questions are the same). The ...
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### System of eigenvalues of partial Laplacians and the Jacquet Langlands correspondence

I'm looking for a precise reference. I could not dig it up in the original paper of Jacquet-Langlands. Let $F$ be a totally real number field with $[F:\mathbb{Q}]=r$, let $B_1=M_2(F)$ be the split ...
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### The spectrum of the Hodge Laplacian on a Riemannian manifold

The Hodge Laplacian operator on differential forms on a (compact?) Riemannian manifold carries useful information about the topology of the manifold. In particular, the multiplicity of the zero ...
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### On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold

Assume that $M$ is a compact Riemannian manifold whose Laplacian is denoted by $\Delta$. Assume that the Euler characteristic of $M$ is zero. Does $M$ admit a non vanishing vector field ...
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### implicit function theorem and harmonic mapping

We are given two Riemannian manifolds $M,N$ of dimension $m$ and $n$ and a function $G \colon M \times N \to \mathbb{R}^n$ which satisfies the assumptions of the implicit function theorem, meaning ...
Suppose $u$ is a sign changing classical solution of the fractional laplacian $$(-\Delta) ^{s} u = 0 \; \text{in } \Omega; u=g \text{ in } \mathbb R^N -\Omega .$$ Is it true that $\|... 2answers 194 views ### Gaps in the spectrum of Laplace-Beltrami operators Let us consider$\mathbb S^d$the unit Euclidean sphere of$\mathbb R^{d+1}$and let$\Delta_{\mathbb S^d}$be the Laplace operator on$\mathbb S^d$. We have $$-\Delta_{\mathbb S^d}=\sum_{k\in \... 0answers 50 views ### Tight bound on spectral gap of compact homogeneous manifold? This paper by Peter Li proves a bound on the spectral gap of the Laplacian on a compact homogeneous manifold of diameter d:$$ \lambda_1 \ge c/d^2, $$where c=\pi^2/4. Can this bound be ... 1answer 141 views ### Spectrum of the Laplacian on the quotient of 3-sphere Given a finite subgroup \Gamma of O(4) acting freely on S^3, is there any reference for the spectrum of Laplacian for the transverse-traceless symmetric 2-tensor on S^3/\Gamma equipped with ... 0answers 82 views ### Algebraic connectivity of the path P_n Let G be a graph with n vertices. Denote by L(G) the Laplacian matrix of G and 0=\lambda_1\leqslant\lambda_2\leqslant...\leqslant\lambda_n its spectrum. The number \lambda_2 is called the ... 0answers 167 views ### Laplace spectra of “half” grid graph Let G=(E,V) be a simple graph. The graph Laplacian is given by$$ L= D-A,$$where D is the degree matrix (diagonal matrix with entries corresponding to the degree of the vertex) and A the ... 0answers 221 views ### Complex factorization of the angular part of the Laplacian Some time ago some research led me to the following equality: \begin{equation} \frac{1}{\sin^2 \phi }\frac{\partial^2 }{\partial \theta^2} +\frac{\partial^2 }{\partial \phi^2} +\cot \phi \frac{\... 0answers 42 views ### On the Variable Coefficient Laplacian This the copy of the question that I had asked in math stackexchange I read about Laplace Operator here. As given in the link, given the metric, we can find the expression for Laplace operator. I am ... 3answers 681 views ### Spectrum of Dirichlet Problem for Laplacian on a Parallelogram Let M \subset \mathbb{R}^2 be parallelogram constructed by putting together two equilateral triangles (so that all sides of the parallelogram have length 1, and the internal angles are 60 and 120).... 1answer 203 views ### Poisson summation formula and its implication for the spectrum of the flat torus I usually ask questions on math.stackexchange but I figure this one is more suited to being asked here. I should preface that I am a complete novice undergraduate, and unlikely to understand answers ... 1answer 359 views ### hyperbolic “Green function” on a product of upper half-planes Let \Delta_{hyp}=\Delta_{hyp,1}=-y^2(\partial_x^2+\partial_y^2) be the hyperbolic Laplacian acting on functions of \mathfrak{h} (the Poincare upper half-plane) and consider its resolvent$$ R(s)=(... 1answer 249 views ### An alternative representation of the principal symbol of the Laplace operator Assume that$(M,g)$is a$n$dimensional Riemannian manifold. We denote by$\Delta$, the Laplacian associated to this Riemannian structure. Are the following two conditions equivalent? First condition ... 1answer 97 views ### A question on nontrivial solution of ODE It is well known that there exists no non-trivial bounded solution of$-u''+u=0$in$\mathbb R.$Is this result even true, the problem $$\bigg(-\frac{d^2}{dx^2}\bigg)^{s} u+u=0$$ has no bounded ... 0answers 62 views ### Green's Function for Fractional Laplacian on the Union of Two Balls I have two disjoint open intervals$B_1, B_2 \subset \mathbb{R}$, and variables$0 < s < 1$and$t \in B_1 \cup B_2$. I want to solve: $$r_{B_1 \cup B_2}(\Delta^{s} f) = \delta_t$$ for$f$. ... 0answers 409 views ### Is Laplacian a surjective operator? For a closed manifold the laplacian is almost surjective operator since the index of$\Delta$is zero and there is no a non constant harmonic function. So the codimension of the image ... 1answer 217 views ### Curvature of the boundary vs. normal derivative of the first eigenfunction Disclaimer. I posted this question in Math.SE, but it haven't received enough attention. Let$\varphi_1$be the first eigenfunction of the zero Dirichlet Laplacian in a planar bounded domain$\Omega$.... 0answers 90 views ### May the heat kernel of a connection Laplacian vanish? Let$M$be a Riemannian manifold and$E \to M$be a Hermitian bundle. If$\nabla$is a Hermitian connection on$E$, one may define the Laplacian$L = \nabla^* \nabla$, and then consider its Friedrichs ... 0answers 589 views ### Are harmonic mappings non-singular outside a set of measure zero? Let$g$be a smooth Riemannian metric on the closed$n$-dimensional unit disk$\mathbb D^n$. Let$f: \mathbb D^n \to \mathbb{R}^n$be a smooth orientation-preserving immersion, and let$\omega :\...
Consider the linear operator $\mathbb{L} : L^2([0,1])\to L^2([0,1])$ defined by $$(\mathbb{L}f)(x) = \int_0^1 xy(f(x)-f(y)) \mathrm{d}y$$ for all $f\in L^2([0,1])$ and $x \in [0,1]$. Is $\mathbb{L}$...