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, 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,$$where D is the degree matrix (diagonal matrix with entries corresponding to the degree of the vertex) and A the ... 0answers 221 views Complex factorization of the angular part of the Laplacian Some time ago some research led me to the following equality: \frac{1}{\sin^2 \phi }\frac{\partial^2 }{\partial \theta^2} +\frac{\partial^2 }{\partial \phi^2} +\cot \phi \frac{\... 0answers 42 views On the Variable Coefficient Laplacian This the copy of the question that I had asked in math stackexchange I read about Laplace Operator here. As given in the link, given the metric, we can find the expression for Laplace operator. I am ... 3answers 681 views Spectrum of Dirichlet Problem for Laplacian on a Parallelogram Let M \subset \mathbb{R}^2 be parallelogram constructed by putting together two equilateral triangles (so that all sides of the parallelogram have length 1, and the internal angles are 60 and 120).... 1answer 203 views Poisson summation formula and its implication for the spectrum of the flat torus I usually ask questions on math.stackexchange but I figure this one is more suited to being asked here. I should preface that I am a complete novice undergraduate, and unlikely to understand answers ... 1answer 359 views hyperbolic “Green function” on a product of upper half-planes Let \Delta_{hyp}=\Delta_{hyp,1}=-y^2(\partial_x^2+\partial_y^2) be the hyperbolic Laplacian acting on functions of \mathfrak{h} (the Poincare upper half-plane) and consider its resolvent$$ 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 $\mathbb{L}$...