# Questions tagged [laplacian]

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

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### The contractivity of the heat semigroup in $L^p$ spaces

Let $M$ be a Riemannian manifold. By functional calculus, it is immediate to show that the heat semigroup is a contraction in $L^2(M)$. I can also show that it is a contraction in any $L^p(M)$ with $p ...

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### Harmonic functions on knot complements

In Axler's Harmonic Function Theory, he and his coauthors develop the theory of harmonic functions on spheres and discs by considering the restrictions of arbitrary polynomials on the sphere $S^{n-1} =...

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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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### Spectrum of Laplace-Beltrami with piecewise constant coefficients

By the Laplace-Beltrami with piecewise constant coefficients I means the operator $-div (f\, \nabla .)$ in the 2-sphere. Where $f$ is a piecewise constant function that takes two values $1$ and $a>...

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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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### Spectrum of the Witten Laplacian on compact Riemannian manifolds

Below I have given what I am calling as the ${\rm Witten{-}Laplacian}_{s,p}$ on a Riemannian manifold $(M,g)$ for any constant $s >0$ and $p \in C^2(M,g)$
How generally is it true that this ${\rm ...

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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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### Laplacian on a manifold with two boundary components

I am interested in the Laplace equation on knot complements. The full complement of a knot $K$ is in $S^3$, but for compactness, we delete an open tubular neighborhood around $K$. The Laplace PDE on $...

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### Fiedler vector, what else?

In the spectral analysis of a graph with 1 connected component, the first non-trivial eigenvector (corresponding to the non-zero smallest eigenvalue) is also called the Fiedler vector. This vector is ...

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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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### 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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### Is $\Delta \phi$ monotone operator on $H^1(\mathbb{R}^d)$ for monotone $\phi$

Let $H^1(\mathbb{R}^d)$ be the usual Sobolev space and let $\phi: \mathbb{R} \to \mathbb{R}$ be a non decreasing Lipschitz function with $\phi(0)=0$. Is the operator $\Delta \phi $ on $H^1(\mathbb{R}^...

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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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### Recent work on Pseudo-Laplacian and Pseudo-cuspform in the spirit of Riemann Hypothesis after the work of Bombieri and Garrett

( This is my first MO question . I'm totally inexperienced on MO so, forgive me for my mistakes .)
Paul Garrett and Enrico Bombieri were (are?) Secretly Working on Pseudo-Laplacians and Pseudo-...

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### Domain of $(-\Delta)^\alpha$, $\alpha \in (0,1)$

Is there any simple way to characterize explicitly the domain of fractional powers for a given operator? For example, the domain of Dirichlet Laplacian on a bounded nice domain $\Omega \subset \mathbb{...

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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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### Pointwise convergence of the eigenfunctions expansion of $f(x)=\frac{1}{|x|}$

Let $\Omega\subset \mathbb{R}^n$ a bounded domain with smooth boundary, $0<\lambda_1\leq \lambda_2 \leq \dots \leq \lambda_k\leq \dots$ the Dirichlet eigenvalues and $\{w_k\}_{k=1}^{+\infty}$ an $L^...

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### Coupled (Solid-Fluid) Heat transfer problem in a Heat Sink

I am trying to solve a coupled heat transfer problem between a solid and fluid (I have under braced the governing equations and labelled them). Eqn. (3) is the partio-integral differential equation I ...

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### Kronecker limit formula for antiperiodic boundary conditions

The celebrated Kronecker limit formula gives the $\zeta$-reguralized determinant of the Laplacian on the torus $\mathbb{R}/(\mathbb{Z}\omega_1+\mathbb{Z}\omega_2)$ in terms of Dedekind eta function of ...

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### Laplacian eigenvalue problem for systems coupled along the boundary

I am looking for references on eigenvalue problems for systems of the following type:
Let $\Omega$ be the region enclosed by a right triangle with legs $\Gamma_1$, $\Gamma_2$, and hypotenuse $\...

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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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### 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 ...

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### maximum principle of fractional laplacian

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 $\|...

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### 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 ...

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### Eigenvectors of graph Laplacian for spectral clustering

I have the following questions regarding the graph Laplacian for spectral clustering:
What is the intuition behind projecting the Laplacian (D-A, where D is the degree matrix and A is the affinity ...

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### 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 ...

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### 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{\...

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### 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 ...

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### What's the full assumption for Laplacian matrix $L=BB^T=\Delta-A$?

Graph with no-selfloop, no-multi-edges, unweighted.
directed
For directed graph Adjacency matrix is a non-symmetric matrix $A_{in}$ considering indegree or $A_{out}$ considering outdegree. Degree ...

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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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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 $f$. ...

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### 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 ...

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### Graph Laplacian Operator

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 $\mathbb{L}$...

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### Laplace Beltrami eigenvalues on surface of polytopes

The recently posted arxiv paper Spectrum of the Laplacian on Regular Polyhedra
by Evan Greif, Daniel Kaplan, Robert S. Strichartz, and Samuel C. Wiese, collects numerical evidence for conjectured ...

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### Do eigenfunctions determine the geometry of a manifold? If so, do finitely many suffice?

Let $X$ be a smooth, Riemannian manifold. It is known that the geometry of $X$ can be recovered from its heat kernel $k_{t}(x,y)$, using Varadhan's Lemma: $\displaystyle\lim_{t \to 0} t \log k_{t}(x,y)...

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### Differential operators on a compact Lie group associated to bracket-generating sets

Let $G$ be a compact connected Lie group of dimension $m$ with Lie algebra $\mathfrak g$.
Let $\{X_1,\dots,X_h\}$ be a linearly independent set of $\mathfrak g$.
Assume that $\{X_1,\dots,X_h\}$ is ...

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### The heat kernel in Hermitian bundles over Riemannian manifolds

In "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne, the authors construct a (unique) heat kernel associated to any generalized Laplacian in a vector bundle over a Riemannian manifold....

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### How does the high-dimensional combinatorial Laplacian work?

When considering the boundary and coboundary maps, we have the common definitions that the boundary map based on the space of chains $C_k(X)$ is $$\partial_k([v_0,...,v_k])=\sum_{i=0}^k (-1)^i[v_0,...,...

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### Bochner formula in different forms

I am looking for a reference (better a book) that contain integral Bochner formulas for domains with boundary (I need it for 1-forms and functions only).
For example I will need the following formula:...

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### Is it possible to compute a valid Laplacian matrix from an effective resistance matrix?

I am wondering whether it is possible to retrieve a node-admittance matrix $G$ (also called Laplacian matrix) in a purely resistive network composed of nets $\{1, \dots, i, \dots, j, \dots, n\}$, from ...

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### Conditions for a matrix to be a Graph Laplacian [closed]

Let M be a symmetric non-negative definite $n\times n$ matrix. Let $K_n$ denote the complete graph on $n$ vertices. Under what conditions is it possible to assign edge weights to $K_n$ in such a way ...

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### Reference on spectral fractional Laplacian

Are there Harnack type inequalities and Schauder type estimates for the spectral fractional Laplacian. References are welcome.

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### Density of squares of radial eigenfunctions

The eigenfunctions of the Laplace operator on the disc can be written in polar coordinates as $f(r,\theta)=R_{nk}(r)e^{ik\theta}$, where $k\in\mathbb Z$ and $n\in\mathbb N$ and the radial function is $...

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### How do eigenvalues of combinatorial Laplacian relates to automorphisms in graphs?

Is there a relation between eigenvalues of the graph Laplacian and the automorphism group of a simple graph?
How are the multiplicities of Laplacian eigenvalues related to the order of the ...