Questions tagged [laplacian]

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

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Nilpotent parts of graph Laplacians

Let $W$ be the adjacency matrix of a directed graph. Let us denote by $D$ the associated in-degree matrix, whose diagonal entries are given by $D_{ii} = \sum_j W_{ij}$. The associated Laplacian $$ L =...
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Inverting laplacian for leverage scores $w_{s,t}({\bf e}_s-{\bf e}_t)^TL^{+}({\bf e}_s-{\bf e}_t)$

Given a graph $G(V,E)$ with edge weights $w_e>0$ and edge-vertex incidence matrix $U$, consider the laplacian $L=U^TWU$, and its Moore-Penrose psuedoinverse $L^{+}$. Here, $W$ is a diagonal matrix ...
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Largest eigenvalue of a Laplacian matrix to lower bound the prime counting function?

Let $L_n$ be the Laplacian matrix of the undirected graph $G_n = (V_n, E_n)$ (which is defined here: Why is this bipartite graph a partial cube, if it is? ) with sorted spectrum: $$\lambda_1 (G_n) \ge ...
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Entropy of eigenvectors of (traceless) laplacian of a bipartite graph

This problem is motivated by the edge states that sometimes appear (and sometimes not) at the level of Huckel hamiltonians for $\pi$-conjugated benzenoid hydrocarbons. If this sentence is cryptic, ...
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Laplace equation with bisection boundary condition on upper side

solve Laplace equation $ U_{xx} + U_{yy} = 0 $ with given boundary condition : $ (x,y) \in (0,a) \times (0,b) $ $\begin{cases} U_x(0,y) = U(x,0) = U_x(a,y) = 0 \\ U(x,b)= c= constant \hspace{3mm} ,...
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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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Prove or disprove the compactness of an operator

Consider $X=L^{2}(0,\pi, \mathbb{R})$. Let $X_{\frac{1}{2}}$ be the domain of $(\Delta)^\frac{1}{2}$ where $\Delta$ is the laplacien operator. We define the operator $K:C([0,a],X_{\frac{1}{2}})\...
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Self-ajointness of the Laplacian over a Riemannian manifold with boundary

I have some doubts about on a passage found on this article (https://arxiv.org/pdf/1510.08136.pdf). Let $(M,g)$ be a Riemannian manifold with boundary; $E\to M$ be an hermitian fiber bundle; $\Delta$ ...
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Interpreting the Higher-order Hodge-Laplace Operator

As an operator on functions, one intuitive way to think about the Laplacian seems to be as an operator that returns the average difference between a function's value at a point and the values of its ...
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Non-linear diffusion on networks

The diffusion equation with constant diffusion $D$ can be represented as: \begin{equation} \frac{\partial \phi(r, t)}{\partial t}=D \Delta \phi(r, t) \end{equation} where $\Delta$ is the Laplace ...
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Fractional Laplacian of smooth cut off functions

Suppose we have a smooth compactly supported function $\phi\in C^{\infty}_c(B_\epsilon(0))$ such that $0\leq \phi \leq 1$, $\phi\equiv 1$ on the unit ball and $\phi$ vanishes outside $B_\epsilon(0).$ ...
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Asymptotics for the first eigenvalue for the Laplace-Beltrami operator on the sphere

I am trying to understand the existence of positive solutions for the following equations, $-\Delta_{\mathbb{S}^n} u + \lambda u = f(u)$ where $f$ is some non-linearity, say $f(t)=t^3.$ By considering ...
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Eigenvalues and eigenfunctions of the Laplace operator on entire plane

According to the answers in the the following questions: How to prove the spectrum of the Laplace operator? and What is spectrum for Laplacian in $\mathbb{R}^n$ , the spectrum of the Laplace operator $...
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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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Spectra of the Laplacian operator on the spherical space-form

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 $...
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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 $L$. ...
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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*} where $c$ does not ...
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5 votes
1 answer
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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 ...
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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 ...
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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} \...
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Eigenvector to zero eigenvalue of general Laplacian

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 ...
4 votes
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290 views

Do Laplace-Beltrami eigenfunctions vary continuously with the metric?

I'm interested in Laplace Beltrami operators $$-\Delta_g:\ \ D(-\Delta_g) \longrightarrow L^2\left(M,\sqrt{|g|}dx\right)$$ on a smooth compact Riemannian Manifold (M,g). Let us fix a unique metric $...
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Green kernel vs fundamental solution

Let $L$ being the Laplacian for a given Lie group $G$. I would like to know what is the difference between the two notions in relation to the operator $L$: The fundamental solution $\Gamma(x)$ of $L$;...
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Optimality condition of the harmonic form representatives of a homology class

In "Hodge theory on metric spaces, Smale et al." the $d$-th harmonic forms of the Hodge Laplacian $\Delta_d=\delta^* \delta+\delta \delta^*$ satisfying $\Delta_d(f)=0$ are claimed to be ...
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Products of eigenfunctions on compact Riemann surfaces

Let $M$ be a compact Riemann surface with genus $g\geq 2$, endowed with the Riemannian metric with constant sectional curvature $-1$. Let $f_1, f_2$ be two (global) eigenfunctions for the Laplace-...
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How to show that $\Delta W \leq −2(n − 4)V$?

I am reading a preprint and trying to understand the proof of Lemma 3.5. On Pg. 19 above eqn (3.49) the authors claim that $\Delta W \leq −2(n − 4)V$ where the functions $W$ and $V$ are defined below, ...
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Is the heat semigroup on a manifold the limit of the heat semigroups associated to a compact exhaustion?

Let $M$ be a paracompact Riemannian manifold, and $E \to M$ a Hermitian vector bundle endowed with a Hermitian connection $\nabla$. Write $M$ as an exhaustion $\bigcup _{j \ge 0} U_j$ with relatively ...
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Relation between the eigenvalues of the weighted laplacian and fractional laplacian?

Consider the eigenvalue problem $-\Delta u = \lambda u\rho$ for $u\in \dot{H}^{1}(\mathbb{R}^n)$ with $n\geq 3$ and weight $\rho\in L^{n/2}(\mathbb{R}^n).$ Let $(\lambda_k, \psi_k)$ be the increasing ...
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Are there any known results for the spectrum of $(-\Delta)^s/V^{p-1}$?

I am interested in generalizing some results known for the $\frac{-\Delta}{U^{p-1}}$ where $U$ is a Talenti bubble to the non-local operator $\frac{(-\Delta)^s}{V^{p-1}}$ where $U$ and $V$ are bubbles ...
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Graph Laplacian definition

Assume that we have node indices $\{1,...,p\}$ and weights between the indices $w_{ji} = w_{ij}\geq 0$ for $i,j \in \{1,...,p\}$ with $w_{ii} = 0$. Then the Laplacian matrix is defined as $L:= \...
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Property about 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} \...
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2 votes
1 answer
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Limit of $e^{-t(-\Delta)^\alpha}$ when $\alpha \to 1$

Let us consider the fractional heat semigroup $\left(e^{-t(-\Delta)^\alpha}\right)_{t\ge 0}$ for $\alpha\in (0,1)$ (the fractional power is taken in whole $\mathbb R^d$). Is there any result about the ...
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Differentiation of functions over graphs

In short: There are various ways to define differentiation over a graph. I am trying to get the big picture, like a more complete and structured bestiary. Definitions. Let $G=(V,E)$ be a directed ...
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1 answer
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Differential inequalities under which a flat function must be identically zero

Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $. Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ ...
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Eigenvalues of Laplacian and eigenvalues of curvature operator

Let $(M^n,g)$ be a compact Riemannian manifold (without boundary). The symmetries of the curvature $R$ of (the Levi-Civita connection associated to) $g$ allow one to realise $R$ as a self-adjoint (...
5 votes
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Line graphs called "graph derivatives": any intuition?

Short version: in several papers, line graphs (and closely related graphs) are called graph derivatives or derived graphs; is there any intuition for such terminologies, in connection with the ...
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Chip-firing clocks

Let $G$ be some outdegree-regular directed graph with $n$ vertices and let $H$ be the Laplacian of $G$, so that the rows of $H$ correspond to chip-firing moves. I’m interested in linear functions $f$ ...
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Deducing elliptic regularity using spherical formula for Laplacian

It is well known that if $\Delta u = f$ and $u\in H_0^1(U),f \in L^2(U)$ for some domain $U,$ then we have $\|u\|_{H^2(U)} \leq C\|f\|_{L^2(U)}$ for some constant $C.$ One way to prove this is to ...
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Neumann/Robin Laplacian semigroup well-known estimate

Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on: $$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \...
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Show that the Laplacian on these domains is isospectral

Let $\Omega_i\subseteq\mathbb R^d$ be bounded and open, $A_i$ denote the weak Laplacian with domain $\mathcal D(A):=\{u\in H_0^1(\Omega_i):\Delta u\in L^2(\Omega_i)\}$ on $L^2(\Omega_i)$ and $$T_i(t)f:...
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Estimate of the norm of the radial part of a function

Consider a function $u\in L^2(\mathbb R^N)$, and another function $\varphi$ which is the unique solution to the Poisson equation $\Delta \varphi = u$ vanishing at $\infty.$ We know that the radial ...
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5 votes
2 answers
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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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7 votes
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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 ...
4 votes
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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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