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We study the graphs associated with Vicsek sets in higher dimensional settings. First, we study the eigenvalues of the Laplacians on the approximating graphs of the Vicsek sets, finding a general spectral decimation function. This is an extension of earlier results on two dimensional Vicsek sets. Second, we study the Vicsek set lattices, which are natural analogues to the Sierpinski lattices. We have a criterion when two different Vicsek set lattices are isomorphic.
We prove that a random group in the triangular density model has, for density larger than 1/3, fixed point properties for actions on $L^p$-spaces (affine isometric, and more generally $(2-2epsilon)^{1/2p}$-uniformly Lipschitz) with $p$ varying in an
In this paper, we show the existence of a sequence of eigenvalues for a Dirichlet problem involving two mixed fractional operators with different orders. We provide lower and upper bounds for the sum of the eigenvalues. Applications of mixed fraction
By introducing a weight function to the Laplace operator, Bakry and Emery defined the drift Laplacian to study diffusion processes. Our first main result is that, given a Bakry-Emery manifold, there is a naturally associated family of graphs whose ei
For $alphain(0,pi)$, let $U_alpha$ denote the infinite planar sector of opening $2alpha$, [ U_alpha=big{ (x_1,x_2)inmathbb R^2: big|arg(x_1+ix_2) big|<alpha big}, ] and $T^gamma_alpha$ be the Laplacian in $L^2(U_alpha)$, $T^gamma_alpha u= -Delta u$,
In this paper we study Lipschitz regularity of elliptic PDEs on geometric graphs, constructed from random data points. The data points are sampled from a distribution supported on a smooth manifold. The family of equations that we study arises in dat