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Spectral boundary conditions for Laplace-type operators, of interest in string and brane theory, are partly Dirichlet, partly Neumann-type conditions, partitioned by a pseudodifferential projection. We give sufficient conditions for…
We study a class of perturbative scalar quantum field theories where dynamics is characterized by Lorentz-invariant or Lorentz-breaking non-local operators of fractional order and the underlying spacetime has a varying spectral dimension.…
We establish stability inequalities for the problem of determining a Dirichlet-Laplace-Beltrami operator from its boundary spectral data. We study the case of complete spectral data as well as the case of partial spectral data.
We discuss quantitative estimates of the local spectral dimension of the two-dimensional Sierpinski gasket. The present arguments were inspired by a previous study of the distribution of the Kusuoka measure by R. Bell, C.-W. Ho, and R. S.…
For the fractional Laplace equation, a surprising observation is the non-uniqueness for the basic Dirichlet type problems. In this paper, a somewhat sharp uniqueness condition for the fractional Laplace equation is established. We derive…
In 2010, the first author of this paper introduced the notion of $\sigma$--stability for a nonempty subset of an $L^0(\mathcal{F},K)$--module in [T.X. Guo, Relations between some basic results derived from two kinds of topologies for a…
We present three equivalent definitions of the fractional $p$-Laplacian $(-\Delta_{\mathbb{H}^{n}})^{s}_{p}$, $0<s<1$, $p>1$, with normalizing constants, on hyperbolic spaces. The explicit values of the constants enable us to study the…
We consider the fractional mean-field equation on the interval $I=(-1,1)$ $$(-\Delta)^\frac{1}{2} u=\rho\frac{e^{u}}{\int_{I}e^{u}dx},$$ subject to Dirichlet boundary conditions, and prove that existence holds if and only if $\rho <2\pi$.…
Galton boards are models of deterministic diffusion in a uniform external field, akin to driven periodic Lorentz gases, here considered in the absence of dissipation mechanism. Assuming a cylindrical geometry with axis along the direction…
In this paper, we first characterize the finiteness of fractal interpolation functions (FIFs) on post critical finite self-similar sets. Then we study the Laplacian of FIFs with uniform vertical scaling factors on Sierpinski gasket (SG). As…
It has been empirically observed that eigenfunctions of Laplace's equation $-\Delta \phi = \lambda \phi$ with Neumann boundary conditions sometimes localize near the boundary of the domain if that boundary is rough (say, fractal). This has…
We consider scaling limits of random quadrangulations obtained by applying the Cori-Vauquelin-Schaeffer bijection to Bienaym\'e-Galton-Watson trees with stably-decaying offspring tails with an exponent $\alpha$ in (1, 2). We show that these…
We construct spectral triples and, in particular, Dirac operators, for the algebra of continuous functions on certain compact metric spaces. The triples are countable sums of triples where each summand is based on a curve in the space.…
We prove and collect numerous explicit and computable results for the fractional Laplacian $(-\Delta)^s f(x)$ with $s>0$ as well as its whole space inverse, the Riesz potential, $(-\Delta)^{-s}f(x)$ with $s\in\left(0,\frac{1}{2}\right)$.…
Efficient and accurate numerical algorithms are developed to solve a generalized Kirchhoff-Love plate model subject to three common physical boundary conditions: (i) clamped; (ii) simply supported; and (iii) free. We solve the model…
We consider fractional directional derivatives and establish some connection with stable densities. Solutions to advection equations involving fractional directional derivatives are presented and some properties investigated. In particular…
A method offering an order of magnitude sensitivity gain is described for using quasar spectra to investigate possible time or space variation in the fine structure constant, alpha. Applying the technique to a sample of 30 absorption…
We study a model for combustion on a boundary. Specifically, we study certain generalized solutions of the equation \[ (-\Delta)^s u = \chi_{\{u>c\}} \] for $0<s<1$ and an arbitrary constant $c$. Our main object of study is the free…
We consider the Dirichlet eigenvalues of the fractional Laplacian $(-\Delta)^s$, with $s\in (0,1)$, related to a smooth bounded domain $\Omega$. We prove that there exists an arbitrarily small perturbation $\tilde\Omega=(I+\psi)(\Omega)$ of…
The (favorable) Falkner-Skan boundary layer profiles are a one parameter ($\beta \in [0,2]$) family of self-similar solutions to the stationary Prandtl system which describes the flow over a wedge with angle $\beta \frac{\pi}{2}$. The most…