Related papers: Anisotropic Green Coordinates
The isotropy of space is not a logical requirement but rather is an empirical question; indeed there is suggestive evidence that universe might be anisotropic. A plausible source of these anisotropies could be quantum gravity corrections.…
In this paper, we study the parabolic and elliptic problems related to the anisotropic $p$-Laplacian operator in the case when it has linear growth on some of the coordinates. In order to define properly a notion of weak solutions and prove…
Unoriented surface reconstructions based on the Gauss formula have attracted much attention due to their elegant mathematical formulation and excellent performance. However, the isotropic characteristics of the formulation limit their…
Multiple scattering methods are widely used to reduce the computational complexity of acoustic or electromagnetic scattering problems when waves propagate through media containing many identical inclusions. Historically, this numerical…
Both analytic and geometric forms of an optimal monotone principle for $L^p$-integral of the Green function of a simply-connected planar domain $\Omega$ with rectifiable simple curve as boundary are established through a sharp…
Quantum entanglement, induced by spatial noncommutativity, is investigated for an anisotropic harmonic oscillator. Exact solutions for the system are obtained after the model is re-expressed in terms of canonical variables, by performing a…
We use an important decoupling property of gravitational field equations in the general relativity theory and modifications, written with respect to nonholonomic frames with 2+2 spacetime decomposition. This allows us to integrate the…
We introduce the Directional Gradient-Curvature (DGC) method, a novel approach for filling gaps in gridded environmental data. DGC is based on an objective function that measures the distance between the directionally segregated normalized…
We introduce a modification of the Fast Marching Algorithm, which solves the generalized eikonal equation associated to an arbitrary continuous riemannian metric, on a two or three dimensional domain. The algorithm has a logarithmic…
This introduction to Green's functions is based on their role as kernels of differential equations. The procedures to construct solutions to a differential equation with an external source or with an inhomogeneity term are put together to…
We study existence and uniqueness of Green functions for the Cheeger $Q$-Laplacian in metric measure spaces that are Ahlfors $Q$-regular and support a $Q$-Poincar\'e inequality with $Q>1$. We prove uniqueness of Green functions both in the…
We prove estimates for the Green's function of the discrete bilaplacian in squares and cubes in two and three dimensions which are optimal except possibly near the corners of the square and the edges and corners of the cube. The main idea…
We present a robust algorithm for the computation of electromagnetic fields radiated by point sources (Hertzian dipoles) in cylindrically stratified media where each layer may exhibit material properties (permittivity, permeability, and…
The response of many materials to applied forces and boundary constraints depends upon internal geometric changes at multiple submacroscopic levels. Hierarchical structured deformations provide a mathematical setting for the description of…
We formulate the theory of nearly autoparallel maps (generalizing conformal transforms) of locally anisotropic spaces and define the nearly autoparallel integration as the inverse operation to both covariant derivation and deformation of…
Adaptive gradient methods have been widely adopted in training large-scale deep neural networks, especially large foundation models. Despite the huge success in practice, their theoretical advantages over classical gradient methods with…
A new set of infinitesimal transformations generalizing scale invariance for strongly anisotropic critical systems is considered. It is shown that such a generalization is possible if the anisotropy exponent \theta =2/N, with N=1,2,3 ...…
Green's function provides an inherent connection between theoretical analysis and numerical methods for elliptic partial differential equations, and general absence of its closed-form expression necessitates surrogate modeling to guide the…
Cellular structures must organize themselves within strict physical constraints, operating with finite resources and well-defined boundaries. Classical systems demonstrate only passive responses to boundaries, from surface energy…
We consider a random, uniformly elliptic coefficient field $a(x)$ on the $d$-dimensional integer lattice $\mathbb{Z}^d$. We are interested in the spatial decay of the quenched elliptic Green function $G(a;x,y)$. Next to stationarity, we…