Related papers: Natural Boundary Conditions for Smoothing in Geome…
Mixed boundary conditions are introduced to finite element exterior calculus. We construct smoothed projections from Sobolev de Rham complexes onto finite element de Rham complexes which commute with the exterior derivative, preserve…
We present a systematic method for constructing static, spherically symmetric regular spacetimes in general relativity satisfying the weak energy condition. Our approach relies on physically reasonable assumptions on the matter energy…
We study smooth linear spectral statistics of twisted Laplacians on random $n$-covers of a fixed compact hyperbolic surface $X$. We consider two aspects of such statistics. The first is the fluctuations of such statistics in a small energy…
Surface parameterizations are widely applied in computer graphics, medical imaging and transformation optics. In this paper, we rigorously derive the gradient vector and Hessian matrix of the discrete conformal energy for spherical…
A model describing cell membranes as optimal shapes with regard to the $L^2$-deficit of their mean curvature to a given constant called spontaneous curvature is considered. It is shown that the corresponding energy functional is lower…
We show that the driving force behind the regularizing effect of Laplacian smoothing on surface elements is the popular mean ratio quality measure. We use these insights to provide natural generalizations to polygons and polyhedra. The…
Consider a lipid membrane with a free exposed edge. The energy describing this membrane is quadratic in the extrinsic curvature of its geometry; that describing the edge is proportional to its length. In this note we determine the boundary…
We generalize the notion of calibrated submanifolds to smooth maps and show that the several examples of smooth maps appearing in the differential geometry become the examples of our situation. Moreover, we apply these notion to give the…
We prove a microlocal lower bound on the mass of high energy eigenfunctions of the Laplacian on compact surfaces of negative curvature, and more generally on surfaces with Anosov geodesic flows. This implies controllability for the…
In this work, we develop new optimization algorithms that use approximate second-order information combined with the gradient regularization technique to achieve fast global convergence rates for both convex and non-convex objectives. The…
In this paper we study the Poisson and heat equations on bounded and unbounded domains with smooth boundary with random Dirichlet boundary conditions. The main novelty of this work is a convenient framework for the analysis of such…
We investigate relations between spatial properties of the free energy and the radius of Gaussian curvature of the underlying curved lattice geometries. For this purpose we derive recurrence relations for the analysis of the free energy…
We study Dirichlet boundary control of Stokes flows in 2D polygonal domains. We consider cost functionals with two different boundary control regularization terms: the $L^2$ norm and an energy space seminorm. We prove well-posedness and…
Machine learning frameworks based on correlations of interatomic positions begin with a discretized description of the density of other atoms in the neighbourhood of each atom in the system. Symmetry considerations support the use of…
In the present work we study classifiers' decision boundaries via Brownian motion processes in ambient data space and associated probabilistic techniques. Intuitively, our ideas correspond to placing a heat source at the decision boundary…
We give a unified statement and proof of a class of wellknown mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on…
We establish the local well-posedness for the free boundary problem for the compressible Euler equations describing the motion of liquid under the influence of Newtonian self-gravity. We do this by solving a tangentially-smoothed version of…
We consider a nonlinear, free boundary fluid-structure interaction model in a bounded domain. The viscous incompressible fluid interacts with a nonlinear elastic body on the common boundary via the velocity and stress matching conditions.…
Smoothness is crucial for attaining fast rates in first-order optimization. However, many optimization problems in modern machine learning involve non-smooth objectives. Recent studies relax the smoothness assumption by allowing the…
Boundary constraints in physical, environmental and engineering models restrict smooth states such as temperature to follow known physical laws at the edges of their spatio-temporal domain. Examples include fixed-state or fixed-derivative…