Related papers: The Hasse Principle for Geometric Variational Prob…
We prove $S$-arithmetic inhomogeneous Khintchine type theorems on analytic nondegenerate manifolds. The divergence case, which constitutes the main substance of this paper, is proved in the general context of Hausdorff measures using…
This paper is concerned with the numerical minimization of energy functionals in Hilbert spaces involving convex constraints coinciding with a semi-norm for a subspace. The optimization is realized by alternating minimizations of the…
A method for the identification of small inhomogeneities from a surface data is presented in the framework of an inverse scattering problem for the Helmholtz equation. Using the assumptions of smallness of the scatterers one reduces this…
We establish an area formula for the spherical measure of intrinsically regular submanifolds of low codimension in Heisenberg groups. The spherical measure is computed with respect to an arbitrary homogeneous distance. Among the arguments…
In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. For this purpose we construct a generalized finite element basis that spans…
We give an exponentially-accurate normal form for a Lagrangian particle moving in a rotating shallow-water system in the semi-geostrophic limit, which describes the motion in the region of an exponentially-accurate slow manifold (a region…
Many economic parameters are identified by ``thin sets'' (submanifolds with Lebesgue measure zero) and hence difficult to recover from data in an ambient space. This paper provides a unified theory for estimation and inference of such…
This treatise concerns shapes in the sense of constellations of points with various automorphisms quotiented out: continuous translations, rotations and dilations, and also discrete mirror image identification and labelling…
Physics-informed neural networks (PINNs), owing to their mesh-free nature, offer a powerful approach for solving high-dimensional partial differential equations (PDEs) in complex geometries, including irregular domains. This capability…
In the real world, we often come across soft objects having spatially varying stiffness, such as human palm or a wart on the skin. In this paper, we propose a novel approach to render thin, deformable objects having spatially varying…
In conventional spectral/finite element methods, the triangulation/quadrilateralization of the domain produces many interior edges which require additional DOF. What if we could directly use the original hull without going to…
We obtain results on approximation of holomorphic maps by algebraic maps, jet transversality theorems for holomorphic and algebraic maps, and the homotopy principle for holomorphic submersions of Stein manifolds to certain algebraic…
The asymptotic formula of the number of integral points in non-compact symmetric homogeneous spaces of semi-simple simply connected algebraic groups is given by the average of the product of the number of local solutions twisted by the…
We consider a smooth system of two homogeneous quadratic equations over the rationals in at least 13 variables. In this case, the Hasse principle is known to hold, thanks to the work of Mordell in 1959. The only local obstruction is over…
Global radial basis function (RBF) collocation methods with inifinitely smooth basis functions for partial differential equations (PDEs) work in general geometries, and can have exponential convergence properties for smooth solution…
We propose HAMSI (Hessian Approximated Multiple Subsets Iteration), which is a provably convergent, second order incremental algorithm for solving large-scale partially separable optimization problems. The algorithm is based on a local…
We develop the theory of Diophantine approximation for systems of simultaneously small linear forms, which coefficients are drawn from any given analytic non-degenerate manifolds. This setup originates from a problem of Sprind\v{z}uk from…
Let $X$ be a connected, compact complex manifold and $S\subset X$ a separating real hypersurface, so that $X$ decomposes as a union of compact complex manifolds with boundary $\bar X^\pm$. Let $\mathcal{M}$ be the moduli space of $S$-framed…
In this note we provide conditions for local invariance of finite dimensional submanifolds for solutions to stochastic partial differential equations (SPDEs) in the framework of the variational approach. For this purpose, we provide a…
In this paper, we study the problem of computing a homotopy from a planar curve $C$ to a point that minimizes the area swept. The existence of such a minimum homotopy is a direct result of the solution of Plateau's problem. Chambers and…