arXiv research

Unified approach for sample aggregation in transfer learning across various divergence measures.

New insights show NAG and FISTA converge linearly without knowing strong convexity modulus.

We present atomistic molecular dynamics simulations of two Polyethylene systems where all entanglements are trapped: a perfect network, and a melt with grafted chain ends. We examine microscopically at what level topological constraints can be considered as a collective entanglement effect, as in tube model theories, o…

Establishes a lower bound for Kähler hyperbolicity modulus in hyperconvex domains and bounded strongly pseudoconvex domains.

Curve shortening flow increases annulus modulus.

The $p$--modulus ${\rm mod}_p(\mathcal{F})$ of a foliation $\mathcal{F}$ on a Riemannian manifold $M$ is a generalization of extremal length of plane curves introduced by L. Ahlfors. We study the variation $t\mapsto{\rm mod}_p(\mathcal{F}_t)$ of the modulus. In particular, we consider product of moduli of orthogonal fo…

We continue the study of the variation of the $p$--modulus of a foliation initiated by the first author. We derive the formula for the second variation which allows to study $p$--stable foliations. We obtain some results concerning codimension one $p$--stable foliations. Moreover, we derive the equation for the critica…

Proves a theorem in sub-Riemannian geometry using Carnot groups.

Study on a metric for disk automorphisms with maximal modulus.

Defines a new modulus for Lipschitz surfaces and proves a homological duality theorem.

For any link and for any modulus $m$ we introduce an equivalence relation on the set of non-trivial m-colorings of the link (an m-coloring has values in Z/mZ). Given a diagram of the link, the equivalence class of a non-trivial m-coloring is formed by each assignment of colors to the arcs of the diagram that is obtaine…

Study on the Euler-Plateau energy with elastic modulus, focusing on minimizers and critical surfaces.

Origami creates flat torus models of any size.

We discuss the maximum modulus principle, and weak unique continuation, for CR functions on an abstract almost CR manifold M. We investigate these matters under the assumption of weak pseudoconcavity, and obtain sharp results about propagation along Sussmann leaves.

Paper proves uniform continuity bounds for complex Monge-Ampère solutions.

We study combinatorial modulus on boundaries of hyperbolic Coxeter groups. We give new examples of hyperbolic groups whose boundary satisfies a combinatorial version of the Loewner property, and prove Cannon's conjecture for Coxeter groups. We also establish some connections with l^p cohomology.

A carpet is a metric space homeomorphic to the Sierpinski carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poinca…

Study on existence and properties of continuous solutions to complex Hessian equations.

Stochastic approximation (SA) is a classical approach for stochastic convex optimization. Previous studies have demonstrated that the convergence rate of SA can be improved by introducing either smoothness or strong convexity condition. In this paper, we make use of smoothness and strong convexity simultaneously to boo…

This article concerns exact results on the minimum number of colors of a Fox coloring over the integers modulo r, of a link with non-null determinant. Specifically, we prove that whenever the least prime divisor of the determinant of such a link and the modulus r is 2, 3, 5, or 7, then the minimum number of colors is 2…

New bounds on the wildness of Bing's involution.

We derive sharp estimates on modulus of continuity for solutions of the heat equation on a compact Riemannian manifold with a Ricci curvature bound, in terms of initial oscillation and elapsed time. As an application, we give an easy proof of the optimal lower bound on the first eigenvalue of the Laplacian on such a ma…

Cannon, Floyd and Parry have studied the modulus of finite subdivision rules extensively. We investigate the properties of the modulus of subdivision rules with linear and exponential growth at every vertex, using barycentric subdivision and a subdivision rule for the Borromean rings as examples. We show that the subdi…

Proves conditions for Fourier transforms in rank 1 symmetric spaces.

We investigate the properties of a modulus of a foliation on a Riemannian manifold. We give necessary and sufficient conditions for the existence of an extremal function and state some of its properties. We obtain the integral formula which, in a sense, combines the integral over the manifold with integral over the lea…

Study on Hölder continuity of complex Monge-Ampère solutions on Stein spaces.

We develop methods to efficiently approximate data in metric spaces without additional assumptions.

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigen…

By adapting methods of \cite{AC} we prove a sharp estimate on the expansion modulus of the gradient of the log of the parabolic kernel to the Schördinger operator with convex potential, which improves an earlier work of Brascamp-Lieb. We also include alternate proofs to the improved log-concavity estimate, and to the f…

By regular tessellation, we mean any hyperbolic 3-manifold tessellated by ideal Platonic solids such that the symmetry group acts transitively on oriented flags. A regular tessellation has an invariant we call the cusp modulus. For small cusp modulus, we classify all regular tessellations. For large cusp modulus, we pr…

The Korányi ellipsoidal ring $\mathcal{E}$ of radii $B$ and $A$, $0<B<A$, is defined as the image of the Korányi spherical ring of the same radii and centred at the origin via a linear contact map $L$ in the Heisenberg group. If $K\ge 1$ is the maximal distortion of $L$ then we prove that the modulus of $\mathcal{E}$ i…

New stretch maps minimize distortion in geometric group theory.

Physics-informed GANs estimate elastic moduli from mechanical tests.

Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.

A set of relations between the modulus and phase is derived for amplitudes of the form $\mels{\hatu(x)}$ where $\hat{U}(x) \in SU(n)$ in the fundamental representation and $x$ denotes the coordinates on the group manifold. An illustration is given for the case $n=2$ as well as a brief discussion of phase singularities …

Stochastic-gradient-based optimization has been a core enabling methodology in applications to large-scale problems in machine learning and related areas. Despite the progress, the gap between theory and practice remains significant, with theoreticians pursuing mathematical optimality at a cost of obtaining specialized…

We prove the Fundamental Gap Conjecture, which states that the difference between the first two Dirichlet eigenvalues (the spectral gap) of a Schrödinger operator with convex potential and Dirichlet boundary data on a convex domain is bounded below by the spectral gap on an interval of the same diameter with zero poten…

Study the discontinuity of functions not embeddable in Euclidean space.

Sharp stability in Almgren problem solved in any dimension.

Sharp estimates derived for quasilinear equations on metric measure spaces.

The study improves fundamental gap estimates for surfaces with non-constant positive curvature.

Let $A \subset \mathbb{R} ^2$ be a smooth doubly connected domain. We consider the Dirichlet energy $E(u)=\int_{A} |\nabla u|^2$, where $u:A \rightarrow \mathbb{C}$, and look for critical points of this energy with prescribed modulus $|u|=1$ on $\partial A$ and with prescribed degrees on the two connected components o…

Quasispheres can be approximated by smooth spheres.

The paper extends a method for numerical conformal mappings to surfaces using Laplace-Beltrami equations.

Study finds numerical moduli in special 2-flags of length 5.

Study on spectral stability of an embedded annulus under curve shortening and Ricci flows.

Let a $A$ be the 1-skeleton of a triangulated topological annulus. We establish bounds on the combinatorial modulus of a refinement $A'$, formed by attaching new vertices and edges to $A$, that depend only on the refinement and not on the structure of $A$ itself. This immediately applies to showing that a disk triangul…

Proposes DMOC for more nuanced neural network robustness.