arXiv research

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

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…

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.

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

Plane Delaunay triangulations are rigid under Luo's discrete conformal change.

Study the discontinuity of functions not embeddable in Euclidean space.

New bounds on the wildness of Bing's involution.

Alternative proof for non-existence of complete curves in differential strata.

A theorem controls the relationship between dimensions of continua.

New calculus on spacetimes for nonlinear differential equations.

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

Recently, Andrews and Clutterbuck [AC13] gave a new proof of the optimal lower eigenvalue bound on manifolds via modulus of continuity for solutions of the heat equation. In this short note, we give an alternative proof of Theorem 2 in [AC13]. More precisely, following Ni's method ([Ni13, Section 6]) we give an ellipti…

Curve shortening flow increases annulus modulus.

In this paper, we prove conformal positive mass theorems for asymptotically flat manifolds with charge. We apply conformal relations to show that if the conformal sum of scalar curvature is not less than the norm square of electric field and electric density, the sum of the mass will not less than the modulus of total …

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…

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

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…

Study on a metric for disk automorphisms with maximal modulus.

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.

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

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

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.

Let $C$ be a subset of $\mathbb{R}^n$ (not necessarily convex), $f:C\to\mathbb{R}$ be a function, and $G:C\to\mathbb{R}^n$ be a uniformly continuous function, with modulus of continuity $ω$. We provide a necessary and sufficient condition on $f$, $G$ for the existence of a convex function $F\in C^{1, ω}(\mathbb{R}^n)$ …

In this paper, we develop a convolutional neural network model to predict the mechanical properties of a two-dimensional checkerboard composite quantitatively. The checkerboard composite possesses two phases, one phase is soft and ductile while the other is stiff and brittle. The ground-truth data used in the training …

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.

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…

Wavelet scattering spectra model non-Gaussian time-series, proving scale invariance for self-similar processes.

We consider a class $\mathscr{X}$ of continuous functions on $[0,1]$ that is of interest from two different perspectives. First, it is closely related to sets of functions that have been studied as generalizations of the Takagi function. Second, each function in $\mathscr{X}$ admits a linear pathwise quadratic variatio…

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 eigenvalues on quaternion-Kähler manifolds with geometric bounds.

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.

Extends three circle theorem to almost Hermitian manifolds.

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…

Study of transitivity in partially hyperbolic maps with expanding linear part.

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.

A maximum principle for C^0 null hypersurfaces is obtained and used to derive a splitting theorem for spacetimes which contain null lines. As a consequence of this null splitting theorem, it is proved that an asymptotically simple vacuum (Ricci flat) spacetime which contains a null line is isometric to Minkowski space.

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