arXiv research

Study natural invariants for third order nonlinear operators on 2D manifolds.

We establish a general Liouville type theorem for conformally invariant fully nonlinear equations.

We present some further results on Liouville type theorems for some conformally invariant fully nonlinear equations.

In this paper we provide a characterization of second order fully nonlinear CR invariant equations on the Heisenberg group, which is the analogue in the CR setting of the result proved in the Euclidean setting by A. Li and the first author (2003). We also prove a comparison principle for solutions of second order fully…

This paper concerns local gradient estimates to solutions of general conformally invariant fully nonlinear elliptic equations of second order.

Study natural invariants for differential operators, simplifying their equivalence problem.

A gauge-invariant form of the nonlinear Hodge equations is studied.

Proposes IIB for domain generalization, overcoming failure modes of IRM.

Nonlinear analysis has played a prominent role in the recent developments in geometry and topology. The study of the Yang-Mills equation and its cousins gave rise to the Donaldson invariants and more recently, the Seiberg-Witten invariants. Those invariants have enabled us to prove a number of striking results for low …

We present some results on a fully nonlinear version of the Yamabe problem and a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations.

We establish Liouville type theorems for degenerate conformally invariant equations.

We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.

Paper extends scattering network for fast, scalable feature extraction.

We study solutions to conformally invariant equations with isolated singularties.

The moving coframe method is applied to solve the local equivalence problem for the class of nonlinear wave equations in two independent variables under an action of the pseudo-group of contact transformations. The structure equations and the complete sets of differential invariants for symmetry groups are found. The s…

In nonlinear latent variable models or dynamic models, if we consider the latent variables as confounders (common causes), the noise dependencies imply further relations between the observed variables. Such models are then closely related to causal discovery in the presence of nonlinear confounders, which is a challeng…

This paper proves Liouville theorems for conformally invariant fully nonlinear equations.

Ancient pancake solutions found for curvature flows.

Proposes iCaRL for nonlinear OOD generalization in causal settings.

We present a conformal deformation involving a fully nonlinear equation in dimension 4, starting with positive scalar curvature. Assuming a certain conformal invariant is positive, one may deform from positive scalar curvature to a stronger condition involving the Ricci tensor. We also give a new conformally invariant …

The paper studies minimal resistance dynamics in radial fields, finding unique solutions for incompressible flows.

The Yamabe problem concerns finding a conformal metric on a given closed Riemannian manifold so that it has constant scalar curvature. This paper concerns mainly a fully nonlinear version of the Yamabe problem and the corresponding Liouville type problem.

We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems - with a reasonable expectation of suc…

In this paper, we prove that nonnegative polyharmonic functions on the upper half space satisfying a conformally invariant nonlinear boundary condition have to be the "\emph{polynomials} plus \emph{bubbles}" form. The nonlinear problem is motivated by the recent studies of boundary GJMS operators and the $Q$-curvature …

Study algebraic invariants from lightning self-attention models.

New metrics for surface shapes incorporating curve properties.

Solves curvature problems on manifolds with negative curvature.

In this investigation, symmetry properties of the nonlinear heat conductivity equations of general form $u_t = [E(x, u)u_x]_x + H(x, u)$ are studied. The point symmetry analysis of these equations is considered as well as an equivalence classification which admits an extension by one dimension of the principal Lie alge…

Study of Dirac equation with non-local nonlinearity on spheres.

Paper proposes a fully data-driven method for Koopman spectral analysis.

This work preserves linear invariants in ensemble filters for non-Gaussian data assimilation.

We define and study the invariant linear and nonlinear horizontal double complexes of a local Lie group.

We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single stand…

Solves open problems for fully nonlinear elliptic equations on manifolds.

Two environments are enough to infer causal graphs and counterfactuals.

Kernel VICReg improves SSL in RKHS, capturing nonlinear structures.

New algorithms extract Koopman invariant subspaces from large-scale data.

Derives representations invariant under crystallographic groups for functions.

New algorithm discovers causal relationships from observational data efficiently.

Legendrian contact homology (LCH) and its associated differential graded algebra are powerful non-classical invariants of Legendrian knots. Linearization makes the LCH computationally tractable at the expense of discarding nonlinear (and noncommutative) information. To recover some of the nonlinear information while pr…

CDANs classify unordered feature sets efficiently and invariantly.

In this paper, we establish two new types of invariant sets for the coupled nonlinear Schrodinger system on $\mathbb{R}^n$, and derive two sharp thresholds of blow-up and global existence for its solution. Some analogous results for the nonlinear Schrodinger system posed on the hyperbolic space $\mathbb{H}^n$ and on th…

We establish interior Lipschitz regularity for continuous viscosity solutions of fully nonlinear, conformally invariant, degenerate elliptic equations. As a by-product of our method, we also prove a weak form of the strong comparison principle, which we refer to as the principle of propagation of touching points, for o…

We provide an easy approach to the geodesic distance on the general linear group GL(n) for left-invariant Riemannian metrics which are also right-O(n)-invariant. The parametrization of geodesic curves and the global existence of length minimizing geodesics are deduced using simple methods based on the calculus of varia…

Unified geometric perspectives on PDEs, torsion invariants, and moduli theory.

Study controls bifurcations in Eulerian flows with multiple Hopf singularities.

Proves existence and compactness of solutions to $σ_2$-Nirenberg problem on sphere.

In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that our results do not assume concavity assumption on the fully nonlinear equations we…