arXiv research

Richberg technique adapted for nonlinear subequations.

We address the restriction problem for viscosity subsolutions of a fully nonlinear PDE on a manifold Z. The constraints on the restrictions of smooth subsolutions to a submanifold X in Z determine a restricted subequation on X. The problem is to show that general (upper semi-continuous) subsolutions restrict to satisfy…

We study the problem of removable singularities for degenerate elliptic equations. Let F be a fully nonlinear second-order partial differential subequation of degenerate elliptic type on a manifold X. We study the question: Which closed subsets E in X have the property that every F-subharmonic function (subsolution) on…

In this paper we study the Dirichlet problem for fully nonlinear second-order equations on a riemannian manifold. As in a previous paper we define equations via closed subsets of the 2-jet bundle. Basic existence and uniqueness theorems are established in a wide variety of settings. However, the emphasis is on starting…

We introduce and investigate the notion of a `generalized equation' of the form $f(D^2 u)=0$, based on the notions of subequations and Dirichlet duality. Precisely, a subset ${\mathbb H}\subset {\rm Sym}^2({\mathbb R}^n)$ is a generalized equation if it is an intersection ${\mathbb H} = {\mathbb E}\cap (-\widetilde{\ma…

Estimates for polynomial operators using determinant majorization and subharmonics.

This article introduces the degenerate special Lagrangian equation (DSL) and develops the basic analytic tools to construct and study its solutions. The DSL governs geodesics in the space of positive graph Lagrangians in $\mathbb{C}^n.$ Existence of geodesics in the space of positive Lagrangians is an important step in…

In this paper, we study the singular sets of $F$-subharmonic functions $u: B_{2}(0^{n})\rightarrow\mathbf{R}$, where $F$ is a subequation. The singular set $\mathcal{S}(u)\subset B_{2}(0^{n})$ has a stratification $\mathcal{S}^{0}(u)\subset\mathcal{S}^{1}(u)\subset\cdots\subset\mathcal{S}^{k}(u)\subset\cdots\subset\mat…

We solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in euclidean space. The main results apply, in particular, to subequations with a Riesz characteristic $p \geq 2$. In this cas…

There is an interesting potential theory associated to each degenerate elliptic, fully nonlinear equation $f(D^2u) = 0$. These include all the potential theories attached to calibrated geometries. This paper begins the study of tangents to the subsolutions in these theories, a topic inspired by the results of Kiselman …

New findings on convexity of special Lagrangian geodesics.

This paper presents a simple, self-contained account of Garding's theory of hyperbolic polynomials, including a recent convexity result of Bauschke-Guler-Lewis-Sendov and an inequality of Gurvits. This account also contains new results, such as the existence of a real analytic arrangement of the eigenvalue functions. I…

We explain how to apply techniques from integrable systems to construct $2k$-soliton homoclinic wave maps from the periodic Minkowski space $S^1\times R^1$ to a compact Lie group, and more generally to a compact symmetric space. We give a correspondence between solutions of the -1 flow equation associated to a compact …

Study of weighted nonlinear flags in symplectic geometry.

This part II of the paper is concerned with questions of existence and uniqueness of tangents in the special case of G-plurisubharmonic functions, where G is a compact subset of the Grassmannian of p-planes in ${\mathbb R}^n$. An upper semi-continuous function u on an open set $Ω$ in ${\mathbb R}^n$ is G-plurisubharmon…

For a system of second order differential equations we determine a nonlinear connection that is compatible with a given generalized Lagrange metric. Using this nonlinear connection, we can find the whole family of metric nonlinear connections that can be associated with a system of SODE and a generalized Lagrange struc…

A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Frechet manifolds of nonlinear flags, in this way generalizing the nonlinear Grassmannians. As an application we describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms that consist of nested symplecti…

We find a convex model for traditional nonlinear regression under L2 loss.

Extends importance sampling to nonlinear models using adjoint operators.

Proposes a new method for nonlinear Bayesian updates using ensemble kernel regression.

Study solves inverse problems for equations with fractional nonlinearities.

Introduces nonlinear splittings on fibre bundles for generalizing connections.

Note on advancements in nonlinear elliptic equations' regularity theory.

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

New methods for constructing submersions between different types of spaces.

The paper introduces a fast algorithm for learning and forecasting nonlinear dynamics from noisy time series data.

AdaKoop efficiently models nonlinear dynamics from nonstationary data streams.

Solves nonlinear problems on metric structures through eigenvalue counting.

Nonlinear MCMC improves Bayesian machine learning sampling.

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…

New method identifies latent sources from nonlinear mixtures without auxiliary variables.

JULIA combines multi-linear and nonlinear models for tensor completion.

Bayesian filtering approach identifies nonlinear restoring forces in dynamic systems.

The purpose of this paper is to construct the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility depending on the option price. We review a method how to transform the problem into a solution of a time depending nonlinear parabolic equation defined on a fixed domain. R…

Unified analysis for nonlinear parametric models in Bayesian optimization.

The paper develops adaptive deep learning methods for nonlinear time series models.

This paper tackles efficient optimization for nonlinear embeddings in similarity learning.

The development of algorithms for unsupervised pattern recognition by nonlinear clustering is a notable problem in data science. Markov clustering (MCL) is a renowned algorithm that simulates stochastic flows on a network of sample similarities to detect the structural organization of clusters in the data, but it has n…

Constructs differential characters on nonlinear Graßmannians.

Stock networks, constructed from stock price time series, are a well-established tool for the characterization of complex behavior in stock markets. Following Mantegna's seminal paper, the linear Pearson's correlation coefficient between pairs of stocks has been the usual way to determine network edges. Recently, possi…

The paper introduces a framework to assess nonlinear causality in financial markets.

Paper connects contrastive learning to MI maximization and establishes robust methods for nonlinear ICA and subspace estimation.

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

Non-Markovian point process shows power-law scaling, similar to nonlinear Markovian process.

Machine learning techniques have recently received significant attention as promising approaches to deal with the optical channel impairments, and in particular, the nonlinear effects. In this work, a machine learning-based classification technique, known as the Parzen window (PW) classifier, is applied to mitigate the…

Paper accelerates nonlinear mapping in online systems with lower time complexity.

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.

New framework IIA identifies innovations in general nonlinear vector autoregressive processes.