arXiv research

Develops a new approach to study nonlinear PDEs and their singularities.

Proposes a new derivative concept for nonlinear DRO problems.

Study of nonlinear PDEs using derived geometry and BV formalism.

Study improves understanding of solutions to complex equations in geometry.

We get decay rate of higher derivatives of nonlinear massless Dirac equations with a kind of "good" spin null form. The method we rely on is similar to that of Li and Zang. However, they only give the decay rate of solution itself to nonlinear massless Dirac system.

Uniform bounds derived for nonlinear statistics.

The paper derives new gradient and Hessian estimates for nonlinear parabolic equations.

Method solves complex optimization problems with high probability bounds.

The paper derives subgradient estimates for a specific nonlinear subparabolic equation on pseudo-Hermitian manifolds.

Data-driven control of robotic systems using Koopman operators with error bounds.

Study moduli spaces of elliptic PDEs using derived $C^{\infty}$-geometry.

We give a unified statement and proof of a class of wellknown mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on the normal derivative at the boundary. These inequalities give rise to an energ…

We present a method to derive local estimates for some classes of fully nonlinear elliptic equations. The advantage of our method is that we derive Hessian estimates directly from $C^0$ estimates. Also, the method is flexible and can be applied to a large class of equations.

Method estimates parameters of complex nonlinear systems.

Paper uses IGA for efficient pricing of financial derivatives, comparing it to FDM and FEM.

A version of the nonlinear Hodge equations is introduced in which the irrotationality condition is weakened. An elliptic estimate for solutions is derived.

We investigate structured sparsity methods for variable selection in regression problems where the target depends nonlinearly on the inputs. We focus on general nonlinear functions not limiting a priori the function space to additive models. We propose two new regularizers based on partial derivatives as nonlinear equi…

Stochastic VB improves nonlinear model inference speed and accuracy.

We show that, locally, all geometric objects of Generalized Kahler Geometry can be derived from a function K, the "generalized Kahler potential''. The metric g and two-form B are determined as nonlinear functions of second derivatives of K. These nonlinearities are shown to arise via a quotient construction from an aux…

Optimal trading strategy derived for nonlinear price impact models.

The paper proves estimates for solutions to nonlinear equations on manifolds with boundary.

Paper presents a new backward deep BSDE method for solving nonlinear FBSDE problems.

Estimates for complex equations on manifolds derived from a conjecture.

The nonlinear equations describing all the nonsingular pencils of metrics of constant Riemannian curvature are derived and the integrability of these nonlinear equations by the method of inverse scattering problem is proved. It is proved that all the nonsingular pairs of compatible metrics of constant Riemannian curvat…

The study sets limits on how well nonlinear models can generalize from training data.

Families of explicit solutions are found to a nonlinear Black-Scholes equation which incorporates the feedback-effect of a large trader in case of market illiquidity. The typical solution of these families will have a payoff which approximates a strangle. These solutions were used to test numerical schemes for solving …

We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. Under the assumption of cone condition, we derive the $L^\infty$ estimate directly.

New insights into nonlinear multiview analysis for better data interpretation.

This research highlights the secrecy potential of nonlinear generative models and their all-or-nothing phase transition.

Derives gradient estimate for a specific nonlinear parabolic equation on Finsler manifolds.

We study a fully nonlinear equation of complex Monge-Ampere type on Hermitian manifolds. We establish the a priori estimates for solutions of the equation up to the second order derivatives with the help of a subsolution.

Unified framework for selecting variables with uncertainty quantification.

In this paper we study some geometrical objects (d-tensors, multi-time semisprays of polymomenta and nonlinear connections) on the dual 1-jet vector bundle $J^{1*}(\cal{T}, M)\to \cal{T}\times M$. Some geometrical formulas, which connect the last two geometrical objects, are also derived. Finally, a canonical nonlinear…

The objective of this paper is to provide a comprehensive study no-arbitrage pricing of financial derivatives in the presence of funding costs, the counterparty credit risk and market frictions affecting the trading mechanism, such as collateralization and capital requirements. To achieve our goals, we extend in severa…

This paper develops a fast algorithm for solving nonlinear PDEs using sparse Cholesky factorization.

We prove several differential Harnack inequalities for positive solutions to nonlinear backward heat equations with different potentials coupled with the Ricci flow. We also derive an interpolated Harnack inequality for the nonlinear heat equation under the $\varepsilon$-Ricci flow on a closed surface. These new Harnac…

The main purposes of this article are to extend our previous results on homogeneous sprays to arbitrary (generalized) sprays, to show that locally diffeomorphic exponential maps can be defined for any (generalized) spray, and to give a (possibly nonlinear) covariant derivative for any (possibly nonlinear) connection. I…

The paper extends Li-Yau-Hamilton estimates to evolving Kähler metrics and nonlinear heat equations.

We derive gradient and second order {\em a priori} estimates for solutions of the Neumann problem for a general class of fully nonlinear elliptic equations on compact Riemannian manifolds with boundary. These estimates yield regularity and existence results.

We study the qualitative behavior of nonlinear Dirac equations arising in quantum field theory on complete Riemannian manifolds. In particular, we derive monotonicity formulas and Liouville theorems for solutions of these equations. Finally, we extend our analysis to Dirac-harmonic maps with curvature term.

Coupled entropy corrects flaws in Tsallis entropy for complex systems.

The paper studies equations in conformal geometry with gradient and existence results.

This work examines the stability of GD and SGD near minima, revealing nonlinear dynamics that differ from linear analysis.

Solves a specific Dirichlet problem on Riemannian manifolds.

We study a class of fully nonlinear elliptic equations on closed Hermitian manifolds. We derive $C^\infty$ {\em a priori} estimates, and then prove the existence of admissible solutions. In the approach, a new Hermitian metic is constructed to launch the method of continuity.

We consider a nonlinear version of the Yamabe problem on locally conformally flat compact manifolds with boundary. The main technique we used is to derive boundary $C^2$ estimates directly from boundary $C^0$ estimates. In particular, the result is a generalization of the work by Escobar.

The paper studies fully nonlinear equations on Hermitian manifolds, proving existence and interior estimates.

We present some new ideas to derive {\em a priori} second order estiamtes for a wide class of fully nonlinear parabolic equations. Our methods, which produce new existence results for the initial-boundary value problems in $\bfR^n$, are powerful enough to work in general Riemannian manifolds.