arXiv research

Uniform bounds derived for fully non-linear equations.

Study shows solutions to certain equations form smooth manifolds.

Paper establishes estimates for solutions on compact manifolds.

Study shows long-term solutions for complex equations on curved spaces.

Paper generalizes sub-slope definition and solves complex equations on compact manifolds.

Unified diffusive bounds for non-linear parabolic equations.

Paper solves complex equations on noncompact manifolds.

The paper establishes boundary estimates for solutions to elliptic equations on Hermitian manifolds.

Veronese webs are rich geometric structures with deep relationships to various domains of mathematics. The PDEs which determine the Veronese web are overdetermined if dim >3, but in the case dim =3 they reduce to a special flavor of a non-linear wave equation. The symmetries embedded in the definition of a Veronese web…

Solves Dirichlet problem for elliptic equations on Hermitian manifolds.

Proves energy estimates for tensorial wave equations, decoupling components for stability proof.

We show that classical Wilczynski--Se-ashi invariants of linear systems of ordinary differential equations are generalized in a natural way to contact invariants of non-linear ODEs. We explore geometric structures associated with equations that have vanishing generalized Wilczynski invariants and establish relationship…

We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex Monge-Ampère, Hessian and inverse Hessian equations. As an application we solve a class of He…

Fiedler and Mallet-Paret prove a version of the classical Poincaré-Bendixson Theorem for scalar parabolic equations. We prove that a similar result holds for bounded solutions of the non-linear Cauchy-Riemann equations. The latter is an application of an abstract theorem for flows with a(n) (unbounded) discrete Lyapuno…

Sharp L∞ estimates for fully non-linear elliptic equations on compact complex manifolds.

Develops equivariant connections for Yang-Mills equations, simplifying interactions modeling.

Paper studies solutions to a specific equation in conformal geometry with singular sets.

Sharp $L^\infty$ estimates proved for complex Monge-Ampère equations.

In this paper, the parallel transport frames over non-lightlike curves in Minkowski 3-space are introduced. Evolution equations of these frames with respect to arc length and time are calculated over the space of these curves. Then the equivalence of the non-linear Schrödinger equation and non-linear heat system to the…

We derive a system of equations governing the coupled gravitational and electromagnetic perturbations of Reissner-Nordström spacetime. The equations are derived in the context of global non-linear stability of Reissner-Nordström under axially symmetric polarized perturbations, as a generalization of the recent work on …

Extends parabolic study to flat hyperkähler manifolds.

New model explains market dynamics with phase transitions and non-linear interactions.

Derives estimates for geometric elliptic equations on complex manifolds.

This paper proposes a new method to learn integration schemes for complex ODEs.

A method for constructing explicit Calabi-Yau metrics in six dimensions in terms of an initial hyperkahler structure is presented. The equations to solve are non linear in general, but become linear when the objects describing the metric depend on only one complex coordinate of the hyperkahler 4-dimensional space and i…

Study on dynamics of non-linear autoencoders learning principal components.

We study two inverse problems on a globally hyperbolic Lorentzian manifold $(M,g)$. The problems are: 1. Passive observations in spacetime: Consider observations in a neighborhood $V\subset M$ of a time-like geodesic $μ$. Under natural causality conditions, we reconstruct the conformal type of the unknown open, relativ…

Study proves global existence and decay for complex wave equations.

Finite time for subsolutions on Riemannian manifolds proved.

Paper models non-linear dynamics from time series data.

Study finds non-uniqueness in sphere metrics with constant fractional curvature.

We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call non-linear affine processes. This is done as follows: given a set of parameters for the process, we construct a corresponding non-linear expectation on the path space of continuous processes. By a general dynamic programm…

The physics informed neural network (PINN) is evolving as a viable method to solve partial differential equations. In the recent past PINNs have been successfully tested and validated to find solutions to both linear and non-linear partial differential equations (PDEs). However, the literature lacks detailed investigat…

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

Solves Dirichlet problem for prescribed scalar curvature in Anti-de Sitter space

In this paper, we pursue the study of second order BSDEs with jumps (2BSDEJs for short) started in our accompanying paper [15]. We prove existence of these equations by a direct method, thus providing complete wellposedness for 2BSDEJs. These equations are a natural candidate for the probabilistic interpretation of som…

New equations use Pin(2) symmetry to study spinor and connection solutions.

We consider a model of linear market impact, and address the problem of replicating a contingent claim in this framework. We derive a non-linear Black-Scholes Equation that provides an exact replication strategy. This equation is fully non-linear and singular, but we show that it is well posed, and we prove existence o…

New model for Knightian uncertainty with jumps.

We consider that the price of a firm follows a non linear stochastic delay differential equation. We also assume that any claim value whose value depends on firm value and time follows a non linear stochastic delay differential equation. Using self-financed strategy and replication we are able to derive a Random Partia…

We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations...etc) in divergence form. This divergence free quantities generalize to target manifolds without symmetries the well known conservation laws for harmonic maps into homogeneous s…

In this paper we study the existence of solutions for a class of non-linear differential equation on compact Riemannian manifolds. We establish a lower and upper solutions' method to show the existence of a smooth positive solution for the equation (EQ1) \begin{equation} \label{E4} Δu \ + \ a(x)u \ = \ f(x)F(u) \ + \ h…

In this paper, we investigate the non-linear Black--Scholes equation: $u_t+ax^2u_{xx}+bx^3u_{xx}^2+c(xu_x-u)=0,\quad a,b>0,\ c\geq0.$ and show that the one can be reduced to the equation $u_t+(u_{xx}+u_x)^2=0$ by an appropriate point transformation of variables. For the resulting equation, we study the group-theore…

In this article, we study a generalisation of the Seiberg-Witten equations, replacing the spinor representation with a hyperKahler manifold equipped with certain symmetries. Central to this is the construction of a (non-linear) Dirac operator acting on the sections of the non-linear fibre-bundle. For hyperKahler manifo…

We propose a new least-squares Monte Carlo algorithm for the approximation of conditional expectations in the presence of stochastic derivative weights. The algorithm can serve as a building block for solving dynamic programming equations, which arise, e.g., in non-linear option pricing problems or in probabilistic dis…

We study the existence of solutions of the non-linear differential equations on the compact Riemannian manifolds $(M^n,g), n\geq 2$, Δ_p u + a(x)u^{p-1} = λf(u,x), (E2) where $Δ_p$ is the $p-$laplacian, with $1<p<n$. The equation (E2) generalizes a equation considered by Aubin, where he has considered, a compact Rieman…

In the neighborhood of a regular point, generalized Kahler geometry admits a description in terms of a single real function, the generalized Kahler potential. We study the local conditions for a generalized Kahler manifold to be a generalized Calabi-Yau manifold and we derive a non-linear PDE that the generalized Kahle…

Improved disability insurance model with collective health claims.