arXiv research

Proves existence of solutions for a specific nonlinear equation on Riemannian manifolds.

Sharp sub-Gaussian bounds for subsolutions of Trudinger's equation on Riemannian manifolds.

Sharp upper bounds found for solutions of a specific equation on Riemannian manifolds.

We prove the concavity of $p$-Rényi entropy power for positive solutions to the doubly nonlinear diffusion equations on $\mathbb{R}^n$ or compact Riemannian manifolds with nonnegative Ricci curvature. As applications, we give new proofs of the sharp $L^p$-Sobolev inequality and $L^p$-Gagliardo-Nirenberg inequalities on…

We study a doubly reflected backward stochastic differential equation (BSDE) with integrable parameters and the related Dynkin game. When the lower obstacle $L$ and the upper obstacle $U$ of the equation are completely separated, we construct a unique solution of the doubly reflected BSDE by pasting local solutions and…

Solitary waves are localized gravity waves that preserve their consistency and henceforth their visibility through properties of nonlinear hydrodynamics. Solitary waves have finite amplitude and spread with constant speed and constant shape. In this paper, we have used Lie group of transformation method to solve (3 + 1…

FNOs learn solution operators of dissipative equations efficiently via spectral methods.

This paper studies gradient flows in asymmetric metric spaces and proves existence results.

Paper confirms Thom's conjecture for nonlinear evolutions on manifolds.

Bäcklund transformations for smooth and ``space discrete'' Hashimoto surfaces are discussed and a geometric interpretation is given. It is shown that the complex curvature of a discrete space curve evolves with the discrete nonlinear Schrödinger equation (NLSE) of Ablowitz and Ladik, when the curve evolves with the Has…

This is the second paper in a series of works devoted to nonholonomic Ricci flows. By imposing non-integrable (nonholonomic) constraints on the Ricci flows of Riemannian metrics we can model mutual transforms of generalized Finsler-Lagrange and Riemann geometries. We verify some assertions made in the first partner pap…

We consider a class of abstract nonlinear evolution equations in supermanifolds (smf's) modelled over Z_2-graded locally convex spaces. We show uniqueness, local existence, smoothness, and an abstract version of causal propagation of the solutions. If an a-priori estimate prevents the solutions from blowing-up then an …

An efficient method to construct Hamiltonian structures for nonlinear evolution equations is described. It is based on the notions of variational Schouten bracket and l*-covering. The latter serves the role of the cotangent bundle in the category of nonlinear evolution PDEs. We first consider two illustrative examples …

Motion of curves and surfaces in $\R^3$ lead to nonlinear evolution equations which are often integrable. They are also intimately connected to the dynamics of spin chains in the continuum limit and integrable soliton systems through geometric and gauge symmetric connections/equivalence. Here we point out the fact that…

Most known examples of doubly periodic minimal surfaces in $\mathbb{R}^3$ with parallel ends limit as a foliation of $\mathbb{R}^3$ by horizontal noded planes, with the location of the nodes satisfying a set of balance equations. Conversely, for each set of points providing a balanced configuration, there is a correspo…

Generalizes Hasimoto transformation to arbitrary flows on space curves.

We propose a new cognitive framework for option price modelling, using quantum neural computation formalism. Briefly, when we apply a classical nonlinear neural-network learning to a linear quantum Schrödinger equation, as a result we get a nonlinear Schrödinger equation (NLS), performing as a quantum stochastic filter…

In this paper, we study the gradient estimates of Li-Yau-Hamilton type for positive solutions to both drifting heat equation and the simple nonlinear heat equation problem $u_t-Δu=au\log u, \ \ u>0$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative (Bakry-Emery)-Ricci curvature. Here…

Paper solves vortex equations on complex surfaces, linking to Higgs bundle stability.

Adaptive wave model for financial option pricing is proposed, as a high-complexity alternative to the standard Black--Scholes model. The new option-pricing model, representing a controlled Brownian motion, includes two wave-type approaches: nonlinear and quantum, both based on (adaptive form of) the Schrödinger equatio…

In this paper, we study the gradient estimate for positive solutions to the following nonlinear heat equation problem $u_t-Δu=au\log u+Vu, \ \ u>0$ on the compact Riemannian manifold $(M,g)$ of dimension $n$ and with non-negative Ricci curvature. Here $a\leq 0$ is a constant, $V$ is a smooth function on $M$ with $-…

Paper presents a novel method to assess boundedness and stability of nonlinear systems with variable delays.

It is shown that the curvature function satisfies a nonlinear evolution equation under the general curve shortening flow and a detailed asymptotic behavior of the closed curves is presented when they contract to a point in finite time.

The paper develops methods for causal inference from single-cell RNA sequencing data with multiple outcomes.

We establish the estimates of modulus of continuity for viscosity solutions of nonlinear evolution equations on manifolds, extending previous work of B. Andrews and J. Clutterbuck for regular solutions on manifolds \cite{AC3} and the first author's recent work for viscosity solutions in Euclidean spaces \cite{me1}.

New algorithm optimizes nonlinear SDEs online with convergence guarantees.

The paper proves an inequality and describes a curve flow in centro-affine geometry.

Geometric analysis of nonlinear dynamics applied to financial time series.

Study curves evolving on hypersurfaces with free boundaries, preserving length.

We briefly review results on nonlinear kinetic equation of Boltzmann type which describe the evolution of wealth in a simple agents market. The mathematical structure of the underlying kinetic equations allows to use well-known techniques of wide use in kinetic theory of rarefied gases to obtain information on the proc…

We prove that conservation of probability for the free heat semigroup on a Riemannian manifold $M$ (namely stochastic completeness), hence a linear property, is equivalent to uniqueness of positive, bounded solutions to nonlinear evolution equations of fast diffusion type on $M$ of the form $u_t=Δφ(u)$, $φ$ being an ar…

We present the Nahm transform of the doubly-periodic instantons introduced in math.DG/9909069, converting them into certain meromorphic solutions of Hitchin's equations over an elliptic curve.

We prove the existence of Ricci flow starting from a class of metrics with unbounded curvature, which are doubly-warped products over an interval with a spherical factor pinched off at an end. These provide a forward evolution from some known and conjectured finite-time local singularities of Ricci flow, generalizing p…

New equations describe surfaces with constant curvature.

We present two approaches to the heat flow on a Finsler manifold $(M,F)$: either as gradient flow on $L^2(M,m)$ for the energy; or as gradient flow on the reverse $L^2$-Wasserstein space $\mathcal{P}_2(M)$ of probability measures on $M$ for the relative entropy. Both approaches depend on the choice of a measure $m$ on …

Enhances DGPs with adaptive RKHS Fourier features for better non-stationary pattern modeling.

Letter analyzes training dynamics of a nonlinear contrastive learning model in high dimensions.

Based on conservation laws for surface layer integrals for critical points of causal variational principles, it is shown how jet spaces can be endowed with an almost-complex structure. We analyze under which conditions the almost-complex structure can be integrated to a canonical complex structure. Combined with the sc…

We recently showed that the S&P500 stock market index is well described by Tsallis non-extensive statistics and nonlinear Fokker-Planck time evolution. We argued that these results should be applicable to a broad range of markets and exchanges where anomalous diffusion and `heavy' tails of the distribution are present.…

Paper explores solving HJB equations using neural networks.

Global solutions found for certain reaction-diffusion equations on specific manifolds.

Study spherical doubly warped spacetimes for stellar collapse and cosmology.

We study the small perturbations of the $1+3$-dimensional Milne model for the Einstein-Klein-Gordon (EKG) system. We prove the nonlinear future stability, and show that the perturbed spacetimes are future causally geodesically complete. For the proof, we work within the constant mean curvature (CMC) gauge and focus on …

We initiate the study of a new nonlinear parabolic equation on a Riemann surface. The evolution equation arises as a reduction of the Anomaly flow on a fibration. We obtain a criterion for long-time existence for this flow, and give a range of initial data where a singularity forms in finite time, as well as a range of…

WeldNet reduces complex dynamics to simpler, manageable segments.

Learning nonlinear dynamics from aggregate data is a challenging problem because the full trajectory of each individual is not available, namely, the individual observed at one time may not be observed at the next time point, or the identity of individual is unavailable. This is in sharp contrast to learning dynamics w…

In a number of physically important cases, the nonholonomically (nonintegrable) constrained Ricci flows can be modelled by exact solutions of Einstein equations with nonhomogeneous (anisotropic) cosmological constants. We develop two geometric methods for constructing such solutions: The first approach applies the form…

Survey on stability of Minkowski spacetime in relativity.