arXiv research

In this paper, we partially settle down the long standing open problem of the finite time blow-up property about the nonlinear Schr$\ddot{o}$dinger equations on some Riemannian manifolds like the standard 2-sphere $S^2$ and the hyperbolic 2-space $H^{2}(-1)$. Using the similar idea, we establish such blow-up results on…

Study shows observability for Schrödinger equations on product manifolds with specific conditions.

Suppose that $G=(V, E)$ is a finite graph with the vertex set $V$ and the edge set $E$. Let $Δ$ be the usual graph Laplacian. Consider the following nonlinear Schr$\ddot{o}$dinger type equation of the form $\left \{ \begin{array}{lcr} -Δu-αu=f(x,u),\\ u\in W^{1,2}(V),\\ \end{array} \right.$ on graph $G$, where $f(x…

The study establishes bounds for Schrödinger operators on Riemannian manifolds.

In this paper, the Dirac, twistor and Killing equations on Weyl manifolds with CSpin structures are investigated. A conformal Schr"odinger-Lichnerowicz formula is presented and used to show integrability conditions for these equations. By introducing the Killing equation for spinors of arbitrary weight, the result of A…

Abstract notes on generative modeling techniques.

Paper introduces a new generative learning model using Schrödinger bridge diffusion in latent space.

Generative model for time series using Schrödinger bridge.

Unified framework for robust, stable, and efficient density ratio estimation.

Characterizes Schrödinger operator boundedness on weighted Riemannian manifolds.

We solved the Schr{ö}dinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus T^n = R^n / Λ is defined as a quotient of the Euclidean space R^n by an arbitrary n-dimensional lattice Λ. The lattice is not neces…

We give a new lower bound for the first gap $λ_2 - λ_1$ of the Dirichlet eigenvalues of the Schr{ö}dinger operator on a bounded convex domain $Ω$ in R$^n$ or S$^n$ and greatly sharpens the previous estimates. The new bound is explicit and computable.

For the spherical Laplacian on the sphere and for the Dirichlet Laplacian in the square}, Antonie Stern claimed in her PhD thesis (1924) the existence of an infinite sequence of eigenvalues whose corresponding eigenspaces contain an eigenfunction with exactly two nodal domains. These results were given complete proofs …

The goal of this article is twofold: in a first part, we prove Gaussian estimates for the heat kernel of Schr{ö}dinger operators delta + V whose potential V is "small at infinity" in an integral sense. In a second part, we prove sharp boundedness result for the associated Riesz transform with potential d(delta+V) --1/2…

New inequalities for spectral zeta kernels on spheres and manifolds.

CMCD sampler connects transport and variational inference for efficient sampling.

Study Galois groupoids of discret Painlevé equations.

Proves solvability of general inverse σ_k equations with constant coefficients.

We present an unsupervised approach for discovering semantic representations of mathematical equations. Equations are challenging to analyze because each is unique, or nearly unique. Our method, which we call equation embeddings, finds good representations of equations by using the representations of their surrounding …

Paper establishes estimates for nonlinear equations on compact manifolds.

Proves C^2,alpha estimates for elliptic equations on hyperkähler manifolds.

The paper generalizes Monge-Ampère equations and their solutions in differential geometry.

We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of the Rabelo equations are found to be related to the sine-Gordon equation. The ot…

Introduces a new PDE involving differential forms for Kähler geometry.

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

In this paper we perform a blow-up and quantization analysis of the following nonlocal Liouville-type equation \begin{equation}(-Δ)^\frac12 u= κe^u-1~\mbox{in $S^1$,} \end{equation} where $(-Δ)^\frac{1}{2}$ stands for the fractional Laplacian and $κ$ is a bounded function. We interpret the above equation as the prescri…

The paper derives gradient estimates for porous medium and fast diffusion equations on metric measure spaces.

Paper solves Hessian equations on Kähler manifolds.

The paper introduces new equations in Kähler geometry and proves their solutions and convexity.

We describe a method to reduce partial differential equations of Monge-Ampère type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Ampère equations and the Plebanski equatio…

Probabilistic grammars improve equation discovery from data.

In this paper, we provide families of second order non-linear partial differential equations, describing pseudospherical surfaces (pss equations), with the property of having local isometric immersions in E^3, with principal curvatures depending on finite-order jets of solutions of the differential equation. These equa…

The paper proves constant rank theorems for special Lagrangian equations.

Study solves HJB equations for time-inconsistent control problems.

We introduce a class of overdetermined systems of partial differential equations of finite type on (pseudo)-Riemannian manifolds that we call the generalised Ricci soliton equations. These equations depend on three real parameters. For special values of the parameters they specialise to various important classes of equ…

In this thesis, we consider the suitability of using the charged cold fluid model in the description of ultra-relativistic beams. The method that we have used is the following. Firstly, the necessary notions of kinetic theory and differential geometry of second order differential equations are explained. Then an averag…

Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.

We determine the Lie point symmetries of the Fokker-Planck equation and provide examples of solutions of this equation. The Fokker-Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation and whose point symmetries give rise to potential symmetries of the Fokker-Planck equ…

Studies projective geometry and partial differential equations prolongation.

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

The paper derives gradient estimates for solutions of certain equations on metric measure spaces.

Study generalizes Hermitian-Einstein equation for cyclic Higgs bundles, proving existence and inequality.

We come up with infinite-dimensional prequantum line bundles and moment map interpretations of three different sets of equations - the generalised Monge-Amp`ere equation, the almost Hitchin system, and the Calabi-Yang-Mills equations. These are all perturbations of already existing equations. Our construction for the g…

Study gradient estimates for nonlinear parabolic equations on Riemannian manifolds.

Study a modified Laplacian equation in spacetime.

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

The paper proves estimates for vortex-type equations on compact Riemann surfaces.

Geometrically interprets two equations, showing their equivalence and providing solutions.