arXiv research

Bell's theorem shows quantum correlations can't be explained by classical causal models, even with some measurement dependence.

Quantum theory challenges traditional cause-effect relations, showing causal influences even without Bell inequality violations.

Many widely studied graphical models with latent variables lead to nontrivial constraints on the distribution of the observed variables. Inspired by the Bell inequalities in quantum mechanics, we refer to any linear inequality whose violation rules out some latent variable model as a "hidden variable test" for that mod…

The ability to witness non-local correlations lies at the core of foundational aspects of quantum mechanics and its application in the processing of information. Commonly, this is achieved via the violation of Bell inequalities. Unfortunately, however, their systematic derivation quickly becomes unfeasible as the scena…

Characterizes causal structure dominance for latent variables.

Bell's Theorem shows that quantum mechanical correlations can violate the constraints that the causal structure of certain experiments impose on any classical explanation. It is thus natural to ask to which degree the causal assumptions -- e.g. locality or measurement independence -- have to be relaxed in order to allo…

Detects causal scenarios with inequality constraints among classical correlations.

It has recently been found that Bell scenarios are only a small subclass of interesting setups for studying the non-classical features of quantum theory within spacetime. We find that it is possible to talk about classical correlations, quantum correlations and other kinds of correlations on any directed acyclic graph,…

Investigates differential smoothness of 3D skew polynomial rings.

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…

We establish a one-parameter family of Harnack inequalities connecting the constrained trace Li-Yau differential Harnack inequality for a nonlinear parabolic equation to the constrained trace Chow-Hamilton Harnack inequality for this nonlinear equation with respect to evolving metrics related to Ricci flow on a 2-dimen…

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.

Paper proves anisotropic Minkowski inequality and related inequalities.

Generative Adversarial Networks generate PXD background noise efficiently.

We present a new proof of the sphere covering inequality in the spirit of comparison geometry, and as a byproduct we find another sphere covering inequality which can be viewed as the dual of the original one. We also prove sphere covering inequalities on surfaces satisfying general isoperimetric inequalities, and disc…

The paper establishes Harnack inequalities for solutions of nonlinear parabolic equations on manifolds with integral Ricci curvature bounds.

DiffSlack learns neural networks with nonlinear constraints via learnable slack variables.

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…

New proof of Penrose inequality using potential theory.

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 prove constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation $ω_t=Δω+aω\ln ω$ on closed manifolds. We also derive a new interpolated Harnack inequality for the equation $ω_t=Δω-ω\lnω+\varepsilon Rω$ on closed surfaces under the $\varepsilon$-Ricci flow. Finally we prove…

Being motivated by the problem of deducing $L^p$-bounds on the second fundamental form of an isometric immersion from $L^p$-bounds on its mean curvature vector field, we prove a (nonlinear) Calderón-Zygmund inequality for maps between complete (possibly noncompact) Riemannian manifolds.

The problem of causal inference is to determine if a given probability distribution on observed variables is compatible with some causal structure. The difficult case is when the causal structure includes latent variables. We here introduce the $\textit{inflation technique}$ for tackling this problem. An inflation of a…

Greedy MI maximization method outperforms existing approaches in nonlinear models.

This is a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. Our purpose is two-fold: Give a concise and geometric review on the birth of weighted Ricci curvature and its applications; Explain recent results from a nonlinear an…

The paper studies frequency monotonicity for solutions of nonlinear equations under Ricci flow.

We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.

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…

Through the study of some elliptic and parabolic fully nonlinear PDEs, we establish conformal versions of quermassintegral inequality, the Sobolev inequality and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on locally conformally flat manifolds.

Soft diamond regularizers improve deep learning performance and sparsity.

Spacelike surfaces in Generalized Robertson-Walker spacetimes whose mean curvature function satisfies a natural nonlinear inequality are analyzed. Several uniqueness and nonexistence results for such compact spacelike surfaces are proved. In the nonparametric case, new Calabi-Bernstein type problems are solved as a con…

New algorithm tackles stochastic optimization with inequality constraints.

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

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

We present proofs of basic results, including those developed by Harold Bell, for the plane fixed point problem: does every map of a non-separating plane continuum have a fixed point? Some of these results had been announced much earlier by Bell but without accessible proofs. We define the concept of the variation of a…

The paper studies nonlinear mass concepts in 3-manifolds with nonnegative scalar curvature.

We classify local minimizers of $\intσ_2+\oint H_2$ among all conformally flat metrics in the Euclidean $(n+1)$-ball, $4\leq n\leq 5$, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local minimizers of the analogous functional in the critical dimension $n+1=4$. If minimiz…

We analyze complexity of financial (and general economic) processes by comparing classical and quantum-like models for randomness. Our analysis implies that it might be that a quantum-like probabilistic description is more natural for financial market than the classical one. A part of our analysis is devoted to study t…

The paper studies solutions to a nonlinear equation on Finsler manifolds with gradient estimates and Harnack inequalities.

We study martingale inequalities from an analytic point of view and show that a general martingale inequality can be reduced to a pair of deterministic inequalities in a small number of variables. More precisely, the optimal bound in the martingale inequality is determined by a fixed point of a simple nonlinear operato…

The paper studies boundedness of pseudo-differential operators on smooth manifolds.

Theoretical study of random forests for nonlinear time series.

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

Though Trudinger-Moser inequalities on compact Riemannian manifolds or Euclidean space are well understood, we know little about them on complete noncompact Riemannian manifolds. In this paper, we established respectively necessary condition and sufficient condition under which Trudinger-Moser inequalities hold on comp…

Paper explores curvature flows on spheres to prove inequalities.

Study on new Monge-Ampère functionals and their variational problems.

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

In this paper we investigate a nonlinear generalization of the Black-Scholes equation for pricing American style call options in which the volatility term may depend on the underlying asset price and the Gamma of the option. We propose a numerical method for pricing American style call options by means of transformatio…